1 / 22

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

Waves

Wave equation

1. Differentiate dD/dt ? d2D/dt2 2. Differentiate

dD/dx ? d2D/dx2 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 doesnt stop
- Changing electric flux
- displacement current
- magnetic circulation

Partial Maxwells equations

Charge ? E field Current ? B field

Faraday Changing B ? E Ampere Changing E ? B

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?

Faraday Ampere

Sub in EE0 sin (kx-wt) and BB0 sin (kx-wt)

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

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 )

Full Maxwell equations inintegral 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

Amperes Law

apply Curl Thm

and the Definition

of current density

to find the Differential form

Integral to differential form

Faradays Law

apply Curl Thm

to find

the Differential form

Finish integral to 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 rv

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!