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Electromagnetism week 9Physical Systems, Tuesday 6.Mar. 2007, EJZ
• 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
Wave equation
1. Differentiate D/t 2D/t2
2. Differentiate D/x 2D/x2
3. Find the speed from 22
222 2 22 2
2
2
2
Dt T f v
D k Tx
Causes and effects of E
Gauss: E fields diverge from charges
Lorentz force: E fields can move charges
0
dq
E A F = q E
Causes and effects of B
Ampere: B fields curl around currents
Lorentz force: B fields can bend moving charges
0dl B I F = q v x B = IL x B
Changing fields create new fields!
Faraday: Changing magnetic flux induces circulating electric field
lE ddt
d B
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
“displacement current” magnetic circulation
lB ddt
d E
00AE dE
Partial Maxwell’s equations
Charge E field Current B field
0
E dAq
0B dl I
lE ddt
d B lB d
dt
d E
00
FaradayChanging B E
AmpereChanging E B
Maxwell eqns electromagnetic waves
Consider waves traveling in the x direction with frequency f=
and wavelength = /k
E(x,t)=E0 sin (kx-t) and
B(x,t)=B0 sin (kx-t)
Do these solve Faraday and Ampere’s laws?
Speed of Maxwellian waves?
Faraday: B0 = k E0 Ampere: 00E0=kB0
Eliminate B0/E0 and solve for v=/k
0=xm = xC2 N/m2
Maxwell equations Light
E(x,t)=E0 sin (kx-t) and B(x,t)=B0 sin (kx-t)solve Faraday’s and Ampere’s laws.
Electromagnetic waves in vacuum have speed c and energy/volume = 1/2 0 E2 = B2 /(20 )
Integral to differential form
0
E dA
v dA v
q
d
dqq d d
d
Gauss’ Law: apply
Divergence Thm: and the
Definition of charge density: to find the
Differential form:
Integral to differential form
0B dl
v dl v dA
I
dII J dA dA
dA
Ampere’s Law: apply
Curl Thm: and the
Definition of current density: to find the
Differential form:
Integral to differential form
dA
v dl v dA
Bd dd
dt dt
E l B
Faraday’s Law: apply
Curl Thm: to find the
Differential form:
Finish integral to differential form…
0
0
d
=
q
E A
E
d 0
0
B A
B
Bdd
dtd
dt
E l
BE
0 0 0
0 0 0
Edd
dtd
dt
B l I
EB J
Maxwell eqns in differential form
0
=
E
0 B
d
dt
BE
0 0 0
d
dt
EB J
Notice the asymmetries – how can we make these symmetric by adding a magnetic monopole?
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!