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Electromagnetism and Energy

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Electromagnetism week 9. Physical Systems, Tuesday 6.Mar. 2007, EJZ. Waves and wave equations. Electromagnetism & Maxwell's eqns. Derive EM wave equation and speed ... – PowerPoint PPT presentation

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Title: Electromagnetism and Energy


1
Electromagnetism week 9 Physical Systems, Tuesday
6.Mar. 2007, EJZ
  • Waves and wave equations
  • Electromagnetism Maxwells eqns
  • Derive EM wave equation and speed of light
  • Derive Max eqns in differential form
  • Magnetic monopole ? more symmetry
  • Next quarter

2
Waves
3
Wave equation
1. Differentiate dD/dt ? d2D/dt2 2. Differentiate
dD/dx ? d2D/dx2 3. Find the speed from
4
Causes and effects of E
  • Gauss E fields diverge from charges
  • Lorentz force E fields can move charges

F q E
5
Causes and effects of B
  • Ampere B fields curl around currents
  • Lorentz force B fields can bend moving charges

F q v x B IL x B
6
Changing fields create new fields!
  • Faraday Changing magnetic flux induces
    circulating electric field

Guess what a changing E field induces?
7
Changing E field creates B field!
  • Current piles charge onto capacitor
  • Magnetic field doesnt stop
  • Changing electric flux
  • displacement current
  • magnetic circulation

8
Partial Maxwells equations
Charge ? E field Current ? B field
Faraday Changing B ? E Ampere Changing E ? B
9
Maxwell eqns ? electromagnetic waves
  • Consider waves traveling in the x direction with
    frequency f w/2p
  • and wavelength l 2p/k
  • E(x,t)E0 sin (kx-wt) and
  • B(x,t)B0 sin (kx-wt)
  • Do these solve Faraday and Amperes laws?

10
Faraday Ampere
11
Sub in EE0 sin (kx-wt) and BB0 sin (kx-wt)
12
Speed of Maxwellian waves?
Faraday wB0 k E0 Ampere
m0e0wE0kB0 Eliminate B0/E0 and solve for vw/k
e0 8.85 x 10-12 C2 N/m2
m0 4 p x 10-7 Tm/A
13
Maxwell equations ? Light
E(x,t)E0 sin (kx-wt) and B(x,t)B0 sin
(kx-wt) solve Faradays and Amperes
laws. Electromagnetic waves in vacuum have speed
c and energy/volume 1/2 e0 E2 B2 /(2m0 )
14
Full Maxwell equations in integral form
15
Integral to differential form
Gauss Law
apply Divergence Thm
and
the Definition of charge density
to find
the Differential form
16
Integral to differential form
Amperes Law
apply Curl Thm
and the Definition
of current density
to find the Differential form
17
Integral to differential form
Faradays Law
apply Curl Thm
to find
the Differential form
18
Finish integral to differential form
19
Finish integral to differential form
20
Maxwell eqns in differential form
Notice the asymmetries how can we make these
symmetric by adding a magnetic monopole?
21
If there were magnetic monopoles
where J rv
22
Next quarter
ElectroDYNAMICS, quantitatively, including Ohms
law, Faradays 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!
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