Waves
Electromagnetism week 9
Physical Systems, Tuesday 6.Mar. 2007, EJZ
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Waves and wave equations
Electromagnetism & Maxwell’s eqns
Derive EM wave equation
and speed of light
λ
Derive Max eqns in differential form
Magnetic monopole Ø more symmetry
• Next quarter
D( x, t ) = DM sin(kx − ωt )
Wave equation
Causes and effects of E
1. Differentiate dD/dt → d2D/dt2
2. Differentiate dD/dx →
Gauss: E fields diverge
from charges
d2D/dx2
2
3. Find the speed from ∂ D ∂t 2 ω 2 ⎛ 2π T ⎞ ⎛ λ ⎞ 2
⎟ = ⎜ ⎟ = (λ f
= 2 =⎜
2
2π
Lorentz force: E fields
can move charges
2
∂ D
∂x 2
k
⎜
⎝
⎟
λ⎠
⎝T ⎠
)
2
= v2
v∫ E ⋅ dA =
q
ε0
F=qE
Causes and effects of B
Ampere: B fields curl
around currents
Changing fields create new fields!
Lorentz force: B fields can
bend moving charges
Faraday: Changing
magnetic flux induces
circulating electric field
dΦ B
= − ∫ E ⋅ dl
dt
v∫ B ⋅ dl = μ I
0
F = q v x B = IL x B
Guess what a changing E field induces?
Partial Maxwell’s equations
Changing E field creates B field!
Current piles charge onto
capacitor
Magnetic field doesn’t stop
dΦ E
μ 0ε 0
= ∫ B ⋅dl
dt
Charge Ø E field
G G q
E
v∫ ⋅ dA =
ε0
Current Ø B field
G G
B
v∫ ⋅ d l = μ0 I
Changing electric flux
Faraday
Changing B Ø E
Ampere
Changing E Ø B
Ø “displacement current”
Ø magnetic circulation
dΦ B
= − ∫ E ⋅ dl
dt
μ 0ε 0
Φ E = ∫ E ⋅ dA
dΦ E
= ∫ B ⋅dl
dt
Faraday + Ampere
Maxwell eqns Ø electromagnetic waves
Consider waves traveling in the x direction
with frequency f= ω/2π
and wavelength λ= 2π/k
E(x,t)=E0 sin (kx-ωt) and
B(x,t)=B0 sin (kx-ωt)
Do these solve Faraday and Ampere’s laws?
dB
dE
=−
dt
dx
μ 0ε 0
dE
dB
=−
dt
dx
Sub in: E=E0 sin (kx-wt) and B=B0 sin (kx-wt)
dΦ B
= − ∫ E ⋅ dl
dt
dB
dE
=−
dt
dx
μ 0ε 0
dΦ E
= ∫ B ⋅dl
dt
μ 0ε 0
dE
dB
=−
dt
dx
Speed of Maxwellian waves?
Faraday: wB0 = k E0
Ampere: m0e0wE0=kB0
Eliminate B0/E0 and solve for v=w/k
m0 = 4 p x 10-7 Tm/A
e0 = 8.85 x 10-12 C2 N/m2
Maxwell equations Ø Light
Full Maxwell equations in
integral form
G G q
v∫ E ⋅ dA =
ε0
G
v∫ B ⋅ dA = 0
E(x,t)=E0 sin (kx-wt) and B(x,t)=B0 sin (kx-wt)
solve Faraday’s and Ampere’s laws.
v∫ E ⋅dl = −
dΦB
dt
v∫ B ⋅dl = μ0ε 0
dΦE
+ μ0 I
dt
Electromagnetic waves in vacuum have speed c
and energy/volume = 1/2 e0 E2 = B2 /(2m0 )
Integral to differential form
G
G
Integral to differential form
q
Gauss’ Law:
v∫ E ⋅ dA = ε
Divergence Thm:
G G
G
v∫ v ⋅ dA = ∫ ( ∇ ⋅ v ) dτ
Ampere’s Law:
apply
0
Differential form:
q = ∫ ρ dτ = ∫
dq
dτ
dτ
G
apply
0
Curl Thm:
and the
Definition of current density:
Definition of charge density:
G
v∫ B ⋅ d l = μ I
to find the
Differential form:
G
G G
G
v
⋅
d
l
=
∇
×
v
⋅
dA
(
)
v∫
∫
I = ∫ J dA = ∫
dI
dA
dA
and the
to find the
Integral to differential form
Faraday’s Law:
Curl Thm:
G
dΦ
d
v∫ E ⋅dl = − dt B = − dt ∫ B ⋅ dA
G
G G
G
v
d
l
v
dA
⋅
=
∇
×
⋅
(
)
v∫
∫
Finish integral to differential form…
apply
to find the
G G q
E
v∫ ⋅ dA =
ε0
G ρ
∇ ⋅ E=
ε0
G
v∫ B ⋅ dA = 0
Differential form:
Maxwell eqns in differential form
ρ
∇ ⋅ E=
ε0
∇⋅B = 0
dΦB
dt
dB
∇×E = −
dt
v∫ E ⋅dl = −
v∫ B ⋅dl = μ ε
0 0
dΦE
+ μ0 I
dt
Next quarter:
ElectroDYNAMICS, quantitatively, including
∇×E = −
dB
dt
Ohm’s law, Faraday’s law and induction, Maxwell equations
Conservation laws, Energy and momentum
Electromagnetic waves
dE
∇ × B = μ 0ε 0
+ μ0 J
dt
Notice the asymmetries – how can we make these symmetric
by adding a magnetic monopole?
Potentials and fields
Electrodynamics and relativity, field tensors
Magnetism is a relativistic consequence of the Lorentz
invariance of charge!