PHYSICS FORMULAS
2426
Electron = -1.602 19 × 10-19 C = 9.11 × 10-31 kg
Proton = 1.602 19 × 10-19 C = 1.67 × 10-27 kg
Neutron = 0 C = 1.67 × 10-27 kg
6.022 × 1023 atoms in one atomic mass unit
e is the elementary charge: 1.602 19 × 10-19 C
Potential Energy, velocity of electron: PE = eV = ½mv2
1V = 1J/C 1N/C = 1V/m 1J = 1 N·m = 1 C·V
1 amp = 6.21 × 1018 electrons/second = 1 Coulomb/second
1 hp = 0.756 kW 1 N = 1 T·A·m 1 Pa = 1 N/m2
Power = Joules/second = I2R = IV [watts W]
Quadratic
Equation: x
b b ac
a
=
- ± 2 - 4
2
Kinetic Energy [J]
KE mv = 12
2
[Natural Log: when eb = x, ln x = b ]
m: 10-3 m: 10-6 n: 10-9 p: 10-12 f: 10-15 a: 10-18
Addition of Multiple Vectors:
r r r r
Rr = Ar+ B +r C r Resultant = Sum of the vectors
R A B C x x x x = + + x-component A A x = cos q
r r r r
R A B C y y y y = + + y-component A A y = sin q
R Rx Ry = 2 + 2 Magnitude (length) of R
qR
y
x
R
R
= tan-1 or tanqR
y
x
R
R
= Angle of the resultant
Multiplication of Vectors:
Cross Product or Vector Product:
i ´ j = k j ´ i = -k
i ´ i = 0
Positive direction:
i
j k
Dot Product or Scalar Product:
i × j = 0 i × i = 1
a × b = abcosq
k
i
j
Derivative of Vectors:
Velocity is the derivative of position with respect to time:
v = + + k = i + j + k
d
dt
x y z
dx
dt
dy
dt
dz
dt
( i j )
Acceleration is the derivative of velocity with respect to
time:
a = + + k = i + j + k
d
dt
v v v
dv
dt
dv
dt
dv
x y z dt
x y z ( i j )
Rectangular Notation: Z = R ± jX where +j represents
inductive reactance and -j represents capacitive reactance.
For example, Z = 8 + j6W means that a resistor of 8W is
in series with an inductive reactance of 6W.
Polar Notation: Z = M Ðq, where M is the magnitude of the
reactance and q is the direction with respect to the
horizontal (pure resistance) axis. For example, a resistor of
4W in series with a capacitor with a reactance of 3W would
be expressed as 5 Ð-36.9° W.
In the descriptions above, impedance is used as an example.
Rectangular and Polar Notation can also be used to
express amperage, voltage, and power.
To convert from rectangular to polar notation:
Given: X - jY (careful with the sign before the ”j”)
Magnitude: X 2 + Y2 = M
Angle:
tanq =
- Y
X
(negative sign carried over
from rectangular notation
in this example)
Note: Due to the way the calculator works, if X is negative,
you must add 180° after taking the inverse tangent. If the
result is greater than 180°, you may optionally subtract
360° to obtain the value closest to the reference angle.
To convert from polar to rectangular (j) notation:
Given: M Ðq
X Value: Mcosq
Y (j) Value: Msinq
In conversions, the j value will have the
same sign as the q value for angles
having a magnitude < 180°.
Use rectangular notation when adding
and subtracting.
Use polar notation for multiplication and division. Multiply in
polar notation by multiplying the magnitudes and adding
the angles. Divide in polar notation by dividing the
magnitudes and subtracting the denominator angle from
the numerator angle.
X
M
Magnitude
q
Y
ELECTRIC CHARGES AND FIELDS
Coulomb's Law: [Newtons N]
F k
q q
r
= 1 2
2
where: F = force on one charge by
the other[N]
k = 8.99 × 109 [N·m2/C2]
q1 = charge [C]
q2 = charge [C]
r = distance [m]
Electric Field: [Newtons/Coulomb or Volts/Meter]
E k
q
r
F
q
= = 2
where: E = electric field [N/C or V/m]
k = 8.99 × 109 [N·m2/C2]
q = charge [C]
r = distance [m]
F = force
Electric field lines radiate outward from
positive charges. The electric field
is zero inside a conductor.
+ -
Relationship of k to Î0:
k =
Î
1
4 0 p
where: k = 8.99 × 109 [N·m2/C2]
Î0 = permittivity of free space
8.85 × 10-12 [C2/N·m2]
Electric Field due to an Infinite Line of Charge: [N/C]
E
r
k
r
=
Î
=
l
p
l
2
2
0
E = electric field [N/C]
l = charge per unit length [C/m}
Î0 = permittivity of free space
8.85 × 10-12 [C2/N·m2]
r = distance [m]
k = 8.99 × 109 [N·m2/C2]
Electric Field due to ring of Charge: [N/C]
E
kqz
z R
=
( 2 + 2 )3/2
or if z >> R, E
kq
z
= 2
E = electric field [N/C]
k = 8.99 × 109 [N·m2/C2]
q = charge [C]
z = distance to the charge [m]
R = radius of the ring [m]
Electric Field due to a disk Charge: [N/C]
E
z
z R
=
Î
-
+
æ
è ç
ö
ø ÷
s
2
1
0
2 2
E = electric field [N/C]
s = charge per unit area
[C/m2}
Î0 = 8.85 × 10-12 [C2/N·m2]
z = distance to charge [m]
R = radius of the ring [m]
Electric Field due to an infinite sheet: [N/C]
E =
Î
s
2 0
E = electric field [N/C]
s = charge per unit area [C/m2}
Î0 = 8.85 × 10-12 [C2/N·m2]
Electric Field inside a spherical shell: [N/C]
E
kqr
R
= 3
E = electric field [N/C]
q = charge [C]
r = distance from center of sphere to
the charge [m]
R = radius of the sphere [m]
Electric Field outside a spherical shell: [N/C]
E
kq
r
= 2
E = electric field [N/C]
q = charge [C]
r = distance from center of sphere to
the charge [m]
Average Power per unit area of an electric or
magnetic field:
W m
E
c
B c m m / 2
2
0
2
0 2 2
= =
m m
W = watts
Em = max. electric field [N/C]
m0 = 4p × 10-7
c = 2.99792 × 108 [m/s]
Bm = max. magnetic field [T]
A positive charge moving in the same direction as the electric
field direction loses potential energy since the potential of
the electric field diminishes in this direction.
Equipotential lines cross EF lines at right angles.
Electric Dipole: Two charges of equal magnitude and
opposite polarity separated by a distance d.
z
-Q
p
d
+Q
E
k
z
=
2
3
p
E
z
=
Î
1
2 0
p 3
p
when z » d
E = electric field [N/C]
k = 8.99 × 109 [N·m2/C2]
Î0 = permittivity of free space 8.85 ×
10-12 C2/N·m2
p = qd [C·m] "electric dipole moment"
in the direction negative to
positive
z = distance [m] from the dipole
center to the point along the
dipole axis where the electric field
is to be measured
Deflection of a Particle in an Electric Field:
2ymv2 = qEL2 y = deflection [m]
m = mass of the particle [kg]
d = plate separation [m]
v = speed [m/s]
q = charge [C]
E = electric field [N/C or V/m
L = length of plates [m]
Potential Difference between two Points: [volts V]
D
D
V V V
PE
q
B A Ed = - = = -
DPE = work to move a charge
from A to B [N·m or J]
q = charge [C]
VB = potential at B [V]
VA = potential at A [V]
E = electric field [N/C or V/m
d = plate separation [m]
Electric Potential due to a Point Charge: [volts V]
V k
q
r
=
V = potential [volts V]
k = 8.99 × 109 [N·m2/C2]
q = charge [C]
r = distance [m]
Potential Energy of a Pair of Charges: [J, N·m or
C·V]
PE q V k
q q
r
= = 2 1
1 2
V1 is the electric potential due to
q1 at a point P
q2V1 is the work required to bring
q2 from infinity to point P
Work and Potential:
DU U U W f i = - = -
U = -W¥
W = F × d = Fd cosq
W q d
i
f
= ò E× s
DV V V
W
q f i = - = -
V d
i
f
= -ò E× s
U = electric potential energy [J]
W = work done on a particle by
a field [J]
W¥ = work done on a particle
brought from infinity (zero
potential) to its present
location [J]
F = is the force vector [N]
d = is the distance vector over
which the force is
applied[m]
F = is the force scalar [N]
d = is the distance scalar [m]
q = is the angle between the
force and distance vectors
ds = differential displacement of
the charge [m]
V = volts [V]
q = charge [C]
Flux: the rate of flow (of an electric field) [N·m2/C]
F = ò E× dA
= ò E(cosq )dA
F is the rate of flow of an electric
field [N·m2/C]
ò integral over a closed surface
E is the electric field vector [N/C]
A is the area vector [m2] pointing
outward normal to the surface.
Gauss' Law:
Î = 0 F qenc
Î ò × = 0 E dA qenc
Î0 = 8.85 × 10-12 [C2/N·m2]
F is the rate of flow of an electric
field [N·m2/C]
qenc = charge within the gaussian
surface [C]
ò integral over a closed surface
E is the electric field vector [J]
A is the area vector [m2] pointing
outward normal to the surface.
CAPACITANCE
Parallel-Plate Capacitor:
C
A
d
= k Î0
C = capacitance [farads F]
k = the dielectric constant (1)
Î0 = permittivity of free space
8.85 × 10-12 C2/N·m2
A = area of one plate [m2]
d = separation between plates [m]
Cylindrical Capacitor:
C
L
b a
= 2 Î0 pk
ln( / )
C = capacitance [farads F]
k = dielectric constant (1)
Î0 = 8.85 × 10-12 C2/N·m2
L = length [m]
b = radius of the outer
conductor [m]
a = radius of the inner
conductor [m]
Spherical Capacitor:
C
ab
b a
= Î
-
4 0 pk
C = capacitance [farads F]
k = dielectric constant (1)
Î0 = 8.85 × 10-12 C2/N·m2
b = radius, outer conductor
[m]
a = radius, inner conductor [m]
Maximum Charge on a Capacitor: [Coulombs C]
Q =VC Q = Coulombs [C]
V = volts [V]
C = capacitance in farads [F]
For capacitors connected in series, the charge Q is equal for
each capacitor as well as for the total equivalent. If the
dielectric constant k is changed, the capacitance is
multiplied by k, the voltage is divided by k, and Q is
unchanged. In a vacuum k = 1, When dielectrics are
used, replace Î0 with k Î0.
Electrical Energy Stored in a Capacitor: [Joules J]
U
QV CV Q
C E = = =
2 2 2
2 2 U = Potential Energy [J]
Q = Coulombs [C]
V = volts [V]
C = capacitance in farads [F]
Charge per unit Area: [C/m2]
s =
q
A
s = charge per unit area [C/m2]
q = charge [C]
A = area [m2]
Energy Density: (in a vacuum) [J/m3]
u = 1 Î E
2 0
2 u = energy per unit volume [J/m3]
Î0 = permittivity of free space
8.85 × 10-12 C2/N·m2
E = energy [J]
Capacitors in Series:
1 1 1
1 2 C C C eff
= + ...
Capacitors in Parallel:
C C C eff = + 1 2 ...
Capacitors connected in series all have the same charge q.
For parallel capacitors the total q is equal to the sum of the
charge on each capacitor.
