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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