Electromagnetism and Energy

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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

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Waves
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Wave equation
1. Differentiate dD/dt ? d2D/dt2 2. Differentiate
dD/dx ? d2D/dx2 3. Find the speed from
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Causes and effects of E

• Gauss E fields diverge from charges

• Lorentz force E fields can move charges

F q E
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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
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Changing fields create new fields!

• Faraday Changing magnetic flux induces
  circulating electric field

Guess what a changing E field induces?
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Changing E field creates B field!

• Current piles charge onto capacitor
• Magnetic field doesnt stop
• Changing electric flux
• displacement current
• magnetic circulation

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Partial Maxwells equations
Charge ? E field Current ? B field
Faraday Changing B ? E Ampere Changing E ? B
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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?

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Faraday Ampere
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Sub in EE0 sin (kx-wt) and BB0 sin (kx-wt)
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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
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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 )
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Full Maxwell equations inintegral form
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Integral to differential form
Gauss Law
apply Divergence Thm
and
the Definition of charge density
to find
the Differential form
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Integral to differential form
Amperes Law
apply Curl Thm
and the Definition
of current density
to find the Differential form
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Integral to differential form
Faradays Law
apply Curl Thm
to find
the Differential form
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Finish integral to differential form
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Finish integral to differential form
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Maxwell eqns in differential form
Notice the asymmetries how can we make these
symmetric by adding a magnetic monopole?
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If there were magnetic monopoles
where J rv
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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!