Time Constant: [seconds]
t = RC t = time it takes the capacitor to reach 63.2%
of its maximum charge [seconds]
R = series resistance [ohms W]
C = capacitance [farads F]
Charge or Voltage after t Seconds: [coulombs C]
charging:
q Q( e ) = 1- -t /t
V V ( e ) S
= 1- -t /t
discharging:
q = Qe-t /t
V V e S
= -t /t
q = charge after t seconds
[coulombs C]
Q = maximum charge [coulombs
C] Q = CV
e = natural log
t = time [seconds]
t = time constant RC [seconds]
V = volts [V]
VS = supply volts [V]
[Natural Log: when eb = x, ln x = b ]
Drift Speed:
I ( )
Q
t
nqv A d = =
D
D
DQ = # of carriers × charge/carrier
Dt = time in seconds
n = # of carriers
q = charge on each carrier
vd = drift speed in meters/second
A = cross-sectional area in meters2
RESISTANCE
Emf: A voltage source which can provide continuous current
[volts]
e = IR + Ir e = emf open-circuit voltage of the battery
I = current [amps]
R = load resistance [ohms]
r = internal battery resistance [ohms]
Resistivity: [Ohm Meters]
r =
E
J
r =
RA
L
r = resistivity [W · m]
E = electric field [N/C]
J = current density [A/m2]
R = resistance [W ohms]
A = area [m2]
L = length of conductor [m]
Variation of Resistance with Temperature:
r - r = r a - 0 0 0 (T T ) r = resistivity [W · m]
r0 = reference resistivity [W · m]
a = temperature coefficient of
resistivity [K-1]
T0 = reference temperature
T - T0 = temperature difference
[K or °C]
CURRENT
Current Density: [A/m2]
i = ò J × dA
if current is uniform
and parallel to dA,
then: i = JA
J ne Vd = ( )
i = current [A]
J = current density [A/m2]
A = area [m2]
L = length of conductor [m]
e = charge per carrier
ne = carrier charge density [C/m3]
Vd = drift speed [m/s]
Rate of Change of Chemical Energy in a Battery:
P = ie P = power [W]
i = current [A]
e = emf potential [V]
Kirchhoff’s Rules
1. The sum of the currents entering a junctions is equal to
the sum of the currents leaving the junction.
2. The sum of the potential differences across all the
elements around a closed loop must be zero.
Evaluating Circuits Using Kirchhoff’s Rules
1. Assign current variables and direction of flow to all
branches of the circuit. If your choice of direction is
incorrect, the result will be a negative number. Derive
equation(s) for these currents based on the rule that
currents entering a junction equal currents exiting the
junction.
2. Apply Kirchhoff’s loop rule in creating equations for
different current paths in the circuit. For a current path
beginning and ending at the same point, the sum of
voltage drops/gains is zero. When evaluating a loop in the
direction of current flow, resistances will cause drops
(negatives); voltage sources will cause rises (positives)
provided they are crossed negative to positive—otherwise
they will be drops as well.
3. The number of equations should equal the number of
variables. Solve the equations simultaneously.
MAGNETISM
André-Marie Ampére is credited with the discovery of
electromagnetism, the relationship between electric
currents and magnetic fields.
Heinrich Hertz was the first to generate and detect
electromagnetic waves in the laboratory.
Magnetic Force acting on a charge q: [Newtons N]
F = qvBsinq
F = qv ´ B
F = force [N]
q = charge [C]
v = velocity [m/s]
B = magnetic field [T]
q = angle between v and B
Right-Hand Rule: Fingers represent the direction of the
magnetic force B, thumb represents the direction of v (at
any angle to B), and the force F on a positive charge
emanates from the palm. The direction of a magnetic field
is from north to south. Use the left hand for a negative
charge.
Also, if a wire is grasped in the right hand with the thumb in
the direction of current flow, the fingers will curl in the
direction of the magnetic field.
In a solenoid with current flowing in the direction of curled
fingers, the magnetic field is in the direction of the thumb.
When applied to electrical flow caused by a changing
magnetic field, things get more complicated. Consider the
north pole of a magnet moving toward a loop of wire
(magnetic field increasing). The thumb represents the
north pole of the magnet, the fingers suggest current flow in
the loop. However, electrical activity will serve to balance
the change in the magnetic field, so that current will
actually flow in the opposite direction. If the magnet was
being withdrawn, then the suggested current flow would be
decreasing so that the actual current flow would be in the
direction of the fingers in this case to oppose the decrease.
Now consider a cylindrical area of magnetic field going into
a page. With the thumb pointing into the page, this would
suggest an electric field orbiting in a clockwise direction. If
the magnetic field was increasing, the actual electric field
would be CCW in opposition to the increase. An electron in
the field would travel opposite the field direction (CW) and
would experience a negative change in potential.
Force on a Wire in a Magnetic Field: [Newtons N]
F = BI lsinq
F = I l ´ B
F = force [N]
B = magnetic field [T]
I = amperage [A]
l = length [m]
q = angle between B and the
direction of the current
Torque on a Rectangular Loop: [Newton·meters N·m]
t = NBIAsinq N = number of turns
B = magnetic field [T]
I = amperage [A]
A = area [m2]
q = angle between B and the
plane of the loop
Charged Particle in a Magnetic Field:
r
mv
qB
=
r = radius of rotational path
m = mass [kg]
v = velocity [m/s]
q = charge [C]
B = magnetic field [T]
Magnetic Field Around a Wire: [T]
B
I
r
=
m
p
0
2
B = magnetic field [T]
m0 = the permeability of free
space 4p×10-7 T·m/A
I = current [A]
r = distance from the center of
the conductor
Magnetic Field at the center of an Arc: [T]
B
i
r
=
m f
p
0
4
B = magnetic field [T]
m0 = the permeability of free
space 4p×10-7 T·m/A
i = current [A]
f = the arc in radians
r = distance from the center of
the conductor
Hall Effect: Voltage across the width of a
conducting ribbon due to a Magnetic Field:
(ne)V h Bi w =
v Bw V d w =
ne = carrier charge density [C/m3]
Vw = voltage across the width [V]
h = thickness of the conductor [m]
B = magnetic field [T]
i = current [A]
vd = drift velocity [m/s]
w = width [m]
Force Between Two Conductors: The force is
attractive if the currents are in the same direction.
F I I
d
1 0 1 2
l 2
=
m
p
F = force [N]
l = length [m]
m0 = the permeability of free
space 4p×10-7 T·m/A
I = current [A]
d = distance center to center [m]
Magnetic Field Inside of a Solenoid: [Teslas T]
B = m nI 0
B = magnetic field [T]
m0 = the permeability of free
space 4p×10-7 T·m/A
n = number of turns of wire per
unit length [#/m]
I = current [A]
Magnetic Dipole Moment: [J/T]
m = NiA m = the magnetic dipole moment [J/T]
N = number of turns of wire
i = current [A]
A = area [m2]
Magnetic Flux through a closed loop: [T·M2 or Webers]
F = BAcosq B = magnetic field [T]
A = area of loop [m2]
q = angle between B and the
perpen-dicular to the plane of
the loop
Magnetic Flux for a changing magnetic field: [T·M2 or
Webers]
F = ò B× dA
B = magnetic field [T]
A = area of loop [m2]
A Cylindrical Changing Magnetic Field
ò E× ds = E r =
d
dt
2p B
F
FB = BA = Bp r 2
d
dt
A
dB
dt
F
=
e = -N
d
dt
F
E = electric field [N/C]
r = radius [m]
t = time [s]
F = magnetic flux [T·m2 or
Webers]
B = magnetic field [T]
A = area of magnetic field
[m2]
dB/dt = rate of change of
the magnetic field [T/s]
e = potential [V]
N = number of orbits
Faraday’s Law of Induction states that the instantaneous
emf induced in a circuit equals the rate of
change of magnetic flux through the circuit. Michael
Faraday made fundamental discoveries in
magnetism, electricity, and light.
e = -N
t
DF
D
N = number of turns
F = magnetic flux [T·m2]
t = time [s]
Lenz’s Law states that the polarity of the induced emf is
such that it produces a current whose magnetic field
opposes the change in magnetic flux through a circuit
Motional emf is induced when a conducting bar moves
through a perpendicular magnetic field.
e = Blv B = magnetic field [T]
l = length of the bar [m]
v = speed of the bar [m/s]
emf Induced in a Rotating Coil:
e = NABw sinwt N = number of turns
A = area of loop [m2]
B = magnetic field [T]
w = angular velocity [rad/s]
t = time [s]
Self-Induced emf in a Coil due to changing current:
e = -L
I
t
D
D
L = inductance [H]
I = current [A]
t = time [s]
Inductance per unit length near the center of a solenoid:
L
n A
l
= m0
2
L = inductance [H]
l = length of the solenoid [m]
m0 = the permeability of free space
4p×10-7 T·m/A
n = number of turns of wire per unit
length [#/m]
A = area [m2]
Amperes' Law:
ò B × ds = ienc m0
B = magnetic field [T]
m0 = the permeability of free space
4p×10-7 T·m/A
ienc = current encircled by the
loop[A]
Joseph Henry, American physicist, made improvements
to the electromagnet.
James Clerk Maxwell provided a theory showing the
close relationship between electric and magnetic
phenomena and predicted that electric and magnetic
fields could move through space as waves.
J. J. Thompson is credited with the discovery of the
electron in 1897.
INDUCTIVE & RCL CIRCUITS
Inductance of a Coil: [H]
L
N
I
=
F N = number of turns
F = magnetic flux [T·m2]
I = current [A]
In an RL Circuit, after one time constant (t = L/R) the
current in the circuit is 63.2% of its final value, e/R.
RL Circuit:
current rise:
I ( )
V
R
= 1 - e-t /t L
current decay:
I
V
R
= e-t /t L
UB = Potential Energy [J]
V = volts [V]
R = resistance [W]
e = natural log
t = time [seconds]
tL = inductive time constant L/R
[s]
I = current [A]
Magnetic Energy Stored in an Inductor:
U LI B = 1
2
2 UB = Potential Energy [J]
L = inductance [H]
I = current [A]
Electrical Energy Stored in a Capacitor: [Joules J]
U
QV CV Q
C E = = =
2 2 2
2 2 UE = Potential Energy [J]
Q = Coulombs [C]
V = volts [V]
C = capacitance in farads [F]
Resonant Frequency: : The frequency at which XL = XC.
In a series-resonant circuit, the impedance is at its
minimum and the current is at its maximum. For a
parallel-resonant circuit, the opposite is true.
f
LC R =
1
2p
w =
1
LC
fR = Resonant Frequency [Hz]
L = inductance [H]
C = capacitance in farads [F]
w = angular frequency [rad/s]
Voltage, series circuits: [V]
V
q
C C = V IR R =
V
X
V
R
X = R = I
V V V R X
2 = 2 + 2
VC = voltage across capacitor [V]
q = charge on capacitor [C]
fR = Resonant Frequency [Hz]
L = inductance [H]
C = capacitance in farads [F]
R = resistance [W]
I = current [A]
V = supply voltage [V]
VX = voltage across reactance [V]
VR = voltage across resistor [V]
Phase Angle of a series RL or RC circuit: [degrees]
tan f = =
X
R
V
V
X
R
cosf = =
V
V
R
Z
R
(f would be negative
in a capacitive circuit)
f = Phase Angle [degrees]
X = reactance [W]
R = resistance [W]
V = supply voltage [V]
VX = voltage across reactance [V]
VR = voltage across resistor [V]
Z = impedance [W]
Impedance of a series RL or RC circuit: [W]
Z R X 2 = 2 + 2
E = I Z
Z
V
X
V
R
V
C
C R
= =
Z = R ± jX
f = Phase Angle [degrees]
X = reactance [W]
R = resistance [W]
V = supply voltage [V]
VX = voltage across reactance [V]
VX = voltage across resistor [V]
Z = impedance [W]
Series RCL Circuits:
The Resultant Phasor X X X L C = - is
in the direction of the larger reactance
and determines whether the circuit is
inductive or capacitive. If XL is larger
than XC, then the circuit is inductive
and X is a vector in the upward
direction.
In series circuits, the amperage is the
reference (horizontal) vector. This is
observed on the oscilloscope by
looking at the voltage across the
resistor. The two vector diagrams at
right illustrate the phase relationship
between voltage, resistance, reactance,
and amperage.
XC
XL
I
R
VL
VC
I
VR
Series RCL
Impedance Z R X X L C
2 = 2 + ( - )2 Z
R
=
cosf
Impedance may be found by adding the components using
vector algebra. By converting the result to polar notation,
the phase angle is also found.
For multielement circuits, total each resistance and reactance
before using the above formula.
