PS 250: Lecture 27 Maxwell’s Equations and Electromagnetic

WavesJ. B. Snively

November 13th, 2015

Today’s Class

Maxwell’s EquationsIntro to WavesSummary

Maxwell’s EquationsI

⇥E · d ⇥A =Q

encl

�o

Gauss’s Law:(E Field)

I�B · d �A = 0

Gauss’s Law:(B Field)

I⇥B · d⇥l = µ

o

✓iC

+ �o

d�E

dt

◆

encl

Ampere’s Law:(B Field)

I�E · d�l = �d�B

dt

Faraday’s Law:(E Field)

Together with the Lorentz Force (F=qE+qvxB), these equations describe classical electromagnetic interactions.

With some math, can be expressed in “point form” or “differential form”, which allows convenient calculation of the electromagnetic wave equations.

Maxwell’s Equations

Maxwell’s EquationsGauss’s Law:

(E Field)

Gauss’s Law:(B Field)

Ampere’s Law:(B Field)

Faraday’s Law:(E Field)

r⇥ ⇤B = µo

⇤J + �

o

⇥ ⇤E

⇥t

!

r⇥ ⇥E = �� ⇥B

�t

� · �B = 0

� · ⇤E =⇥

�o

Electromagnetic Radiation (i.e., Radiation of Waves)

Acceleration of Charge

Changing Currentor

Occurs as a result of:

Today’s Class

Maxwell’s EquationsIntro to WavesSummary

Electromagnetic Waves

Transverse – E and B fields are perpendicular to the direction of propagation. Propagates in direction of ExB.

In a vacuum (Free Space), the wave propagates at the speed of light “c”.

Magnitudes of E and B are related by

Electromagnetic Waves Wavelength and Frequency

Propagate at the speed of light:

c =1

p�o

µo

�

k= c

k=wavenumber, where wavelength: � =2⇥

k

and frequency (Hertz) is given by:

We can thus write: � =c

f

f =⇥

2�

Electromagnetic Waves The Spectrum

Electromagnetic Plane Waves

Electromagnetic Waves Sinusoidal Solutions...

E = Emax

cos(kx� �t)

B = Bmax

cos(kx� �t)

Emax

= cBmax

Field amplitudes determined by speed of light:

c =1

p�o

µo

For a plane wave traveling in the x-direction:

Electromagnetic Waves Sinusoidal Solutions...

⇥E =

ˆjEmax

cos(kx� �t)⇥B =

ˆkBmax

cos(kx� �t)

⇥B = �ˆkBmax

cos(kx+ �t)

⇥E =

ˆjEmax

cos(kx+ �t)

Electromagnetic Waves Materials other than “Free Space”

Define wave speed v, where v≤c:

v =1

p�µ

� = K�o

µ = Km

µo

Recall that: ,

v =1

p�µ

=1p

KKm

1p�o

µo

=cp

KKm

Electromagnetic Waves “Index of Refraction”

Can define index of refraction, relating the wave speed v in a material with speed of light c:

n =c

v=

pKKm

Keep in mind, however, that dielectric “constant” K varies with frequency.

(Also note that Km = 1 for many materials.)

Summary / Next Class:

Mastering Physics for Monday.

Homework for next-next Wednesday