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Physics PHYS 354
Electricity and Magnetism II
Problem Set #7
1.
Coulomb and Lorentz Gauges
a)
Show that Maxwell’s (time-dependent form) equations imply


  

0 A
,
B    A and E   
4 t
where  is a scalar potential.
b)
Show that the gauge transformations

 

,
A  A   and    
t
where  is a scalar function, do not affect the gauge-independent values of


E or B .
c)
d)

Suppose that A and  do not satisfy the Coulomb gauge condition. Show
that the Coulomb gauge can be reached by selecting a suitable , which is
 
the solution to the differential equation 2    A .

Suppose that A and  do not satisfy the Lorentz gauge condition, which is
  1 
 A 2
 0.
c t
Show that the Lorentz gauge can be reached by selecting a suitable ,
which is the solution to the differential equation
2 
  1 
1  2



 A 2
.
c2  t 2
c t
2.
Relativistic generalization of Ohm's Law
In the rest frame of a medium one has


J  E 


where J  is the current density, E  is the electric field, and  is the conductivity
(measured in the rest frame). This is called Ohm's Law.
a)
Why is this equation not relativistic?
b)
Try setting
1
J   KF  u (Eq. 1)
c
where u is the 4-velocity of the medium. Find K such that in the rest

frame u   c,0 the above equation reduces to Ohm's Law.
 
c)
The above equation implies
u  J  0 (Eq. 2)
Why? Show by going to the rest frame that Eq. 2 is not generally true.
d)
Fix up Eq. 1 by adding an extra term
J 
M

u J  u  F  u
2
c
Show that M 1 .
e)
 
If the medium has a velocity  c with respect to an inertial frame,
show that the current in that frame is given by




     

J  E  B    E   (Eq. 3)
 
where E , B , and  are measured in this inertial frame, and show
J0 
1     
   J  E  (Eq. 4).
 
 2 
Hint: show Eq. 4 first, then use it to get Eq. 3.
2
f)
(Harder) Assume the medium is uncharged in its rest frame   0 .
Using the Lorentz transformation and Eq. 3 show that

3.
c

Moving Wire
An infinitely long thin wire along the z axis moves with a speed v along the z axis
and carries a charge per unit length of q, and so a current qv.
a)
b)
c)
d)
4.


  
  
 E and J  E  B .
Calculate the charge density and current as seen by an observer in S'
moving with a speed v along the z axis.
Calculate the electric and magnetic fields seen by this observer.
Transform these fields to find the fields in the original frame.
Check that these agree with a direct calculation in the original frame S.
Maxwell's Equations in a Dielectric Medium
We want to construct Maxwell's equations in a uniformly moving dielectric
medium.
Assume
i)
ii)


that G  is constructed from D and H just as F  is constructed


from E and B and transforms like a tensor,
that, therefore, Maxwell's equations are written ,
 G
0 J 

x
iii)
and that in the rest frame of the medium
 1


D  E and H  B .

In general, the 4-velocity of the medium is u  . The relation between G  and
F  must be
G  KF  
L
u u F u u F  
c2
where K and L are constants depending on  and  . Find K and L.
3
5.
Relativistic Equations of Motion
a)
The relativistic equations of motion of a charged particle in a given
electromagnetic field are
m
d 2 x
dx
qF   
2
d
d
where d is the proper time interval d 
dt

. Show that the force in a
particular reference frame therefore reduces to the Lorentz force,
 dp
  
F  q E v B
dt




where p mv is the relativistic momentum.
b)
A particle travels with varying speed v t  in a circle of fixed radius R,
with equation
r  x 2  y 2 R and z 0 .

The components of dv dt are then dv dt along the tangent and v 2 R

towards the center. On the circular path, E is along the tangent and has a

magnitude E and B is in the z direction and has a magnitude B. Write
down the components of the equation of motion in the radial and
tangential directions. You may find that it helps to write

d  d 
v   v  dv .
dt
dt
dt
c)
Show that the equations in (b) can both be satisfied if
E t   R
dBt 
dt


where E and B are both functions of time alone. Show that this equation
is a form a Faraday's law (from general physics) assuming that the average
magnetic flux through the circular loop is twice that on the boundary.
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