ElectromagnetismGiancoli Ch.21-29

Physics of Astronomy, winter week 7

• Review of waves and wave equations• Summary of EM & Maxwell’s eqns• Derive EM wave equation and speed of light!• Magnetic monopole more symmetry

• Today: to American Physical Society meeting!• Next Friday 1:00-2:00 in 1050 Lab I: Science

Carnival poster session

Waves

( , ) sin( )MD x t D kx t

Wave equation

1. Differentiate D/t 2D/t2

2. Differentiate D/x 2D/x2

3. Find the speed from 22

222 2 22 2

2

2

2

Dt T f v

D k Tx

Causes and effects of E

Gauss: E fields diverge from charges

Lorentz force: E fields can move charges

F = q E

0

AdEq

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 I0ldB

Changing fields create new fields!

Faraday: Changing magnetic flux induces circulating electric field

Guess what a changing E field induces?

lE ddt

d B

Changing E field creates B field!

Current piles charge onto capacitor

Magnetic field doesn’t stop

Changing electric flux

“displacement current” magnetic circulation

lB ddt

d E

00AE dE

Partial Maxwell’s equations

Charge E field Current B field

FaradayChanging B E

AmpereChanging E B

0

AdEq

I0ldB

lE ddt

d B lB d

dt

d E

00

Maxwell eqns electromagnetic waves

Consider waves traveling in the x direction with frequency f=

and wavelength = /k

E(x,t)=E0 sin (kx-t) and

B(x,t)=B0 sin (kx-t)

Do these solve Faraday and Ampere’s laws?

Faraday + Ampere

lE ddt

d B

dx

dB

dt

dE00

dx

dE

dt

dB

lB ddt

d E

00

dx

dB

dt

dE00

dx

dE

dt

dB

Sub in: E=E0 sin (kx-t) and B=B0 sin (kx-t)

Speed of Maxwellian waves?

Faraday: -B0 = k E0 Ampere: 00E0=kB0

Eliminate B0/E0 and solve for v=/k

0=xm = xC2 N/m2

Maxwell equations Light

E(x,t)=E0 sin (kx-t) and B(x,t)=B0 sin (kx-t)solve Faraday’s and Ampere’s laws.

Electromagnetic waves in vacuum have speed c and energy/volume = 1/2 0 E2 = B2 /(20 )

Full Maxwell equations

0

AdEq

B dA 0

Bdd

dt

E l

0 0 0Ed

d Idt

B l

Notice the asymmetries – how can we make these symmetric by adding a magnetic monopole?