Damped Oscillations in an RCL Series Circuit:
q = Qe- Rt / 2L cos(w¢t + f)
where
w ¢ = w 2 - (R / 2L)2
w = 1/ LC
When R is small and w¢ » w:
U
Q
C
= e-Rt L
2
2
/
q = charge on capacitor [C]
Q = maximum charge [C]
e = natural log
R = resistance [W]
L = inductance [H]
w = angular frequency of the
undamped oscillations
[rad/s]
w = angular frequency of the
damped oscillations
[rad/s]
U = Potential Energy of the
capacitor [J]
C = capacitance in farads [F]
Parallel RCL Circuits:
I I I I T R C L = 2 + ( - )2
tanf =
I - I
I
C L
R
V
IL
I
R
C
I
To find total current and phase angle in multielement circuits,
find I for each path and add vectorally. Note that when
converting between current and resistance, a division will
take place requiring the use of polar notation and resulting
in a change of sign for the angle since it will be divided into
(subtracted from) an angle of zero.
Equivalent Series Circuit: Given the Z in polar notation of a
parallel circuit, the resistance and reactance of the
equivalent series circuit is as follows:
R ZT = cosq X ZT = sinq
AC CIRCUITS
Instantaneous Voltage of a Sine Wave:
V =V ft max sin 2p V = voltage [V]
f = frequency [Hz]
t = time [s]
Maximum and rms Values:
I
I= m
2
V
V= m
2
I = current [A]
V = voltage [V]
RLC Circuits:
V V V V R L C = 2 + ( - )2 Z R X X L C = 2 + ( - )2
tanf =
X - X
R
L C P IV avg = cosf
PF = cosf
Conductance (G): The
reciprocal of resistance in
siemens (S).
Susceptance (B, BL, BC): The
reciprocal of reactance in
siemens (S).
Admittance (Y): The reciprocal
of impedance in siemens (S).
B Y
Susceptance
Conductance
Admittance
G
ELECTROMAGNETICS
WAVELENGTH
c = lf
c = E / B
1Å = 10-10m
c = speed of light 2.998 × 108 m/s
l = wavelength [m]
f = frequency [Hz]
E = electric field [N/C]
B = magnetic field [T]
Å= (angstrom) unit of wavelength
equal to 10-10 m
m = (meters)
WAVELENGTH SPECTRUM
BAND METERS ANGSTROMS
Longwave radio 1 - 100 km 1013 - 1015
Standard Broadcast 100 - 1000 m 1012 - 1013
Shortwave radio 10 - 100 m 1011 - 1012
TV, FM 0.1 - 10 m 109 - 1011
Microwave 1 - 100 mm 107 - 109
Infrared light 0.8 - 1000 mm 8000 - 107
Visible light 360 - 690 nm 3600 - 6900
violet 360 nm 3600
blue 430 nm 4300
green 490 nm 4900
yellow 560 nm 5600
orange 600 nm 6000
red 690 nm 6900
Ultraviolet light 10 - 390 nm 100 - 3900
X-rays 5 - 10,000 pm 0.05 - 100
Gamma rays 100 - 5000 fm 0.001 - 0.05
Cosmic rays < 100 fm < 0.001
Intensity of Electromagnetic Radiation [watts/m2]:
I
P
r
= s
4p 2
I = intensity [w/m2]
Ps = power of source [watts]
r = distance [m]
4pr2 = surface area of sphere
Force and Radiation Pressure on an object:
a) if the light is totally
absorbed:
F
IA
c
= P
I
c r =
b) if the light is totally
reflected back along the
path:
F
IA
c
=
2
P
I
c r =
2
F = force [N]
I = intensity [w/m2]
A = area [m2]
Pr = radiation pressure [N/m2]
c = 2.99792 × 108 [m/s]
Poynting Vector [watts/m2]:
S = EB = E
1 1
0 0
2
m m
cB = E
m0 = the permeability of free
space 4p×10-7 T·m/A
E = electric field [N/C or V/M]
B = magnetic field [T]
c = 2.99792 × 108 [m/s]
LIGHT
Indices of Refraction: Quartz: 1.458
Glass, crown 1.52
Glass, flint 1.66
Water 1.333
Air 1.000 293
Angle of Incidence: The angle measured from the
perpendicular to the face or from the perpendicular to the
tangent to the face
Index of Refraction: Materials of greater density have
a higher index of refraction.
n
c
v
º
n = index of refraction
c = speed of light in a vacuum 3 × 108 m/s
v = speed of light in the material [m/s]
n
n
=
l
l
0 l0 = wavelength of the light in a vacuum [m]
ln = its wavelength in the material [m]
Law of Refraction: Snell’s Law
n n 1 1 2 2 sinq = sinq n = index of refraction
q = angle of incidence
traveling to a region of
lesser density: q q 2 1 >
refracted
Source
n
q
n1
2
1
q2
traveling to a region of
greater density:
q q 2 1 <
refracted
Source
n
q
n1
2
1
q2
Critical Angle: The maximum
angle of incidence for which light
can move from n1 to n2
sinqc
n
n
= 2
1
for n1 > n2
Sign Conventions: When M is
negative, the image is inverted. p is positive when the
object is in front of the mirror, surface, or lens. Q is
positive when the image is in front of the mirror or in back
of the surface or lens. f and r are positive if the center of
curvature is in front of the mirror or in back of the surface
or lens.
Magnification by spherical mirror or thin lens. A
negative m means that the image is inverted.
M
h
h
i
p
=
¢
= -
h’ = image height [m]
h = object height [m]
i = image distance [m]
p = object distance [m]
reflected
refracted
q
Source
q
n1
n2
Plane Refracting Surface:
plane refracting surface:
n
p
n
i
1 = - 2
p = object distance
i = image distance [m]
n = index of refraction
Lensmaker’s Equation for a thin lens in air:
1 1 1 ( )
1
1 1
1 2 f p i
n
r r
= + = - -
æ
è ç
ö
ø ÷
r1 = radius of surface nearest the
object[m]
r2 = radius of surface nearest the
image [m]
f = focal length [m]
i = image distance [m]
p = object distance [m]
n = index of refraction
Virtual Image
C2
r2
F1
r1
F2 C1
C2
F1
p
F2
i
C1
Real Image
Thin Lens when the thickest part is thin compared to p.
i is negative on the left, positive on the right
f
r
=
2
f = focal length [m]
r = radius [m]
Converging Lens
f is positive (left)
r1 and r2 are positive in
this example
Diverging Lens
f is negative (right)
r1 and r2 are negative in
this example
Two-Lens System Perform the calculation in steps.
Calculate the image produced by the first lens, ignoring the
presence of the second. Then use the image position
relative to the second lens as the object for the second
calculation ignoring the first lens.
Spherical Refracting Surface This refers to two
materials with a single refracting surface.
n
p
n
i
n n
r
1 + 2 = 2 1
-
M
h
h
n i
n p
=
¢
= - 1
2
p = object distance
i = image distance [m] (positive for real
images)
f = focal point [m]
n = index of refraction
r = radius [m] (positive when facing a
convex surface, unlike with mirrors)
M = magnification
h' = image height [m]
h = object height [m]
Constructive and Destructive Interference by Single
and Double Slit Defraction and Circular Aperture
Young’s double-slit experiment (bright fringes/dark fringes):
Double Slit
Constructive:
DL = d sinq = ml
Destructive:
DL d m = = + sin ( ) q l 12
d = distance between the slits [m]
q = the angle between a normal
line extending from midway
between the slits and a line
extending from the midway
point to the point of ray
Intensity:
I Im =
æ
è ç
ö
ø ÷
(cos )
sin 2
2
b
a
a
b
p
l
= q
d
sin
a
p
l
= q
a
sin
Single-Slit
Destructive:
a sinq = ml
Circular Aperture
1st Minimum:
sin .
.
q
g
= 122
dia
intersection.
m = fringe order number [integer]
l = wavelength of the light [m]
a = width of the single-slit [m]
DL = the difference between the
distance traveled of the two
rays [m]
I = intensity @ q [W/m2]
Im = intensity @ q = 0 [W/m2]
d = distance between the slits [m]
In a circular aperture, the 1st
minimum is the point at which
an image can no longer be
resolved.
A reflected ray undergoes a phase shift of 180°
when the reflecting material has a greater index of
refraction n than the ambient medium. Relative to the
same ray without phase shift, this constitutes a path
difference of l/2.
Interference between Reflected and Refracted rays
from a thin material surrounded by another medium:
Constructive:
2 12
nt = (m+ )l
Destructive:
2nt = ml
n = index of refraction
t = thickness of the material [m]
m = fringe order number [integer]
l = wavelength of the light [m]
If the thin material is between two different media, one with a
higher n and the other lower, then the above constructive
and destructive formulas are reversed.
Wavelength within a medium:
l
l
n n
=
c n f n = l
l = wavelength in free space [m]
ln = wavelength in the medium [m]
n = index of refraction
c = the speed of light 3.00 × 108 [m/s]
f = frequency [Hz]
Polarizing Angle: by Brewster’s Law, the angle of
incidence that produces complete polarization in the
reflected light from an amorphous material such as glass.
tanqB
n
n
= 2
1
q q r B + = 90°
n = index of refraction
qB = angle of incidence
producing a 90° angle
between reflected and
refracted rays.
qr = angle of incidence of the
refracted ray.
partially polarized
qr
n
2
1
n
qb qb
non-polarized
Source
polarized
Intensity of light passing through a polarizing lense:
[Watts/m2]
initially unpolarized: I = 1 I
2 0
initially polarized:
I = I0
cos2 q
I = intensity [W/m2]
I0 = intensity of source [W/m2]
q = angle between the polarity
of the source and the lens.
Tom Penick tomzap@eden.com www.teicontrols.com/notes 1/31/99
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Tuesday, December 24, 2013
Physics Formulas List
Learning physics is all about applying concepts to solve problems. This article provides a comprehensive physics formulas list, that will act as a ready reference, when you are solving physics problems. You can even use this list, for a quick revision before an exam.
Physics is the most fundamental of all sciences. It is also one of the toughest sciences to master. Learning physics is basically studying the fundamental laws that govern our universe. I would say that there is a lot more to ascertain than just remember and mug up the physics formulas. Try to understand what a formula says and means, and what physical relation it expounds. If you understand the physical concepts underlying those formulas, deriving them or remembering them is easy. This Buzzle article lists some physics formulas that you would need in solving basic physics problems.
Physics Formulas
Mechanics
Friction
Moment of Inertia
Newtonian Gravity
Projectile Motion
Simple Pendulum
Electricity
Thermodynamics
Electromagnetism
Optics
Quantum Physics
Derive all these formulas once, before you start using them. Study physics and look at it as an opportunity to appreciate the underlying beauty of nature, expressed through natural laws. Physics help is provided here in the form of ready to use formulas. Physics has a reputation for being difficult and to some extent that's true, due to the mathematics involved. If you don't wish to think on your own and apply basic physics principles, solving physics problems is always going to be tough. Our physics formulas list is aimed at helping you out in solving problems. The joy of having solved a physics problem on your own, is worth all the effort! Understanding physics concepts challenges your imagination and thinking potential, wherein, if you visualize a problem, then you can come up with a solution. So here is the promised list which will help you out.
Mechanics
Mechanics is the oldest branch of physics. Mechanics deals with all kinds and complexities of motion. It includes various techniques, which can simplify the solution of a mechanical problem.
Motion in One Dimension
The formulas for motion in one dimension (Also called Kinematical equations of motion) are as follows. (Here 'u' is initial velocity, 'v' is final velocity, 'a' is acceleration and t is time):
s = ut + ½ at2
v = u + at
v2 = u2 + 2as
vav (Average Velocity) = (v+u)/2
Momentum, Force and Impulse
Formulas for momentum, impulse and force concerning a particle moving in 3 dimensions are as follows (Here force, momentum and velocity are vectors ):
Momentum is the product of mass and velocity of a body. Momentum is calculate using the formula: P = m (mass) x v (velocity)
Force can defined as something which causes a change in momentum of a body. Force is given by the celebrated newton's law of motion: F = m (mass) x a (acceleration)
Impulse is a large force applied in a very short time period. The strike of a hammer is an impulse. Impulse is given by I = m(v-u)
Pressure
Pressure is defined as force per unit area:
Pressure (P) = Force (F)Area (A)
Density
Density is the mass contained in a body per unit volume.
