# Electromagnetism week 9

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16 Νοε 2013 (πριν από 4 χρόνια και 6 μήνες)

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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

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

 
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

2
2
2
2
2
2
2
2
2
2
2
2
D
t
T
f v
D
k T
x

 

 
 

 
    
 

 
 
 

Causes and effects of E

Gauss: E fields diverge
from charges

Lorentz force: E fields
can move charges

0
d
q

 

E A
F

= q
E

Causes and effects of B

Ampere: B fields curl
around currents

Lorentz force: B fields can
bend moving charges

0
dl

 

B I
F

= q
v x B = IL x B

Changing fields create new fields!

magnetic flux induces
circulating electric field

l
E
d
dt
d
B

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

l
B
d
dt
d
E

0
0

A
E
d
E

Partial Maxwell’s equations

Charge

E field

Current

B field

0
E dA
q

 

0
B dl
I

 

l
E
d
dt
d
B

l
B
d
dt
d
E

0
0

Changing B

E

Ampere

Changing E

B

Maxwell eqns

electromagnetic waves

Consider waves traveling in the x direction
with frequency f=
/2

and wavelength

=
2
/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?

l
E
d
dt
d
B

dx
dB
dt
dE

0
0

dx
dE
dt
dB

l
B
d
dt
d
E

0
0

dx
dB
dt
dE

0
0

dx
dE
dt
dB

Sub in: E=E
0

sin (kx
-
w
t) and B=B
0

sin (kx
-
w
t)

Speed of Maxwellian waves?

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)

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

0
E dA
q

 

B dA 0
 

B
d
d
dt

  

E l
0 0 0
E
d
d I
dt
 

  

B l
Integral to differential form

0
E dA
v dA v
q
d
dq
q d d
d

 

 
  
 

 
 
Gauss’ Law
:

apply

Divergence Thm:

and the

Definition of charge density:
to
find the

Differential form
:

Integral to differential form

0
B dl
v dl v dA
I
dI
I J dA dA
dA

 
   
 

 
 
Ampere’s Law
:

apply

Curl Thm:

and the

Definition of current density:
to
find the

Differential form
:

Integral to differential form

dA
v dl v dA
B
d
d
d
dt dt

    
   
 
 
E l B
:

apply

Curl Thm:

to
find the

Differential form
:

Finish integral to differential form

0
0
E dA
E=
q

 


B dA 0
 

B
d
d
dt
d
dt

  
  

E l
B
E
0 0 0
E
d
d I
dt
 

  

B l
Finish integral to differential form…

0
0
d
=
q

 


E A
E
d 0
0
 
 

B A
B
B
d
d
dt
d
dt

  
  

E l
B
E
0 0 0
0 0 0
E
d
d
dt
d
dt
 
 

  
  

B l I
E
B J
Maxwell eqns in differential form

0
=


E
0
 
B
d
dt
  
B
E
0 0 0
d
dt
 
  
E
B J
Notice the asymmetries

how can we make these symmetric
?

If there were magnetic monopoles…

0
=
e


E
0
m

 
B
0
m
d
dt

   
B
E J
0 0 0
d
dt
 
  
E
B J
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!