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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 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 d D/ d t d 2 D/ d t 2 2. Differentiate d D/ d x d 2 D/ d x 2 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 doesn’t stop Changing electric flux ®“displacement current” ® magnetic circulation

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Partial Maxwell’s equations Charge E fieldCurrent 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= and wavelength = /k E(x,t)=E 0 sin (kx- t) and B(x,t)=B 0 sin (kx- t) Do these solve Faraday and Ampere’s laws?

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Faraday + Ampere

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Sub in: E=E 0 sin (kx- w t) and B=B 0 sin (kx- w t)

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Speed of Maxwellian waves? Faraday: w B 0 = k E 0 Ampere: m 0 e 0 w E 0 =kB 0 Eliminate B 0 /E 0 and solve for v= w /k m 0 = 4 p x 10 -7 T m /A e 0 = 8.85 x 10 -12 C 2 N/m 2

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Maxwell equations Light E(x,t)=E 0 sin (kx- w t) and B(x,t)=B 0 sin (kx- w t) solve Faraday’s and Ampere’s laws. Electromagnetic waves in vacuum have speed c and energy/volume = 1/2 e 0 E 2 = B 2 /(2 m 0 )

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Full Maxwell equations in integral 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 Ampere’s Law: apply Curl Thm: and the Definition of current density: to find the Differential form:

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Integral to differential form Faraday’s Law: apply Curl Thm: to find the 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 = v

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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!

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