The formula for density is:
Density (D) = Mass(M)Volume (V)
Angular Momentum
Angular momentum is an analogous quantity to linear momentum in which the body is undergoing rotational motion. The formula for angular momentum (J) is given by:
J = r x p
where J denotes angular momentum, r is radius vector and p is linear momentum.
Torque
Torque can be defined as moment of force. Torque causes rotational motion. The formula for torque is: τ = r x F, where τ is torque, r is the radius vector and F is linear force.
Circular Motion
The formulas for circular motion of an object of mass 'm' moving in a circle of radius 'r' at a tangential velocity 'v' are as follows:
Centripetal force (F) = mv2r
Centripetal Acceleration (a) = v2r
Center of Mass
General Formula for Center of mass of a rigid body is :
R = ΣNi = 1 miriΣNi = 1mi
where R is the position vector for center of mass, r is the generic position vector for all the particles of the object and N is the total number of particles.
Reduced Mass for two Interacting Bodies
The physics formula for reduced mass (μ) is :
μ = m1m2m1 + m2
where m1 is mass of the first body, m2 is the mass of the second body.
Work and Energy
Formulas for work and energy in case of one dimensional motion are as follows:
W (Work Done) = F (Force) x D (Displacement)
Energy can be broadly classified into two types, Potential Energy and Kinetic Energy. In case of gravitational force, the potential energy is given by
P.E.(Gravitational) = m (Mass) x g (Acceleration due to Gravity) x h (Height)
The transitional kinetic energy is given by ½ m (mass) x v2(velocity squared)
Power
Power is, work done per unit time. The formula for power is given as
Power (P) = V2R =I2R
where P=power, W = Work, t = time.
[Back to Formula Index]
Friction
Friction can be classified to be of two kinds : Static friction and dynamic friction.
Static Friction: Static friction is characterized by a coefficient of static friction μ . Coefficient of static friction is defined as the ratio of applied tangential force (F) which can induce sliding, to the normal force between surfaces in contact with each other. The formula to calculate this static coefficient is as follows:
μ = Applied Tangential Force (F)Normal Force(N)
The amount of force required to slide a solid resting on flat surface depends on the co efficient of static friction and is given by the formula:
FHorizontal = μ x M(Mass of solid) x g (acceleration)
Dynamic Friction:
Dynamic friction is also characterized by the same coefficient of friction as static friction and therefore formula for calculating coefficient of dynamic friction is also the same as above. Only the dynamic friction coefficient is generally lower than the static one as the applied force required to overcome normal force is lesser.
[Back to Formula Index]
Moment of Inertia
Here are some formulas for Moments of Inertia of different objects. (M stands for mass, R for radius and L for length):
Object Axis Moment of Inertia
Disk Axis parallel to disc, passing through the center MR2/2
Disk Axis passing through the center and perpendicular to disc MR2/2
Thin Rod Axis perpendicular to the Rod and passing through center ML2/12
Solid Sphere Axis passing through the center 2MR2/5
Solid Shell Axis passing through the center 2MR2/3
[Back to Formula Index]
Newtonian Gravity
Here are some important formulas, related to Newtonian Gravity:
Newton's Law of universal Gravitation:
Fg = Gm1m2r2
where
m1, m2 are the masses of two bodies
G is the universal gravitational constant which has a value of 6.67300 × 10-11 m3 kg-1 s-2
r is distance between the two bodies
Formula for escape velocity (vesc) = (2GM / R)1/2where,
M is mass of central gravitating body
R is radius of the central body
[Back to Formula Index]
Projectile Motion
Here are two important formulas related to projectile motion:
(v = velocity of particle, v0 = initial velocity, g is acceleration due to gravity, θ is angle of projection, h is maximum height and l is the range of the projectile.)
Maximum height of projectile (h) = v0 2sin2θ2g
Horizontal range of projectile (l) = v0 2sin 2θ / g
[Back to Formula Index]
Simple Pendulum
The physics formula for the period of a simple pendulum (T) = 2π √(l/g)where
l is the length of the pendulum
g is acceleration due to gravity
Conical Pendulum
The Period of a conical pendulum (T) = 2π √(lcosθ/g)
where
l is the length of the pendulum
g is acceleration due to gravity
Half angle of the conical pendulum
[Back to Formula Index]
Electricity
Here are some formulas related to electricity.
Ohm's Law
Ohm's law gives a relation between the voltage applied a current flowing across a solid conductor:
V (Voltage) = I (Current) x R (Resistance)
Power
In case of a closed electrical circuit with applied voltage V and resistance R, through which current I is flowing,
Power (P) = V2R
= I2R. . . (because V = IR, Ohm's Law)
Kirchoff's Voltage Law
For every loop in an electrical circuit:
ΣiVi = 0
where Vi are all the voltages applied across the circuit.
Kirchoff's Current Law
At every node of an electrical circuit:
ΣiIi = 0
where Ii are all the currents flowing towards or away from the node in the circuit.
Resistance
The physics formulas for equivalent resistance in case of parallel and series combination are as follows:
Resistances R1, R2, R3 in series:
Req = R1 + R2 + R3
Resistances R1 and R2 in parallel:
Req = R1R2R1 + R2
For n number of resistors, R1, R2...Rn, the formula will be:
1/Req = 1/R1 + 1/R2 + 1/R3...+ 1/Rn
Capacitors
A capacitor stores electrical energy, when placed in an electric field. A typical capacitor consists of two conductors separated by a dielectric or insulating material. Here are the most important formulas related to capacitors. Unit of capacitance is Farad (F) and its values are generally specified in mF (micro Farad = 10 -6 F).
Capacitance (C) = Q / V
Energy Stored in a Capacitor (Ecap) = 1/2 CV2 = 1/2 (Q2 / C) = 1/2 (QV)
Current Flowing Through a Capacitor I = C (dV / dt)
Equivalent capacitance for 'n' capacitors connected in parallel:
Ceq (Parallel) = C1 + C2 + C3...+ Cn = Σi=1 to n Ci
Equivalent capacitance for 'n' capacitors in series:
1 / Ceq (Series) = 1 / C1 + 1 / C2...+ 1 / Cn = Σi=1 to n (1 / Ci)
Here
C is the capacitance
Q is the charge stored on each conductor in the capacitor
V is the potential difference across the capacitor
Parallel Plate Capacitor Formula:
C = kε0 (A/d)
Where
k = dielectric constant (k = 1 in vacuum)
ε0 = Permittivity of Free Space (= 8.85 × 10-12 C2 / Nm2)
A = Plate Area (in square meters)
d = Plate Separation (in meters)
Cylinrical Capacitor Formula:
C = 2π kε0 [L / ln(b / a)]
Where
k = dielectric constant (k = 1 in vacuum)
ε0 = Permittivity of Free Space (= 8.85 × 10-12 C2 / Nm2)
L = Capacitor Length
a = Inner conductor radius
b = Outer conductor radius
Spherical Capacitor Formula:
C = 4π kε0 [(ab)/(b-a)]
Where
k = dielectric constant (k = 1 in vacuum)
ε0 = Permittivity of Free Space (= 8.85 × 10-12 C2 / Nm2)
a = Inner conductor radius
b = Outer conductor radius
Inductors
An inductor is an electrical component which resists the flow of electrons or electric current through it. This property of inductance, in these devices, is caused by the electromotive force, created by magnetic field induced in them. The unit of inductance is Henry. Here are some important formulas associated with inductors.
Energy Stored in Inductor (Estored) = 1/2 (LI2)
Where, L is inductance and I is the current flowing through the inductor.
Inductance of a cylindrical air core Coil (L) = (m0KN2A / l)
Where
L is inductance measured in Henries
N is the number of turns on the coil
A is cross-sectional area of the coil
m0 is the permeability of free space (= 4π × 10-7 H/m)
K is the Nagaoka coefficient
l is the length of coil
Inductors in a Series Network
For inductors, L1, L2...Ln connected in series,
Leq = L1 + L2...+ Ln (L is inductance)
Inductors in a Parallel Network
For inductors, L1, L2...Ln connected in parallel,
1 / Leq = 1 / L1 + 1 / L2...+ 1 / Ln
[Back to Formula Index]
Thermodynamics Formulas
Thermodynamics is a vast field providing an analysis of the behavior of matter in bulk. It's a field focused on studying matter and energy in all their manifestations. Here are some of the most important formulas associated with classical thermodynamics and statistical physics.
First Law of Thermodynamics
dU = dQ + dW
where, dU is the change in internal energy, dQ is the heat absorbed by the system and dW is the work done on the system.
Thermodynamic Potentials
All of thermodynamical phenomena can be understood in terms of the changes in five thermodynamic potentials under various physical constraints. They are Internal Energy (U), Enthalpy (H), Helmholtz Free Energy (F), Gibbs Free Energy (G), Landau or Grand Potential (Φ). Each of these scalar quantities represents the potentiality of a thermodynamic system to do work of various kinds under different types of constraints on its physical parameters.
Thermodynamic Potential Defining Equation
U is Energy
T is temperature
S is Entropy
N is particle number
µ is Chemical Potential
p is Pressure
V is Volume
H is Enthalpy
G is Gibbs Free Energy
Φ is Grand Potential
F is Helmholtz Free Energy
Internal Energy (U) dU = TdS − pdV + µdN
Enthalpy (H) H = U + pV
dH = TdS + Vdp + µdN
Gibbs Free Energy (G) G = U - TS + pV = F + pV = H - TS
dG = -SdT + Vdp + µdN
Helmholtz Free Energy (F) F = U - TS
dF = - SdT - pdV + µdN
Landau or Grand Potential Φ = F - µN
dΦ = - SdT - pdV - Ndµ
Ideal Gas Equations
An ideal gas is a physicist's conception of a perfect gas composed of non-interacting particles which are easier to analyze, compared to real gases, which are much more complex, consisting of interacting particles. The resulting equations and laws of an ideal gas conform with the nature of real gases under certain conditions, though they fail to make exact predictions due the interactivity of molecules, which is not taken into consideration. Here are some of the most important physics formulas and equations, associated with ideal gases. Let's begin with the prime ideal gas laws and the equation of state of an ideal gas.
Law Equation
P is Pressure
V is Volume
T is Temperature
n is the number of moles
R is the ideal gas constant [= 8.3144621(75) J / K mol]
N is the number of particles
k is the Boltzmann Constant (= 1.3806488(13)×10-23)
Boyle's Law PV = Constant
or
P1V1 = P2V2
(At Constant Temperature)
Charles's Law V / T = Constant
or
V1 / T1 = V2 / T2
(At Constant Pressure)
Amontons' Law of Pressure-Temperature P / T = Constant
or
P1 / T1 = P2 / T2
(At Constant Volume)
Equation of State For An Ideal Gas PV = nRT = NkT
Kinetic Theory of Gases
Based on the primary assumptions that the volume of atoms or molecules is negligible, compared to the container volume and the attractive forces between molecules are negligible, the kinetic theory describes the properties of ideal gases. Here are the most important physics formulas related to the kinetic theory of monatomic gases.
Pressure (P) = 1/3 (Nm v2)
Here, P is pressure, N is the number of molecules and v2 is the mean squared particle velocity.
Internal Energy (U) = 3/2 (NkT)
Heat Capacities
Heat Capacity at Constant Pressure (Cp) = 5/2 Nk = Cv + Nk
Heat Capacity at Constant Volume (Cv) = 3/2 Nk
Ratio of Heat Capacities (γ) = Cp / Cv = 5/3
Velocity Formulas
Mean Molecular Velocity (Vmean) = [(8kT)/(πm)]1/2
Root Mean Square Velocity of a Molecule (Vrms) = (3kT/m)1/2
Most Probable Velocity of a Molecule (Vprob) = (2kT/m)1/2
Mean Free Path of a Molecule (λ) = (kT)/√2πd2P (Here P is in Pascals)
Here N is the number of molecules, k is the Boltzmann constant, P is pressure, d is the molecular diameter, m is mass of the molecule and T is the gas temperature.
