Transcript Electromagnetism week 9

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

Derive Max eqns in differential form
Magnetic monopole  more symmetry
• Next quarter
Waves
D( x, t )  DM sin(kx  t )
Wave equation
1. Differentiate dD/dt  d2D/dt2
2. Differentiate dD/dx  d2D/dx2
2

D
3. Find the speed from
2
 2 

t 
T        f 2  v 2


 2 D 2 k 2  2   T 
 
x
2
2
2
Causes and effects of E
Gauss: E fields diverge
from charges
Lorentz force: E fields
can move charges
q
 E  dA  
F=qE
0
Causes and effects of B
Ampere: B fields curl
around currents
 B  dl   I
0
Lorentz force: B fields can
bend moving charges
F = q v x B = IL x B
Changing fields create new fields!
Faraday: Changing
magnetic flux induces
circulating electric field
d B
   E  dl
dt
Guess what a changing E field induces?
Changing E field creates B field!
Current piles charge onto
capacitor
Magnetic field doesn’t stop
Changing electric flux
d E
 0 0
  B dl
dt
 E   E  dA
 “displacement current”
 magnetic circulation
Partial Maxwell’s equations
Charge  E field
 E  dA 
q
0
Current  B field
 Bdl   I
0
Faraday
Changing B  E
Ampere
Changing E  B
d B
   E  dl
dt
d E
 0 0
  B dl
dt
Maxwell eqns  electromagnetic waves
Consider waves traveling in the x direction
with frequency f= /2
and wavelength = 2/k
E(x,t)=E0 sin (kx-t) and
B(x,t)=B0 sin (kx-t)
Do these solve Faraday and Ampere’s laws?
Faraday + Ampere
d B
   E  dl
dt
dB
dE

dt
dx
d E
 0 0
  B dl
dt
dE
dB
 0 0

dt
dx
dB
dE

dt
dx
dE
dB
 0 0

dt
dx
Sub in: E=E0 sin (kx-wt) and B=B0 sin (kx-wt)
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  Light
E(x,t)=E0 sin (kx-wt) and B(x,t)=B0 sin (kx-wt)
solve Faraday’s and Ampere’s laws.
Electromagnetic waves in vacuum have speed c
and energy/volume = 1/2 e0 E2 = B2 /(2m0 )
Full Maxwell equations in
integral form
q
 E  dA  
0
 B  dA  0
dB
 E dl   dt
dE
 B dl  0 0 dt  0 I
Integral to differential form
q
Gauss’ Law:
 E  dA  
Divergence Thm:
 v  dA      v  d
Definition of charge density:
to find the
dq
q    d  
d
d
Differential form:
0
apply
and the
Integral to differential form
Ampere’s Law:
 Bdl   I
apply
0
Curl Thm:
Definition of current density:
Differential form:
 v  d l      v   dA
dI
I   J dA  
dA
dA
and the
to find the
Integral to differential form
Faraday’s Law:
Curl Thm:
Differential form:
dB
d
 E dl   dt   dt  B  dA
 v  d l      v   dA
apply
to find the
Finish integral to differential form…
 E  dA 
q
0

  E=
0
 B  dA  0
dB
 E dl   dt
dB
E  
dt
dE
 B dl  0 0 dt  0 I
Finish integral to differential form…
 E  dA 
q
0

  E=
0
 Bd A 0
B  0
dB
 E dl   dt
dB
E  
dt
dE
 B dl  0 0 dt  0I
dE
  B   0 0
 0 J
dt
Maxwell eqns in differential form

  E=
0
 B  0
dB
E  
dt
dE
  B   0 0
 0 J
dt
Notice the asymmetries – how can we make these symmetric
by adding a magnetic monopole?
If there were magnetic monopoles…
e
  E=
0
dB
  E   0 J m 
dt
  B  0  m
dE
  B   0 0
 0 J
dt
where J = v
Next quarter:
ElectroDYNAMICS, quantitatively, including
Ohm’s law, Faraday’s law and induction, Maxwell equations
Conservation laws, Energy and momentum
Electromagnetic waves
Potentials and fields
Electrodynamics and relativity, field tensors
Magnetism is a relativistic consequence of the Lorentz
invariance of charge!