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Special Relativity
For any pair of space-time events measured from two inertial reference frames S and S
′ , where S
′ is moving relative
to S with constant velocity v in the positive x direction:
γ ≡
v
2
c
2
∆x = γ(∆x
′
′
) ∆x
′
= γ(∆x − v∆t)
∆y = ∆y
′ ∆y
′ = ∆y
∆z = ∆z
′
∆z
′
= ∆z
∆t = γ(∆t
′
v
c
2
∆x
′
) ∆t
′
= γ(∆t −
v
c
2
∆x)
If ∆t
′ is the proper time between two events, then the time interval measured from another frame will be
∆t = γ∆t
′
If ∆x
′ is the proper length of an object, then the length measured from another frame will be
∆x =
∆x
′
γ
For a particle moving with velocity ~u relative to S (and velocity ~u
′ relative to S
′ ),
ux =
u
′
x
v
c
2
u
′
x
u
′
x
u x
− v
v
c
2
u x
u y
u
′
y
γ(1 +
v
c
2 u
′
x
u
′
y
uy
γ(1 −
v
c
2 ux)
u z
u
′
z
γ(1 +
v
c
2
u
′
x
u
′
z
u z
γ(1 −
v
c
2
u x
Electrostatics
F = q
E (electric force on a particle with charge q)
The electric field at point P due to a small element of charge dq is
d
E =
4 π 0
dq
r
2
ˆr
where ~r (= rrˆ) is the displacement vector that points from dq to P.
For any closed surface S, ∮
S
E · d
A =
Q
enclosed by S
(Gauss’s Law)
Known Electric Field Strengths
E =
Q
4 π 0 r
2
(point charge Q)
E =
λ
2 π 0 r
(infinite line of uniform charge density λ C/m)
E =
σ
(infinite sheet of uniform charge density σ C/m
2 )
Electric Potential
U = qV (electrostatic potential energy of a particle with charge q)
V
b
− V
a
~r b
~ra
E · d
l (finding electric potential from electric field)
E = −
∇V (finding electric field from electric potential)
The electrostatic potential at point P due to a small element of charge dq, relative to V (r = ∞) = 0, is
dV =
4 π 0
dq
r
where r is the distance from dq to P.
Capacitance
Q = CV (definition of capacitance)
C =
0
A
d
(for a vacuum- or air-filled parallel-plate capacitor with area A and separation d)
U =
CV
2
(energy stored in a capacitor)
Electric Field Energy
Electric fields store energy with a density (J/m
3 ) of
u =
0
E
2
Dielectrics
E =
E
0
K
(electric field inside a dielectric material placed in an external field E 0
C =
A
d
= K 0
A
d
(for a parallel-plate capacitor filled with a dielectric material)
Mutual Inductance
M =
B thru 2 due 1
I
1
B thru 1 due 2
I
2
E
1
= −M
dI 2
dt
E
2
= −M
dI 1
dt
Self Inductance
L =
ΦB
I
E = −L
dI
dt
U =
LI
2 (energy stored in an inductor)
Magnetic Field Energy
Magnetic fields store energy with a density (J/m
3 ) of
u =
B
2
μ 0
Circuit Elements
Type Relationship Series Combination Parallel Combination
Resistor V = IR Req = R 1 + R 2
1
Req
1
R 1
1
R 2
Capacitor I = C
dV
dt
1
Ceq
1
C 1
1
C 2
Ceq = C 1 + C 2
Inductor V = L
dI
dt
L
eq
= L
1
+ L
2
1
Leq
1
L 1
1
L 2
Electric Power
P = V I (power consumed by any electrical device)
P = I
2 R (power dissipated as heat in a resistor)
U =
CV
2
(energy stored in a capacitor)
U =
LI
2
(energy stored in an inductor)
Maxwell’s Equations
Integral Form
E · d
A =
Q
E · d
l =
−dΦ B
dt
∮
B · d
A = 0
B · d
l = μ 0 I + μ 0 0
dΦ E
dt
Differential Form
E =
ρ
∇ ×
E =
B
∂t
B = 0
∇ ×
B = μ 0
J + μ 0
0
E
∂t
Electromagnetic Waves
f (x, t) = f (x − vt, 0) = g(x − vt) (general form of a wave moving in the +x direction)
2 f
∂x
2
v
2
2 f
∂t
2
(1-D wave equation to which f (x, t) is a solution)
f (x, t) = F 0
sin(kx − ωt) = F 0
sin k(x − vt) (a sinusoidal wave moving in the +x direction)
v =
ω
k
(speed of propagation)
λ =
2 π
k
(wavelength)
T =
2 π
ω
(period)
λf = v (relation between speed, wavelength, and frequency)
c =
μ 0
0
= 3 × 10
8 m/s (speed of EM waves in a vacuum)
B =
E
c
(relation between magnitudes of E and B in an EM plane wave)
Energy in EM Waves
The power density (energy transported per unit time per unit area) of an EM wave is given by
S =
μ 0
E ×
B (Poynting vector)
Intensity is the time-averaged power density
S
avg
E
0
B
0
2 μ 0
(intensity of an EM plane wave)
Math Corner
Solutions to Common Differential Equations
Decaying Exponential
The differential equation
τ
df (t)
dt
has solutions of the form
f (t) = F 0
−t/τ
where:
τ is called the time constant
A is an arbitrary constant that depends on the initial conditions
Geometry
A sphere of radius R has volume
4
3
πR
3 and surface area 4πR
2 .
A cylinder of radius R and height h has volume πR
2 h and surface area 2πRh + 2πR
2 (the first term is the area
around the side, the second term is the area of the top and bottom).
Trigonometry
sin
2 θ + cos
2 θ = 1 sin
2 θ =
1
2
1
2
cos(2θ) sin 45
◦ = cos 45
◦
1 √
2
sin(θ + φ) = sin θ cos φ + cos θ sin φ cos
2 θ =
1
2
1
2
cos(2θ) sin 30
◦ = cos 60
◦
1
2
cos(θ + φ) = cos θ cos φ − sin θ sin φ sin(2θ) = 2 sin θ cos θ sin 60
◦ = cos 30
◦
√
3
2
Quadratic Formula
If ax
2
−b ±
b
2 − 4 ac
2 a