[Back to Formula Index]
Electromagnetism
Here are some of the basic formulas from electromagnetism.
The coulombic force between two charges at rest is
(F) = q1q24πε0r2
Here,
q1, q2 are charges
ε0 is the permittivity of free space
r is the distance between the two charges
Lorentz Force
The Lorentz force is the force exerted by an electric and/or magnetic field on a charged particle.
(Lorentz Force) F = q (E + v x B)
where
q is the charge on the particle
E and B are the electric and magnetic field vectors
Relativistic Mechanics
Here are some of the most important relativistic mechanics formulas. The transition from classical to relativistic mechanics is not at all smooth, as it merges space and time into one by taking away the Newtonian idea of absolute time. If you know what is Einstein's special theory of relativity, then the following formulas will make sense to you.
Lorentz Transformations
Lorentz transformations can be perceived as rotations in four dimensional space. Just as rotations in 3D space mixes the space coordinates, a Lorentz transformation mixes time and space coordinates. Consider two, three dimensional frames of reference S(x,y,z) and S'(x',y',z') coinciding with each other.
Now consider that frame S' starts moving with a constant velocity v with respect to S frame. In relativistic mechanics, time is relative! So the time coordinate for the S' frame will be t' while that for S frame will be t.
Consider
γ = 1√(1 - v2/c2)
The coordinate transformations between the two frames are known as Lorentz transformations and are given as follows:
Lorentz Transformations of Space and Time
x = γ (x' + vt') and x' = γ (x - vt)
y = y'
z= z'
t = γ(t' + vx'/c2) and t' = γ(t - vx/c2)
Relativistic Velocity Transformations
In the same two frames S and S', the transformations for velocity components will be as follows (Here (Ux, Uy, Uz) and (Ux', Uy', Uz') are the velocity components in S and S' frames respectively):
Ux = (Ux' + v) / (1 + Ux'v / c2)
Uy = (Uy') / γ(1 + Ux'v / c2)
Uz = (Uz') / γ(1 + Ux'v / c2) and
Ux' = (Ux - v) / (1 - Uxv / c2)
Uy' = (Uy) / γ(1 - Uxv / c2)
Uz' = (Uz) / γ(1 - Uxv / c2)
Momentum and Energy Transformations in Relativistic Mechanics
Consider the same two frames (S, S') as in case of Lorentz coordinate transformations above. S' is moving at a velocity 'v' along the x-axis. Here again γis the Lorentz factor. In S frame (Px, Py, Pz) and in S' frame (Px', Py', Pz') are momentum components. Now we consider formulas for momentum and energy transformations for a particle, between these two reference frames in relativistic regime.
Component wise Momentum Transformations and Energy Transformations
Px = γ(Px' + vE' / c2)
Py = Py'
Pz = Pz'
E = γ(E' + vPx)
and
Px' = γ(Px - vE' / c2)
Py' = Py
Pz' = Pz
E' = γ(E - vPx)
Physical Formulas for Quantities in Relativistic Dynamics
All the known quantities in classical mechanics get modified, when we switch over to relativistic mechanics which is based on the special theory of relativity. Here are formulas of quantities in relativistic dynamics.
Relativistic momentum p = γm0v
where m0 is the rest mass of the particle.
Rest mass energy E = m0c2
Total Energy (Relativistic) E = √(p2c2 + m02c4))
[Back to Formula Index]
Optics
Optics is one of the oldest branches of physics. There are many important optics physics formulas, which we need frequently in solving physics problems. Here are some of the important and frequently needed optics formulas.
Snell's Law
Sin i Sin r = n2n1 = v1v2
where i is angle of incidence
r is the angle of refraction
n1 is refractive index of medium 1
n2 is refractive index of medium 2
v1, v2 are the velocities of light in medium 1 and medium 2 respectively
Gauss Lens Formula: 1/u + 1/v = 1/f
where
u - object distance
v - image distance
f - Focal length of the lens
Lens Maker's Equation
The most fundamental property of any optical lens is its ability to converge or diverge rays of light, which is measued by its focal length. Here is the lens maker's formula, which can help you calculate the focal length of a lens, from its physical parameters.
1 / f = [n-1][(1 / R1) - (1 / R2) + (n-1) d / nR1R2)]
Here,
n is refractive index of the lens material
R1 is the radius of curvature of the lens surface, facing the light source
R2 is the radius of curvature of the lens surface, facing away from the light source
d is the lens thickness
If the lens is very thin, compared to the distances - R1 and R2, the above formula can be approximated to:
(Thin Lens Approximation) 1 / f ≈ (n-1) [1 / R1 - 1 / R2]
Compound Lenses
The combined focal length (f) of two thin lenses, with focal length f1 and f2, in contact with each other:
1 / f = 1 / f1 + 1 / f2
If the two thin lenses are separated by distance d, their combined focal length is provided by the formula:
1 / f = 1 / f1 + 1 / f2 - (d / f1 - f2))
Newton's Rings Formulas
Here are the important formulas for Newton's rings experiment which illustrates diffraction.
nth Dark ring formula: r2n = nRλ
nth Bright ring formula: r2n = (n + ½) Rλ
where
nth ring radius
Radius of curvature of the lens
Wavelength of incident light wave
[Back to Formula Index]
Quantum Physics
Quantum physics is one of the most interesting branches of physics, which describes atoms and molecules, as well as atomic sub-structure. Here are some of the formulas related to the very basics of quantum physics, that you may require frequently.
De Broglie Wave
De Broglie Wavelength:
λ = hp
where, λ- De Broglie Wavelength, h - Planck's Constant, p is momentum of the particle.
Bragg's Law of Diffraction: 2a Sin θ = nλ
where
a - Distance between atomic planes
n - Order of Diffraction
θ - Angle of Diffraction
λ - Wavelength of incident radiation
Planck Relation
The plank relation gives the connection between energy and frequency of an electromagnetic wave:
E = hv = hω2π
where h is Planck's Constant, v the frequency of radiation and ω = 2πv
Uncertainty Principle
Uncertainty principle is the bedrock on which quantum mechanics is based. It exposes the inherent limitation that nature imposes on how precisely a physical quantity can be measured. Uncertainty relation holds between any two non-commuting variables. Two of the special uncertainty relations are given below.
Position-Momentum Uncertainty
What the position-momentum uncertainty relation says is, you cannot predict where a particle is and how fast it is moving, both, with arbitrary accuracy. The more precise you are about the position, more uncertain will you be about the particle's momentum and vice versa. The mathematical statement of this relation is given as follows:
Δx.Δp ≥ h2π
where Δx is the uncertainty in position and Δp is the uncertainty in momentum.
Energy-Time Uncertainty
This is an uncertainty relation between energy and time. This relation gives rise to some astounding results like, creation of virtual particles for arbitrarily short periods of time! It is mathematically stated as follows:
ΔE.Δt ≥ h2π
where ΔE is the uncertainty in energy and Δt is the uncertainty in time.
[Back to Formula Index]
This concludes my review of some of the important physics formulas. This list, is only representative and is by no means anywhere near complete. Physics is the basis of all sciences and therefore its domain extends over all sciences. Every branch of physics theory abounds with countless formulas. If you resort to just mugging up all these formulas, you may pass exams, but you will not be doing real physics. If you grasp the underlying theory behind these formulas, physics will be simplified. To view physics through the formulas and laws, you must be good at maths. There is no way you can run away from it. Mathematics is the language of nature!
The more things we find out about nature, more words we need to describe them. This has led to increasing jargonization of science with fields and sub-fields getting generated. You could refer to a glossary of science terms and scientific definitions for any jargon that is beyond your comprehension.
If you really want to get a hang of what it means to be a physicist and get an insight into physicist's view of things, read 'Feynman Lectures on Physics', which is highly recommended reading, for anyone who loves physics. It is written by one of the greatest physicists ever, Prof. Richard Feynman. Read and learn from the master. Solve as many problems as you can, on your own, to get a firm grasp of the subject.
By Omkar Phatak
Last Updated: March 7, 2012
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the Existence from the beginning to our days.
=.
I offer this scheme of ‘Fundamental Theory 0f Existence’:
1 The infinite vacuum T=0K
2 The particle: C/D = pi, R/N= k , E = Mc^2 = kc^2 , h = 0 , i^2= -1
3 The spins: h =E/t , h =kb, h* = h/2pi
4 The photon, the inertia
5 The electron: e^2 = h*ca, E = h*f , electromagnetic field
6 The gravitation, the star, the time and space
7 The proton
8 The atom(s)
9 The cell(s)
10 The Laws
a) The Law of conservation and transformation energy/mass
b) The Heisenberg Uncertainty Principle / Law
c) The Pauli Exclusion Principle/ Law
11 The test
==.
Everybody can check and explain why, for example,
the number one (1) or any other is wrong.
===.
Best wishes.
Israel Sadovnik Socratus
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Read more at Buzzle: http://www.buzzle.com/articles/physics-formulas-list.html
Learning physics is all about applying concepts to solve problems. This article provides a comprehensive physics formulas list, that will act as a ready reference, when you are solving physics problems. You can even use this list, for a quick revision before an exam.
Physics is the most fundamental of all sciences. It is also one of the toughest sciences to master. Learning physics is basically studying the fundamental laws that govern our universe. I would say that there is a lot more to ascertain than just remember and mug up the physics formulas. Try to understand what a formula says and means, and what physical relation it expounds. If you understand the physical concepts underlying those formulas, deriving them or remembering them is easy. This Buzzle article lists some physics formulas that you would need in solving basic physics problems.
Physics Formulas
Mechanics
Friction
Moment of Inertia
Newtonian Gravity
Projectile Motion
Simple Pendulum
Electricity
Thermodynamics
Electromagnetism
Optics
Quantum Physics
Derive all these formulas once, before you start using them. Study physics and look at it as an opportunity to appreciate the underlying beauty of nature, expressed through natural laws. Physics help is provided here in the form of ready to use formulas. Physics has a reputation for being difficult and to some extent that's true, due to the mathematics involved. If you don't wish to think on your own and apply basic physics principles, solving physics problems is always going to be tough. Our physics formulas list is aimed at helping you out in solving problems. The joy of having solved a physics problem on your own, is worth all the effort! Understanding physics concepts challenges your imagination and thinking potential, wherein, if you visualize a problem, then you can come up with a solution. So here is the promised list which will help you out.
Mechanics
Mechanics is the oldest branch of physics. Mechanics deals with all kinds and complexities of motion. It includes various techniques, which can simplify the solution of a mechanical problem.
Motion in One Dimension
The formulas for motion in one dimension (Also called Kinematical equations of motion) are as follows. (Here 'u' is initial velocity, 'v' is final velocity, 'a' is acceleration and t is time):
s = ut + ½ at2
v = u + at
v2 = u2 + 2as
vav (Average Velocity) = (v+u)/2
Momentum, Force and Impulse
Formulas for momentum, impulse and force concerning a particle moving in 3 dimensions are as follows (Here force, momentum and velocity are vectors ):
Momentum is the product of mass and velocity of a body. Momentum is calculate using the formula: P = m (mass) x v (velocity)
Force can defined as something which causes a change in momentum of a body. Force is given by the celebrated newton's law of motion: F = m (mass) x a (acceleration)
Impulse is a large force applied in a very short time period. The strike of a hammer is an impulse. Impulse is given by I = m(v-u)
Pressure
Pressure is defined as force per unit area:
Pressure (P) = Force (F)Area (A)
Density
Density is the mass contained in a body per unit volume.
The formula for density is:
Density (D) = Mass(M)Volume (V)
Angular Momentum
Angular momentum is an analogous quantity to linear momentum in which the body is undergoing rotational motion. The formula for angular momentum (J) is given by:
J = r x p
where J denotes angular momentum, r is radius vector and p is linear momentum.
