 # Electromagnetism week 9 Physical Systems, Tuesday 6.Mar. 2007, EJZ Waves and wave equations Electromagnetism & Maxwell’s eqns Derive EM wave equation and.

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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

Waves

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

Causes and effects of E Gauss: E fields diverge from charges Lorentz force: E fields can move charges F = q E

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

Changing fields create new fields! Faraday: Changing magnetic flux induces circulating electric field 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

Partial Maxwell’s equations Charge  E fieldCurrent  B field Faraday Changing B  E Ampere Changing E  B

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?

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

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

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 )

Full Maxwell equations in integral form

Integral to differential form Gauss’ Law: apply Divergence Thm: and the Definition of charge density: to find the Differential form:

Integral to differential form Ampere’s Law: apply Curl Thm: and the Definition of current density: to find the Differential form:

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

Finish integral to differential form…

Maxwell eqns in differential form Notice the asymmetries – how can we make these symmetric by adding a magnetic monopole?

If there were magnetic monopoles… 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!

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