FARADAY’S LAW
It is observed experimentally that if the magnetic flux through a circuit is changed a
“voltage” is produced around the circuit in such a direction as to oppose the change. The
magnetic flux is defined in the same way as electric flux:
 
M   B  dA
We put “voltage” in quotes since it is quite different than the voltage we have discussed before –
that voltage would be zero around a closed loop since E is a conservative field. This “voltage”
has the same effect on a charge – it gives the work/charge in moving between two points. In this
case, it is moving around a closed loop. Since work is given by:
 

dW  F  dr  qE  dr
we define “voltage” (to which we give the name EMF) as:
 
EMF   E  d 
Thus
EMF  
dM
d  
   B  dA
dt
dt
The first question is whether there is really anything new here. To see why this is a
legitimate question, consider the Lorentz force law:

 
F  qv  B
This means that a charge moving in a magnetic field will experience a force equivalent to that
produced by an electric field:
  
E  vB
Thus it should experience an EMF around the loop of:


  
 
EMFM    v  B   d       v  B   dA (Stokes Theorem)
MOTIONAL EMF
As we will see this EMF will be non-zero in certain circumstances even without making
any changes in the Maxwell equations we have so far considered. To see this we note first that


the EMF will be zero unless at least one of B or v is not constant. Recalling that:
  


     
 
   A  B   B    A  B   A    A     B  A   B
we have

    
     
EMF    B    v  B    v    v    B  v    B   dA


We note that the first two terms are concerned only with variation in v with B constant, while


the last two hold v constant while B varies. We will treat the two sets differently.

Consider first the contribution to this integral due to changes in B . That part of the curl
is:
     
  
  v    B  v    B    v    B
 
(recall that   B  0 ). But





dB B B dx B dy B dz




t x dt y dt z dt
dt





B B
B
B
B     


 v B
vx 
vy 
vz 
z
dt x
dy
dt
Thus:
  
 v   B


is that part of d B /dt arising from the loop’s motion in an inhomogeneous B .

Next consider the contribution due changes in v . These will result in a distortion in the
shape or orientation of the loop. To find their effect we go back to the original form for the EMF:


 
  
 
  v  B   d      B  v   d     B   v  d  

(we can interchange · and ×). But the change in area of the loop due to the shift of d  is:

dA  vd sin  dt ˆ
where ̂ is in the direction of the change in area. We recall that the vector area is perpendicular

to the surface and directed outward. The new chunk of area at d  is the parallelogram defined by


d  and v . The outward direction is out of board (recall the right hand rule), and hence the

 
 
direction of ̂ is that of v  d  and dA   v  d   dt . Hence

dA  
 v  d
dt


gives the change in A at d  . Thus the change in the integral due to distortion of the loop is:



  dA

  v  B   d    B  
 dt
Thus putting the two results together we get:

EMFM 
  dB
  
  dt


d    B 
B    d 

 dA
  dA  B   dA    B  dA  
dt
dt
t 
 dt
Comparing this with Faraday’s Law we have:


 B
d   

EMFF    B  dA  EMFM  
 dA
dt 
 t
Thus the only thing that is new is the term:




B 
 dA
t
To see the change this makes in Maxwell’s equations we note that since it involves only change

in B for a fixed loop we can write:



 
 
 B
 E  d      E   dA   

dA




 t
Then since the loop is arbitrary we must have:

 
B
E  
t
Thus a magnetic field which changes in time produces a non-conservative electric field. This has
profound consequences, as we shall see.
Note that there are now two sources of EMF – a time varying magnetic field, and a loop
which changes shape or orientation. The first is new, the second is not. Ultimately the EMF is
always given by:
EMF  
d M
dt
but this can be calculated in several different ways. For example consider a coil of wire rotating
in a constant magnetic field:
We can calculate the EMF around the loop either by finding M directly or by calculating:

 
  v  B   d 
If B were a function of time and we used the second method we would have to add a term:



Let’s see how this works.

B 
 dA
t
First find M directly. We have:
  
M   B  dS  Byˆ   2   xˆ sin t  yˆ cos t    2B cos t
where
Area   2
EMF   2B  sin t   2
B
cos t
dt
Next calculate:

 
  v  B  d 
around the loop. Here we must be careful. We go around the loop keeping it to our left.



On sides (1) and (3) v is in or out of the paper resulting in v × B in the + or – z direction.
Hence:


 v  B  d   0


On (2) v is into the paper and of magnitude (l/2). Since B is in the ŷ direction we get:
  
vB 
B sin  t   zˆ 
2
on (2) and
  
vB 
B sin  t   zˆ 
2
0n (4). (Recall the diagram)

But d   zˆ on (2) and ẑ on (4). Thus

  
v
  B  d 


  2

B sin  t   2   2 B sin  t 
 
 2

Hence method two is off by the term:
2
B
cos  t 
t

If d B /dt = 0, the methods give the same result.
MAXWELL’S EQUATIONS
We now have the equations:
  
E 
,0

 
E
E  
t
 
B  0
 

  B  0 J
Maxwell recognized that these are inconsistent with charge conservation:
  
J 
0
t
To see this take the divergence of:
 

  
 
  B   0 J       B   0   0  J
But
 

J  
0
t
To fix this he added a term:
0

t
to the right hand side to get:
  
    
     B   0    J  
t 

But from
  
E 
,0
we have

 E 
  

,0   E ,0  
t
t t
Thus

  
 E 
 
     B    0    J  ,0  

t 


 

E
  D 
  B  0 J  0 ,0
 0  J 

t
t 

Hence the final form of the Maxwell equations is:
  
E 
,0

 
B
E  
t
 
B  0

 

E 
  B   0  J  ,0 
t 

In a material medium these become:
 
D  

 
B
E  
t
 
B  0

   D
H  J 
t
 
 
To use these we must have D(E) and H(B) .