Torque
Torque can be defined as moment of force. Torque causes rotational motion. The formula for torque is: τ = r x F, where τ is torque, r is the radius vector and F is linear force.
Circular Motion
The formulas for circular motion of an object of mass 'm' moving in a circle of radius 'r' at a tangential velocity 'v' are as follows:
Centripetal force (F) = mv2r
Centripetal Acceleration (a) = v2r
Center of Mass
General Formula for Center of mass of a rigid body is :
R = ΣNi = 1 miriΣNi = 1mi
where R is the position vector for center of mass, r is the generic position vector for all the particles of the object and N is the total number of particles.
Reduced Mass for two Interacting Bodies
The physics formula for reduced mass (μ) is :
μ = m1m2m1 + m2
where m1 is mass of the first body, m2 is the mass of the second body.
Work and Energy
Formulas for work and energy in case of one dimensional motion are as follows:
W (Work Done) = F (Force) x D (Displacement)
Energy can be broadly classified into two types, Potential Energy and Kinetic Energy. In case of gravitational force, the potential energy is given by
P.E.(Gravitational) = m (Mass) x g (Acceleration due to Gravity) x h (Height)
The transitional kinetic energy is given by ½ m (mass) x v2(velocity squared)
Power
Power is, work done per unit time. The formula for power is given as
Power (P) = V2R =I2R
where P=power, W = Work, t = time.
[Back to Formula Index]
Friction
Friction can be classified to be of two kinds : Static friction and dynamic friction.
Static Friction: Static friction is characterized by a coefficient of static friction μ . Coefficient of static friction is defined as the ratio of applied tangential force (F) which can induce sliding, to the normal force between surfaces in contact with each other. The formula to calculate this static coefficient is as follows:
μ = Applied Tangential Force (F)Normal Force(N)
The amount of force required to slide a solid resting on flat surface depends on the co efficient of static friction and is given by the formula:
FHorizontal = μ x M(Mass of solid) x g (acceleration)
Dynamic Friction:
Dynamic friction is also characterized by the same coefficient of friction as static friction and therefore formula for calculating coefficient of dynamic friction is also the same as above. Only the dynamic friction coefficient is generally lower than the static one as the applied force required to overcome normal force is lesser.
[Back to Formula Index]
Moment of Inertia
Here are some formulas for Moments of Inertia of different objects. (M stands for mass, R for radius and L for length):
Object Axis Moment of Inertia
Disk Axis parallel to disc, passing through the center MR2/2
Disk Axis passing through the center and perpendicular to disc MR2/2
Thin Rod Axis perpendicular to the Rod and passing through center ML2/12
Solid Sphere Axis passing through the center 2MR2/5
Solid Shell Axis passing through the center 2MR2/3
[Back to Formula Index]
Newtonian Gravity
Here are some important formulas, related to Newtonian Gravity:
Newton's Law of universal Gravitation:
Fg = Gm1m2r2
where
m1, m2 are the masses of two bodies
G is the universal gravitational constant which has a value of 6.67300 × 10-11 m3 kg-1 s-2
r is distance between the two bodies
Formula for escape velocity (vesc) = (2GM / R)1/2where,
M is mass of central gravitating body
R is radius of the central body
[Back to Formula Index]
Projectile Motion
Here are two important formulas related to projectile motion:
(v = velocity of particle, v0 = initial velocity, g is acceleration due to gravity, θ is angle of projection, h is maximum height and l is the range of the projectile.)
Maximum height of projectile (h) = v0 2sin2θ2g
Horizontal range of projectile (l) = v0 2sin 2θ / g
[Back to Formula Index]
Simple Pendulum
The physics formula for the period of a simple pendulum (T) = 2π √(l/g)where
l is the length of the pendulum
g is acceleration due to gravity
Conical Pendulum
The Period of a conical pendulum (T) = 2π √(lcosθ/g)
where
l is the length of the pendulum
g is acceleration due to gravity
Half angle of the conical pendulum
[Back to Formula Index]
Electricity
Here are some formulas related to electricity.
Ohm's Law
Ohm's law gives a relation between the voltage applied a current flowing across a solid conductor:
V (Voltage) = I (Current) x R (Resistance)
Power
In case of a closed electrical circuit with applied voltage V and resistance R, through which current I is flowing,
Power (P) = V2R
= I2R. . . (because V = IR, Ohm's Law)
Kirchoff's Voltage Law
For every loop in an electrical circuit:
ΣiVi = 0
where Vi are all the voltages applied across the circuit.
Kirchoff's Current Law
At every node of an electrical circuit:
ΣiIi = 0
where Ii are all the currents flowing towards or away from the node in the circuit.
Resistance
The physics formulas for equivalent resistance in case of parallel and series combination are as follows:
Resistances R1, R2, R3 in series:
Req = R1 + R2 + R3
Resistances R1 and R2 in parallel:
Req = R1R2R1 + R2
For n number of resistors, R1, R2...Rn, the formula will be:
1/Req = 1/R1 + 1/R2 + 1/R3...+ 1/Rn
Capacitors
A capacitor stores electrical energy, when placed in an electric field. A typical capacitor consists of two conductors separated by a dielectric or insulating material. Here are the most important formulas related to capacitors. Unit of capacitance is Farad (F) and its values are generally specified in mF (micro Farad = 10 -6 F).
Capacitance (C) = Q / V
Energy Stored in a Capacitor (Ecap) = 1/2 CV2 = 1/2 (Q2 / C) = 1/2 (QV)
Current Flowing Through a Capacitor I = C (dV / dt)
Equivalent capacitance for 'n' capacitors connected in parallel:
Ceq (Parallel) = C1 + C2 + C3...+ Cn = Σi=1 to n Ci
Equivalent capacitance for 'n' capacitors in series:
1 / Ceq (Series) = 1 / C1 + 1 / C2...+ 1 / Cn = Σi=1 to n (1 / Ci)
Here
C is the capacitance
Q is the charge stored on each conductor in the capacitor
V is the potential difference across the capacitor
Parallel Plate Capacitor Formula:
C = kε0 (A/d)
Where
k = dielectric constant (k = 1 in vacuum)
ε0 = Permittivity of Free Space (= 8.85 × 10-12 C2 / Nm2)
A = Plate Area (in square meters)
d = Plate Separation (in meters)
Cylinrical Capacitor Formula:
C = 2π kε0 [L / ln(b / a)]
Where
k = dielectric constant (k = 1 in vacuum)
ε0 = Permittivity of Free Space (= 8.85 × 10-12 C2 / Nm2)
L = Capacitor Length
a = Inner conductor radius
b = Outer conductor radius
Spherical Capacitor Formula:
C = 4π kε0 [(ab)/(b-a)]
Where
k = dielectric constant (k = 1 in vacuum)
ε0 = Permittivity of Free Space (= 8.85 × 10-12 C2 / Nm2)
a = Inner conductor radius
b = Outer conductor radius
Inductors
An inductor is an electrical component which resists the flow of electrons or electric current through it. This property of inductance, in these devices, is caused by the electromotive force, created by magnetic field induced in them. The unit of inductance is Henry. Here are some important formulas associated with inductors.
Energy Stored in Inductor (Estored) = 1/2 (LI2)
Where, L is inductance and I is the current flowing through the inductor.
Inductance of a cylindrical air core Coil (L) = (m0KN2A / l)
Where
L is inductance measured in Henries
N is the number of turns on the coil
A is cross-sectional area of the coil
m0 is the permeability of free space (= 4π × 10-7 H/m)
K is the Nagaoka coefficient
l is the length of coil
Inductors in a Series Network
For inductors, L1, L2...Ln connected in series,
Leq = L1 + L2...+ Ln (L is inductance)
Inductors in a Parallel Network
For inductors, L1, L2...Ln connected in parallel,
1 / Leq = 1 / L1 + 1 / L2...+ 1 / Ln
[Back to Formula Index]
Thermodynamics Formulas
Thermodynamics is a vast field providing an analysis of the behavior of matter in bulk. It's a field focused on studying matter and energy in all their manifestations. Here are some of the most important formulas associated with classical thermodynamics and statistical physics.
First Law of Thermodynamics
dU = dQ + dW
where, dU is the change in internal energy, dQ is the heat absorbed by the system and dW is the work done on the system.
Thermodynamic Potentials
All of thermodynamical phenomena can be understood in terms of the changes in five thermodynamic potentials under various physical constraints. They are Internal Energy (U), Enthalpy (H), Helmholtz Free Energy (F), Gibbs Free Energy (G), Landau or Grand Potential (Φ). Each of these scalar quantities represents the potentiality of a thermodynamic system to do work of various kinds under different types of constraints on its physical parameters.
Thermodynamic Potential Defining Equation
U is Energy
T is temperature
S is Entropy
N is particle number
µ is Chemical Potential
p is Pressure
V is Volume
H is Enthalpy
G is Gibbs Free Energy
Φ is Grand Potential
F is Helmholtz Free Energy
Internal Energy (U) dU = TdS − pdV + µdN
Enthalpy (H) H = U + pV
dH = TdS + Vdp + µdN
Gibbs Free Energy (G) G = U - TS + pV = F + pV = H - TS
dG = -SdT + Vdp + µdN
Helmholtz Free Energy (F) F = U - TS
dF = - SdT - pdV + µdN
Landau or Grand Potential Φ = F - µN
dΦ = - SdT - pdV - Ndµ
Ideal Gas Equations
An ideal gas is a physicist's conception of a perfect gas composed of non-interacting particles which are easier to analyze, compared to real gases, which are much more complex, consisting of interacting particles. The resulting equations and laws of an ideal gas conform with the nature of real gases under certain conditions, though they fail to make exact predictions due the interactivity of molecules, which is not taken into consideration. Here are some of the most important physics formulas and equations, associated with ideal gases. Let's begin with the prime ideal gas laws and the equation of state of an ideal gas.
Law Equation
P is Pressure
V is Volume
T is Temperature
n is the number of moles
R is the ideal gas constant [= 8.3144621(75) J / K mol]
N is the number of particles
k is the Boltzmann Constant (= 1.3806488(13)×10-23)
Boyle's Law PV = Constant
or
P1V1 = P2V2
(At Constant Temperature)
Charles's Law V / T = Constant
or
V1 / T1 = V2 / T2
(At Constant Pressure)
Amontons' Law of Pressure-Temperature P / T = Constant
or
P1 / T1 = P2 / T2
(At Constant Volume)
Equation of State For An Ideal Gas PV = nRT = NkT
Kinetic Theory of Gases
Based on the primary assumptions that the volume of atoms or molecules is negligible, compared to the container volume and the attractive forces between molecules are negligible, the kinetic theory describes the properties of ideal gases. Here are the most important physics formulas related to the kinetic theory of monatomic gases.
Pressure (P) = 1/3 (Nm v2)
Here, P is pressure, N is the number of molecules and v2 is the mean squared particle velocity.
Internal Energy (U) = 3/2 (NkT)
Heat Capacities
Heat Capacity at Constant Pressure (Cp) = 5/2 Nk = Cv + Nk
Heat Capacity at Constant Volume (Cv) = 3/2 Nk
Ratio of Heat Capacities (γ) = Cp / Cv = 5/3
Velocity Formulas
Mean Molecular Velocity (Vmean) = [(8kT)/(πm)]1/2
Root Mean Square Velocity of a Molecule (Vrms) = (3kT/m)1/2
Most Probable Velocity of a Molecule (Vprob) = (2kT/m)1/2
Mean Free Path of a Molecule (λ) = (kT)/√2πd2P (Here P is in Pascals)
Here N is the number of molecules, k is the Boltzmann constant, P is pressure, d is the molecular diameter, m is mass of the molecule and T is the gas temperature.
[Back to Formula Index]
Electromagnetism
Here are some of the basic formulas from electromagnetism.
The coulombic force between two charges at rest is
(F) = q1q24πε0r2
Here,
q1, q2 are charges
ε0 is the permittivity of free space
r is the distance between the two charges
Lorentz Force
The Lorentz force is the force exerted by an electric and/or magnetic field on a charged particle.
