Transcript Lecture25

Maxwell’s Equations
If we combine all the laws we know about electromagnetism, then
we obtain Maxwell’s equations.
These four equations plus a force law form the basis for
all of electromagnetism!
Thesbe laws predict that accelerating charges will radiate
electromagnetic waves!
The fact that classical models of the atom contradicted Maxwell’s
equations motivated quantum mechanics.
Maxwell’s Equations
Integral Form
qin
 E  dA  
S
 B  dA  0
S
0
dE
 B  ds  0 I  0 0 dt
dB
 E  ds   dt
Gauss’s laws, Ampere’s law and Faraday’s law all combined!
They are nearly symmetric with respect to magnetism and electricity.
The lack of magnetic monopoles is the main reason
why they are not completely symmetric.
Maxwell’s Equations
Integral Form
qin
 E  dA  
S
 B  dA  0
S
0
dE
 B  ds  0 I  0 0 dt
dB
 E  ds   dt
Differential Form

E 
0
 B  0
dE
  B  0 J  0 0
dt
dB
 E  
dt
Maxwell and Lorentz Force Law
Differential Form

E 
0
 B  0
dE
  B  0 J  0 0
dt
dB
 E  
dt
~
~ ~ ~
F  qE  qv  B
FYI, These are connected to the integral equations via
the generalized stokes equation
Derivatives and Partial Derivatives
•When you have multiple variables, and you need to take
the derivative, you use a partial derivative
•Partial derivatives are like ordinary derivatives, but all
other variables are treated as constants
•{We have done this before; remember the gradient}
f  sin  kx  t 
f
 k cos  kx   t 
x
Vector Derivatives: Dot products in
Cartesian Coordinates



  xˆ  yˆ  zˆ
x
y
z
 B
Nambla: a vector derivative
is the divergence of B.
 

 
  B   xˆ 
yˆ  zˆ    Bx xˆ  By yˆ  Bz zˆ 
y
z 
 x
 Bx By Bz 



0
y
z 
 x
Vector Derivatives: Cross products
in Cartesian Coordinates
 B



  xˆ  yˆ  zˆ
x
y
z
Nambla: a vector derivative
is the curl of B.
 

 
  B   xˆ  yˆ  zˆ    Bx xˆ  By yˆ  Bz zˆ 
y
z 
 x
 Bz By   Bz Bz 
 By Bx 




 xˆ  
 zˆ
 yˆ  
z   z
x 
y 
 y
 x
dE
 0 J  0 0
dt
Vector Derivatives: In other
coordinates



needs to be converted
  xˆ  yˆ  zˆ Nambla
if we change coordinates
x
y
z

1  ˆ
1
 ˆ
  rˆ 

 Spherical:
r
r 
r sin  

1  ˆ
1
 ˆ
  B  rˆ 

  Br rˆ  Bˆ  Bˆ
r
r 
r sin  

Br 1 B
1 B ˆ



 0
r r  r sin  

Two of Maxwell’s Equations
dE
  B  0 J  0 0
dt
dB
 E  
dt



  xˆ  yˆ  zˆ
x
y
z
Nambla: a vector derivative
 

 
  B   xˆ  yˆ  zˆ    Bx xˆ  By yˆ  Bz zˆ 
y
z 
 x
 Bz By   Bx Bz 
 By Bx 




 xˆ  
 zˆ
 yˆ  
z   z
x 
y 
 y
 x
Two of Maxwell’s Equations
dE
  B  0 J  0 0
dt
dB
 E  
dt



  xˆ  yˆ  zˆ
x
y
z
Nambla: a vector derivative
 

 
 E   xˆ  yˆ  zˆ    Ex xˆ  E y yˆ  Ez zˆ 
y
z 
 x
 Ez E y   Ex Ez 
 E y Ex 




 xˆ  
 zˆ
 yˆ  
z   z
x 
y 
 y
 x
Waves from Electromagnetism
•Consider electric fields (pointing in the y-direction) that
depend only on x and t
•Consider magnetic fields (pointing in the z-direction) that
depend only on x and t
•Consider vacuum , aka free space, so J=0
Plane waves
(We could be more general)
Using Maxwell’s Equations
E y
Ex
Bz


x
y
t
By
By
Ex Ez


z
x
t
E y
Bx Bz

 0 0
z
x
t
Bx
Ez E y


y
z
t
Ex
Bz By

 0 0
y
z
t
Bx
Ez

 0 0
x
y
t
Electromagnetic Waves
E y
Bz

x
t
E y
Bz

 0 0
x
t
•These equations look like sin functions will solve them.
Ey  E0 cos  kx  t 
Bz  B0 cos  kx  t 
 kE0 sin  kx   t    B0 sin  kx   t 
kB0 sin  kx   t   0 0 E0 sin  kx   t 
kE0   B0
kB0  0 0 E0