Massachusetts Institute of technology
Department of Physics
8.022 Fall 2004/11/09
Quiz #2: Formula sheet
Work on the problems you know how to solve first!
Exam is closed book and closed notes. Useful math formulae are provided below. No calculators will be needed. a
Potential: φ (a ) − φ (b) = − ∫ E ⋅ ds
b
1
1
ρφ dV =
E 2 dV .
∫
2 V
8π ∫
Pressure: A layer of surface charge density σ exerts a pressure P = 2πσ 2 .
Energy of E: The energy of an electrostatic configuration U =
Current density: J = ρ v .
Current: I = dQ / dt = ∫ J ⋅ da ( I is the current through surface S ).
S
∂ρ
∂t
Ohm’s law: J = σ cE (microscopic form): V=IR (macroscopic form)
Q2 1
Capacitance: Q=CV. Energy stored in capacitor: UC =
= CV 2
2C 2
v
Lorentz force: F = qE + q × B
c
I
I
Magnetic force on current: F = dl × B ; or F / L = × B
c
c
Continuity: ∇ ⋅ J = −
Vector potential: B = ∇ × A ; A =
I
c
∫
dl
r
Biot-Savart law: dB = Idl × r̂ /(cr 2 )
Maxwell’s equations in differential form (so far!!!): ∇ ⋅ E = 4πρ (Gauss's law)
∇⋅B = 0
∇× E = −
∇× B =
1 ∂B
(Faraday's law)
c ∂t
4π
J (Ampere's law)
c
Maxwell’s equations in integral form (so far):
∫ E ⋅ da = 4π Q (Gauss's law. Q is charge enclosed by surface S)
S
∫
C
E ⋅ ds = −
1 ∂φ B
=e.m.f. (Faraday's law. φ B is B flux through surface bounded by C.)
c ∂t
4π
I (Ampere's law. I is current enclosed by contour C.)
C
c
Self inductance: ε = −LdI / dt
Mutual Inductance: ε 1 = −M 12 dI 2 / dt ; ε 2 = −M 21dI 1 / dt ; M 12 = M 21
1
B 2 dV
Magnetic energy: U =
8π ∫
1
Energy stored in an inductor: UL = LI 2
2
∫
B ⋅ ds =
Time dilation: Moving clocks run slow: ∆tstationary = γ∆tmoving
Length contraction: Moving rulers are shortened: Lstationary = Lmoving / γ
Transformation of fields: denotes parallel to v , ⊥ denotes perpendicular to v
v
E' = E
E⊥' = γ (E⊥ + × B⊥ )
c
v
B' = B
B⊥' = γ (B⊥ − × E⊥ )
c
Useful Math
Cartesian.
∂t
∂t
∂t
x̂ +
ŷ +
ẑ
∂x
∂y
∂z
∂ v ∂ v y ∂ vz
Divergence: ∇ ⋅ v = x +
+
∂x ∂y ∂z
∂ v y ∂ vx
∂v ∂v
∂ v ∂ vy
) x̂ + ( x − z ) ŷ + (
−
)ẑ
Curl: ∇ × v = ( z −
∂y ∂z
∂z ∂x
∂x ∂y
Gradient: ∇t =
Laplacian: ∇ 2 t =
∂ 2t ∂ 2t ∂ 2 t
+
+
∂ x2 ∂ y2 ∂ z2
Spherical.
∂t
1 ∂t
1 ∂t
r̂ +
θ̂ +
φ̂
∂r
r ∂θ
r sin θ ∂ φ
∂
1 ∂
1
1 ∂ vφ
( r 2 v r )+
Divergence: ∇ ⋅ v = 2
(sin θvθ )+
r sin θ ∂ θ
r sin θ ∂ φ
r ∂r
Gradient: ∇t =
Curl:
∇×v =
∂ (sin θvφ ) ∂ vθ
1
1 1 ∂ v r ∂ (rvφ )
1 ∂ (rvθ ) ∂ v r
−
−
−
]θ̂ + [
]φ̂
[
]r̂ + [
∂θ
∂φ
∂r
∂θ
r ∂r
r sin θ
r sin θ ∂ φ
∂t
∂ 2t
1 ∂ 2 ∂t
1
1
∂
(r
)+
(sin θ
)+
Laplacian: ∇ t = 2
∂ r r 2 sin θ ∂θ
∂ θ r 2 sin 2 θ ∂ φ 2
r ∂r
2
Cylindrical.
∂t
∂t
1 ∂t
ρ̂ +
φ̂ +
ẑ
∂ρ
ρ ∂φ
∂z
1 ∂
1 ∂ vφ ∂ v z
Divergence: ∇ ⋅ v =
( ρv ρ ) +
+
ρ ∂ρ
ρ ∂φ
∂z
∂v ∂v
1 ∂ vz ∂ vφ
1 ∂ ( ρ vφ ) ∂ vρ
Curl: ∇ × v = [
−
−
]ρ̂ + [ ρ − z ]φ̂ + [
]ẑ
ρ ∂φ ∂ z
∂z ∂ρ
ρ ∂ρ
∂θ
∂t
1 ∂ 2t ∂ 2 t
1 ∂
Laplacian: ∇ 2 t =
(ρ
)+ 2
+
ρ ∂ρ ∂ ρ ρ ∂φ 2 ∂ z 2
Gradient: ∇t =
Binomial expansion:
(1 ± x )n = 1 ±
nx
1!
Stokes’ theorem:
Gauss’ theorem:
+
n (n − 1)x 2
2!
±
(x 2 < 1);(1 ± x ) −n = 1 ∓
∫ F ⋅ ds = ∫ curl F ⋅ dA
∫ F ⋅ dA = ∫ div FdV
C
S
S
V
nx
1!
+
n (n + 1)x 2
2!
∓
(x 2 < 1)