(Lorentz Force) F = q (E + v x B)
where
q is the charge on the particle
E and B are the electric and magnetic field vectors
Relativistic Mechanics
Here are some of the most important relativistic mechanics formulas. The transition from classical to relativistic mechanics is not at all smooth, as it merges space and time into one by taking away the Newtonian idea of absolute time. If you know what is Einstein's special theory of relativity, then the following formulas will make sense to you.
Lorentz Transformations
Lorentz transformations can be perceived as rotations in four dimensional space. Just as rotations in 3D space mixes the space coordinates, a Lorentz transformation mixes time and space coordinates. Consider two, three dimensional frames of reference S(x,y,z) and S'(x',y',z') coinciding with each other.
Now consider that frame S' starts moving with a constant velocity v with respect to S frame. In relativistic mechanics, time is relative! So the time coordinate for the S' frame will be t' while that for S frame will be t.
Consider
γ = 1√(1 - v2/c2)
The coordinate transformations between the two frames are known as Lorentz transformations and are given as follows:
Lorentz Transformations of Space and Time
x = γ (x' + vt') and x' = γ (x - vt)
y = y'
z= z'
t = γ(t' + vx'/c2) and t' = γ(t - vx/c2)
Relativistic Velocity Transformations
In the same two frames S and S', the transformations for velocity components will be as follows (Here (Ux, Uy, Uz) and (Ux', Uy', Uz') are the velocity components in S and S' frames respectively):
Ux = (Ux' + v) / (1 + Ux'v / c2)
Uy = (Uy') / γ(1 + Ux'v / c2)
Uz = (Uz') / γ(1 + Ux'v / c2) and
Ux' = (Ux - v) / (1 - Uxv / c2)
Uy' = (Uy) / γ(1 - Uxv / c2)
Uz' = (Uz) / γ(1 - Uxv / c2)
Momentum and Energy Transformations in Relativistic Mechanics
Consider the same two frames (S, S') as in case of Lorentz coordinate transformations above. S' is moving at a velocity 'v' along the x-axis. Here again γis the Lorentz factor. In S frame (Px, Py, Pz) and in S' frame (Px', Py', Pz') are momentum components. Now we consider formulas for momentum and energy transformations for a particle, between these two reference frames in relativistic regime.
Component wise Momentum Transformations and Energy Transformations
Px = γ(Px' + vE' / c2)
Py = Py'
Pz = Pz'
E = γ(E' + vPx)
and
Px' = γ(Px - vE' / c2)
Py' = Py
Pz' = Pz
E' = γ(E - vPx)
Physical Formulas for Quantities in Relativistic Dynamics
All the known quantities in classical mechanics get modified, when we switch over to relativistic mechanics which is based on the special theory of relativity. Here are formulas of quantities in relativistic dynamics.
Relativistic momentum p = γm0v
where m0 is the rest mass of the particle.
Rest mass energy E = m0c2
Total Energy (Relativistic) E = √(p2c2 + m02c4))
[Back to Formula Index]
Optics
Optics is one of the oldest branches of physics. There are many important optics physics formulas, which we need frequently in solving physics problems. Here are some of the important and frequently needed optics formulas.
Snell's Law
Sin i Sin r = n2n1 = v1v2
where i is angle of incidence
r is the angle of refraction
n1 is refractive index of medium 1
n2 is refractive index of medium 2
v1, v2 are the velocities of light in medium 1 and medium 2 respectively
Gauss Lens Formula: 1/u + 1/v = 1/f
where
u - object distance
v - image distance
f - Focal length of the lens
Lens Maker's Equation
The most fundamental property of any optical lens is its ability to converge or diverge rays of light, which is measued by its focal length. Here is the lens maker's formula, which can help you calculate the focal length of a lens, from its physical parameters.
1 / f = [n-1][(1 / R1) - (1 / R2) + (n-1) d / nR1R2)]
Here,
n is refractive index of the lens material
R1 is the radius of curvature of the lens surface, facing the light source
R2 is the radius of curvature of the lens surface, facing away from the light source
d is the lens thickness
If the lens is very thin, compared to the distances - R1 and R2, the above formula can be approximated to:
(Thin Lens Approximation) 1 / f ≈ (n-1) [1 / R1 - 1 / R2]
Compound Lenses
The combined focal length (f) of two thin lenses, with focal length f1 and f2, in contact with each other:
1 / f = 1 / f1 + 1 / f2
If the two thin lenses are separated by distance d, their combined focal length is provided by the formula:
1 / f = 1 / f1 + 1 / f2 - (d / f1 - f2))
Newton's Rings Formulas
Here are the important formulas for Newton's rings experiment which illustrates diffraction.
nth Dark ring formula: r2n = nRλ
nth Bright ring formula: r2n = (n + ½) Rλ
where
nth ring radius
Radius of curvature of the lens
Wavelength of incident light wave
[Back to Formula Index]
Quantum Physics
Quantum physics is one of the most interesting branches of physics, which describes atoms and molecules, as well as atomic sub-structure. Here are some of the formulas related to the very basics of quantum physics, that you may require frequently.
De Broglie Wave
De Broglie Wavelength:
λ = hp
where, λ- De Broglie Wavelength, h - Planck's Constant, p is momentum of the particle.
Bragg's Law of Diffraction: 2a Sin θ = nλ
where
a - Distance between atomic planes
n - Order of Diffraction
θ - Angle of Diffraction
λ - Wavelength of incident radiation
Planck Relation
The plank relation gives the connection between energy and frequency of an electromagnetic wave:
E = hv = hω2π
where h is Planck's Constant, v the frequency of radiation and ω = 2πv
Uncertainty Principle
Uncertainty principle is the bedrock on which quantum mechanics is based. It exposes the inherent limitation that nature imposes on how precisely a physical quantity can be measured. Uncertainty relation holds between any two non-commuting variables. Two of the special uncertainty relations are given below.
Position-Momentum Uncertainty
What the position-momentum uncertainty relation says is, you cannot predict where a particle is and how fast it is moving, both, with arbitrary accuracy. The more precise you are about the position, more uncertain will you be about the particle's momentum and vice versa. The mathematical statement of this relation is given as follows:
Δx.Δp ≥ h2π
where Δx is the uncertainty in position and Δp is the uncertainty in momentum.
Energy-Time Uncertainty
This is an uncertainty relation between energy and time. This relation gives rise to some astounding results like, creation of virtual particles for arbitrarily short periods of time! It is mathematically stated as follows:
ΔE.Δt ≥ h2π
where ΔE is the uncertainty in energy and Δt is the uncertainty in time.
[Back to Formula Index]
This concludes my review of some of the important physics formulas. This list, is only representative and is by no means anywhere near complete. Physics is the basis of all sciences and therefore its domain extends over all sciences. Every branch of physics theory abounds with countless formulas. If you resort to just mugging up all these formulas, you may pass exams, but you will not be doing real physics. If you grasp the underlying theory behind these formulas, physics will be simplified. To view physics through the formulas and laws, you must be good at maths. There is no way you can run away from it. Mathematics is the language of nature!
The more things we find out about nature, more words we need to describe them. This has led to increasing jargonization of science with fields and sub-fields getting generated. You could refer to a glossary of science terms and scientific definitions for any jargon that is beyond your comprehension.
If you really want to get a hang of what it means to be a physicist and get an insight into physicist's view of things, read 'Feynman Lectures on Physics', which is highly recommended reading, for anyone who loves physics. It is written by one of the greatest physicists ever, Prof. Richard Feynman. Read and learn from the master. Solve as many problems as you can, on your own, to get a firm grasp of the subject.
By Omkar Phatak
Last Updated: March 7, 2012
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All these formulas explain some part of the real world.
The problem is to put them in one logical line in order to explain
the Existence from the beginning to our days.
=.
I offer this scheme of ‘Fundamental Theory 0f Existence’:
1 The infinite vacuum T=0K
2 The particle: C/D = pi, R/N= k , E = Mc^2 = kc^2 , h = 0 , i^2= -1
3 The spins: h =E/t , h =kb, h* = h/2pi
4 The photon, the inertia
5 The electron: e^2 = h*ca, E = h*f , electromagnetic field
6 The gravitation, the star, the time and space
7 The proton
8 The atom(s)
9 The cell(s)
10 The Laws
a) The Law of conservation and transformation energy/mass
b) The Heisenberg Uncertainty Principle / Law
c) The Pauli Exclusion Principle/ Law
11 The test
==.
Everybody can check and explain why, for example,
the number one (1) or any other is wrong.
===.
Best wishes.
Israel Sadovnik Socratus
===================. - Israel Socratus [October 1, 2011]
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Crit Care. 2003; 7(6): 451–459.
Published online 2003 November 5.
PMCID: PMC374386
Statistics review 7: Correlation and regression
1
This article has been cited by other articles in PMC.
Abstract
The
present review introduces methods of analyzing the relationship between
two quantitative variables. The calculation and interpretation of the
sample product moment correlation coefficient and the linear regression
equation are discussed and illustrated. Common misuses of the techniques
are considered. Tests and confidence intervals for the population
parameters are described, and failures of the underlying assumptions are
highlighted.
Keywords: coefficient of determination, correlation coefficient, least squares regression line
Introduction
The
most commonly used techniques for investigating the relationship
between two quantitative variables are correlation and linear
regression. Correlation quantifies the strength of the linear
relationship between a pair of variables, whereas regression expresses
the relationship in the form of an equation. For example, in patients
attending an accident and emergency unit (A&E), we could use
correlation and regression to determine whether there is a relationship
between age and urea level, and whether the level of urea can be
predicted for a given age.
Scatter diagram
When
investigating a relationship between two variables, the first step is
to show the data values graphically on a scatter diagram. Consider the
data given in Table Table1.1.
These are the ages (years) and the logarithmically transformed
admission serum urea (natural logarithm [ln] urea) for 20 patients
attending an A&E. The reason for transforming the urea levels was to
obtain a more Normal distribution [1]. The scatter diagram for ln urea and age (Fig. (Fig.1)1) suggests there is a positive linear relationship between these variables.
Correlation
On
a scatter diagram, the closer the points lie to a straight line, the
stronger the linear relationship between two variables. To quantify the
strength of the relationship, we can calculate the correlation
coefficient. In algebraic notation, if we have two variables x and y,
and the data take the form of n pairs (i.e. [x1, y1], [x2, y2], [x3, y3] ... [xn, yn]), then the correlation coefficient is given by the following equation:
where 
is the mean of the x values, and 
is the mean of the y values.
is the mean of the x values, and is the mean of the y values.
This
is the product moment correlation coefficient (or Pearson correlation
coefficient). The value of r always lies between -1 and +1. A value of
the correlation coefficient close to +1 indicates a strong positive
linear relationship (i.e. one variable increases with the other; Fig. Fig.2).2).
A value close to -1 indicates a strong negative linear relationship
(i.e. one variable decreases as the other increases; Fig. Fig.3).3). A value close to 0 indicates no linear relationship (Fig. (Fig.4);4); however, there could be a nonlinear relationship between the variables (Fig. (Fig.55).
For
the A&E data, the correlation coefficient is 0.62, indicating a
moderate positive linear relationship between the two variables.
Hypothesis test of correlation
We
can use the correlation coefficient to test whether there is a linear
relationship between the variables in the population as a whole. The
null hypothesis is that the population correlation coefficient equals 0.
The value of r can be compared with those given in Table Table2,2, or alternatively exact P values
can be obtained from most statistical packages. For the A&E data, r
= 0.62 with a sample size of 20 is greater than the value highlighted
bold in Table Table22 for P = 0.01, indicating a P value
of less than 0.01. Therefore, there is sufficient evidence to suggest
that the true population correlation coefficient is not 0 and that there
is a linear relationship between ln urea and age.
5%
and 1% points for the distribution of the correlation coefficient under
the null hypothesis that the population correlation is 0 in a
two-tailed test
Confidence interval for the population correlation coefficient
Although
the hypothesis test indicates whether there is a linear relationship,
it gives no indication of the strength of that relationship. This
additional information can be obtained from a confidence interval for
the population correlation coefficient.
To calculate a
confidence interval, r must be transformed to give a Normal distribution
making use of Fisher's z transformation [2]:
The standard error [3] of zr is approximately:
and hence a 95% confidence interval for the true population value for the transformed correlation coefficient zr is given by zr - (1.96 × standard error) to zr + (1.96 × standard error). Because zr is Normally distributed, 1.96 deviations from the statistic will give a 95% confidence interval.
For the A&E data the transformed correlation coefficient zr between ln urea and age is:
The standard error of zr is:
The 95% confidence interval for zr is therefore 0.725 - (1.96 × 0.242) to 0.725 + (1.96 × 0.242), giving 0.251 to 1.199.
We
must use the inverse of Fisher's transformation on the lower and upper
limits of this confidence interval to obtain the 95% confidence interval
for the correlation coefficient. The lower limit is:
giving 0.25 and the upper limit is:
giving 0.83. Therefore, we are 95% confident that the population correlation coefficient is between 0.25 and 0.83.
The
width of the confidence interval clearly depends on the sample size,
and therefore it is possible to calculate the sample size required for a
given level of accuracy. For an example, see Bland [4].
Misuse of correlation
There are a number of common situations in which the correlation coefficient can be misinterpreted.
One
of the most common errors in interpreting the correlation coefficient
is failure to consider that there may be a third variable related to
both of the variables being investigated, which is responsible for the
apparent correlation. Correlation does not imply causation. To
strengthen the case for causality, consideration must be given to other
possible underlying variables and to whether the relationship holds in
other populations.
A nonlinear relationship may exist
between two variables that would be inadequately described, or possibly
even undetected, by the correlation coefficient.
A data
set may sometimes comprise distinct subgroups, for example males and
females. This could result in clusters of points leading to an inflated
correlation coefficient (Fig. (Fig.6).6). A single outlier may produce the same sort of effect.
Subgroups in the data resulting in a misleading correlation. All data: r = 0.57; males: r = -0.41; females: r = -0.26.
It
is important that the values of one variable are not determined in
advance or restricted to a certain range. This may lead to an invalid
estimate of the true correlation coefficient because the subjects are
not a random sample.
Another situation
in which a correlation coefficient is sometimes misinterpreted is when
comparing two methods of measurement. A high correlation can be
incorrectly taken to mean that there is agreement between the two
methods. An analysis that investigates the differences between pairs of
observations, such as that formulated by Bland and Altman [5], is more appropriate.
Regression
In
the A&E example we are interested in the effect of age (the
predictor or x variable) on ln urea (the response or y variable). We
want to estimate the underlying linear relationship so that we can
predict ln urea (and hence urea) for a given age. Regression can be used
to find the equation of this line. This line is usually referred to as
the regression line.
Note that in a scatter diagram the response variable is always plotted on the vertical (y) axis.
Equation of a straight line
The
equation of a straight line is given by y = a + bx, where the
coefficients a and b are the intercept of the line on the y axis and the
gradient, respectively. The equation of the regression line for the
A&E data (Fig. (Fig.7)7)
is as follows: ln urea = 0.72 + (0.017 × age) (calculated using the
method of least squares, which is described below). The gradient of this
line is 0.017, which indicates that for an increase of 1 year in age
the expected increase in ln urea is 0.017 units (and hence the expected
increase in urea is 1.02 mmol/l). The predicted ln urea of a patient
aged 60 years, for example, is 0.72 + (0.017 × 60) = 1.74 units. This
transforms to a urea level of e1.74 = 5.70 mmol/l. The y
intercept is 0.72, meaning that if the line were projected back to age =
0, then the ln urea value would be 0.72. However, this is not a
meaningful value because age = 0 is a long way outside the range of the
data and therefore there is no reason to believe that the straight line
would still be appropriate.
Method of least squares
The
regression line is obtained using the method of least squares. Any line
y = a + bx that we draw through the points gives a predicted or fitted
value of y for each value of x in the data set. For a particular value
of x the vertical difference between the observed and fitted value of y
is known as the deviation, or residual (Fig. (Fig.8).8).
The method of least squares finds the values of a and b that minimise
the sum of the squares of all the deviations. This gives the following
formulae for calculating a and b:
Usually, these values would be calculated using a statistical package or the statistical functions on a calculator.
Hypothesis tests and confidence intervals
We
can test the null hypotheses that the population intercept and gradient
are each equal to 0 using test statistics given by the estimate of the
coefficient divided by its standard error.
The
test statistics are compared with the t distribution on n - 2 (sample
size - number of regression coefficients) degrees of freedom [4].
The 95% confidence interval for each of the population coefficients are calculated as follows: coefficient ± (tn-2 × the standard error), where tn-2 is the 5% point for a t distribution with n - 2 degrees of freedom.
For the A&E data, the output (Table (Table3)3) was obtained from a statistical package. The P value
for the coefficient of ln urea (0.004) gives strong evidence against
the null hypothesis, indicating that the population coefficient is not 0
and that there is a linear relationship between ln urea and age. The
coefficient of ln urea is the gradient of the regression line and its
hypothesis test is equivalent to the test of the population correlation
coefficient discussed above. The P value for the constant of
0.054 provides insufficient evidence to indicate that the population
coefficient is different from 0. Although the intercept is not
significant, it is still appropriate to keep it in the equation. There
are some situations in which a straight line passing through the origin
is known to be appropriate for the data, and in this case a special
regression analysis can be carried out that omits the constant [6].
Regression parameter estimates, P values and confidence intervals for the accident and emergency unit data
Analysis of variance
As
stated above, the method of least squares minimizes the sum of squares
of the deviations of the points about the regression line. Consider the
small data set illustrated in Fig. Fig.9.9.
This figure shows that, for a particular value of x, the distance of y
from the mean of y (the total deviation) is the sum of the distance of
the fitted y value from the mean (the deviation explained by the
regression) and the distance from y to the line (the deviation not
explained by the regression).
The regression line for these data is given by y = 6 + 2x. The observed, fitted values and deviations are given in Table Table4.4.
The sum of squared deviations can be compared with the total variation
in y, which is measured by the sum of squares of the deviations of y
from the mean of y. Table Table44
illustrates the relationship between the sums of squares. Total sum of
squares = sum of squares explained by the regression line + sum of
squares not explained by the regression line. The explained sum of
squares is referred to as the 'regression sum of squares' and the
unexplained sum of squares is referred to as the 'residual sum of
squares'.
Small data set with the fitted values from the regression, the deviations and their sums of squares
This partitioning of the total sum of squares can be presented in an analysis of variance table (Table (Table5).5).
The total degrees of freedom = n - 1, the regression degrees of freedom
= 1, and the residual degrees of freedom = n - 2 (total - regression
degrees of freedom). The mean squares are the sums of squares divided by
their degrees of freedom.
If
there were no linear relationship between the variables then the
regression mean squares would be approximately the same as the residual
mean squares. We can test the null hypothesis that there is no linear
relationship using an F test. The test statistic is calculated as the
regression mean square divided by the residual mean square, and a P value may be obtained by comparison of the test statistic with the F distribution with 1 and n - 2 degrees of freedom [2]. Usually, this analysis is carried out using a statistical package that will produce an exact P value.
In fact, the F test from the analysis of variance is equivalent to the t
test of the gradient for regression with only one predictor. This is
not the case with more than one predictor, but this will be the subject
of a future review. As discussed above, the test for gradient is also
equivalent to that for the correlation, giving three tests with
identical P values. Therefore, when there is only one predictor variable it does not matter which of these tests is used.
The analysis of variance for the A&E data (Table (Table6)6) gives a P value of 0.006 (the same P value as obtained previously), again indicating a linear relationship between ln urea and age.
Coefficent of determination
Another useful quantity that can be obtained from the analysis of variance is the coefficient of determination (R2).
It is the proportion of the total variation in y accounted for by the regression model. Values of R2 close to 1 imply that most of the variability in y is explained by the regression model. R2 is the same as r2 in regression when there is only one predictor variable.
For the A&E data, R2 = 1.462/3.804 = 0.38 (i.e. the same as 0.622),
and therefore age accounts for 38% of the total variation in ln urea.
This means that 62% of the variation in ln urea is not accounted for by
age differences. This may be due to inherent variability in ln urea or
to other unknown factors that affect the level of ln urea.
Prediction
The
fitted value of y for a given value of x is an estimate of the
population mean of y for that particular value of x. As such it can be
used to provide a confidence interval for the population mean [3]. The fitted values change as x changes, and therefore the confidence intervals will also change.
The 95% confidence interval for the fitted value of y for a particular value of x, say xp, is again calculated as fitted y ± (tn-2 × the standard error). The standard error is given by:
Fig. Fig.1010
shows the range of confidence intervals for the A&E data. For
example, the 95% confidence interval for the population mean ln urea for
a patient aged 60 years is 1.56 to 1.92 units. This transforms to urea
values of 4.76 to 6.82 mmol/l.
Regression line, its 95% confidence interval and the 95% prediction interval for individual patients.
The fitted value for y also provides a predicted value for an individual, and a prediction interval or reference range [3] can be obtained (Fig. (Fig.10).10). The prediction interval is calculated in the same way as the confidence interval but the standard error is given by:
For
example, the 95% prediction interval for the ln urea for a patient aged
60 years is 0.97 to 2.52 units. This transforms to urea values of 2.64
to 12.43 mmol/l.
Both confidence intervals and prediction intervals become wider for values of the predictor variable further from the mean.
Assumptions and limitations
The
use of correlation and regression depends on some underlying
assumptions. The observations are assumed to be independent. For
correlation both variables should be random variables, but for
regression only the response variable y must be random. In carrying out
hypothesis tests or calculating confidence intervals for the regression
parameters, the response variable should have a Normal distribution and
the variability of y should be the same for each value of the predictor
variable. The same assumptions are needed in testing the null hypothesis
that the correlation is 0, but in order to interpret confidence
intervals for the correlation coefficient both variables must be
Normally distributed. Both correlation and regression assume that the
relationship between the two variables is linear.
A
scatter diagram of the data provides an initial check of the assumptions
for regression. The assumptions can be assessed in more detail by
looking at plots of the residuals [4,7].
Commonly, the residuals are plotted against the fitted values. If the
relationship is linear and the variability constant, then the residuals
should be evenly scattered around 0 along the range of fitted values
(Fig. (Fig.1111).
(a) Scatter diagram of y against x suggests that the relationship is nonlinear. (b) Plot of residuals against fitted values in panel a; the curvature of the relationship is shown more clearly. (c) Scatter diagram of y against x suggests that the variability ...
In
addition, a Normal plot of residuals can be produced. This is a plot of
the residuals against the values they would be expected to take if they
came from a standard Normal distribution (Normal scores). If the
residuals are Normally distributed, then this plot will show a straight
line. (A standard Normal distribution is a Normal distribution with mean
= 0 and standard deviation = 1.) Normal plots are usually available in
statistical packages.
Figs Figs1212 and and1313
show the residual plots for the A&E data. The plot of fitted values
against residuals suggests that the assumptions of linearity and
constant variance are satisfied. The Normal plot suggests that the
distribution of the residuals is Normal.
When
using a regression equation for prediction, errors in prediction may
not be just random but also be due to inadequacies in the model. In
particular, extrapolating beyond the range of the data is very risky.
A
phenomenon to be aware of that may arise with repeated measurements on
individuals is regression to the mean. For example, if repeat measures
of blood pressure are taken, then patients with higher than average
values on their first reading will tend to have lower readings on their
second measurement. Therefore, the difference between their second and
first measurements will tend to be negative. The converse is true for
patients with lower than average readings on their first measurement,
resulting in an apparent rise in blood pressure. This could lead to
misleading interpretations, for example that there may be an apparent
negative correlation between change in blood pressure and initial blood
pressure.
Conclusion
Both
correlation and simple linear regression can be used to examine the
presence of a linear relationship between two variables providing
certain assumptions about the data are satisfied. The results of the
analysis, however, need to be interpreted with care, particularly when
looking for a causal relationship or when using the regression equation
for prediction. Multiple and logistic regression will be the subject of
future reviews.
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