Download Physics PHYS 354 Electricity and Magnetism II Problem Set #4

Physics PHYS 354
Electricity and Magnetism II
Problem Set #4
1.
Vector Transformations
Consider two right-handed, orthogonal coordinate systems, their axes being
defined by the unit vectors ê1 , ê2 , ê3 and ê1 , ê2 , ê3 respectively. The two
systems have a common origin O. The position vector of point P may then be
written as

r xi eˆi xieˆi .
It is clear that every ê can be written as a linear combination of the ê 's:
eˆi Rij eˆ j .
a)
Establish the relations Rij eˆi eˆ j , which imply that eˆi eˆi eˆ j  eˆ j .
b)
Show that it follows at once that eˆi eˆj R ji .
c)
Use the results of (a) and (b) to establish the transformation rules for the
vector components x and x':
xi Rij x j and xi xj R ji .
d)
Use the orthogonality relations for the basis vectors ê and ê :
eˆi ˆe j  ij and eˆi eˆj  ij
to establish the orthogonality relations for the tensors R ij :
Rin R jn  ij and Rni Rnj  ij .
e)
The three unit vectors êi (i=1, 2, 3), as well as the êi , satisfy the relations
eˆ eˆ eˆ
i
j
k
 ijk
where  ijk is the totally antisymmetric tensor. Use this property to show
that the determinant of the transformation matrix Rij has the value 1:
det Rij 1 . It may be helpful to show that det Rij   ijk R1i R2 j R3k first.
This can be done by brute force.
2.
Binomial Expansion of 
P
 
x-y

The electric potential  x  for

the charge distribution  x  is
given by


x


1
 x 
3
x  
d
y
 
4 0 
xy

y
where the vectors are defined
according to the diagram at
right.
a)

o
Show, using the binomial expansion, that
xˆ
1
1
1
ˆˆ
 j yj 
  
3 xi x j 3 yi y j  ij yk yk   .
x y
xi xi xi xi
2xk xk 2


when x  y .
b)
Show then that the electric potential can be written in the form:
xˆ
xˆ xˆ

q
 x  
 j p j  i j 3 Qij   .
xi xi xi xi
2xk xk 2
c)
Write down expressions for the charge q , the dipole moment p j , and the
quadrupole moment tensor Qij .
d)
e)

  
Now let the axes be translated by Y , that is, y  y Y . Find the
expressions for q , pi , and Qij .
Show that the quadrupole moment of a spherically symmetric charge
distribution is zero, assuming that the origin is located at the center of the
distribution. Is the quadrupole moment still zero if the origin is located
elsewhere?
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f)
We showed in part (b) that the electric potential due to a quadrupole is
xˆ xˆ

  x   i j 3 Qij .
2xk xk 2
This is obtained when a charge distribution has a net charge of zero, and a
dipole moment of zero. An object consisting of two positive charges +e
flanking a negative charge -2e, each at a distance L has this property.
 
 
Using the relationship E x   x  , show that the electric field is given
by:
Ek 
 Qkm xˆm 5 xˆi xˆ j Qij
4 
 xk .
2 x5
x
3.
Triple Cross Product Identity
     

Derive the vector identity    a   a  2a using index notation. Hint:
use the fact that  ijk lmk  il jm  im jl .
4.
Magnetic Dipole
 
The magnetic vector potential of a magnetic dipole at x  0 is given by

  
   x
A x  m   3 ,
x

where m is the magnetic moment. Show that the magnetic field is given by
  
  3nm
n  m
Bx  
3
x


where n is a unit vector in the same direction as x .
5.
Electric Multipole Expansion

 
A charge density  x  is confined to a small region near x 0 . It is acted on by
an electric field due to distant charges which therefore varies slowly in space over

the region of  . Describe this external electric field by a potential  x  with a
 
Taylor series expansion near x 0 :
3

x  0 i xi  12 ij xi x j 
where ij  ji .
a)
 ii 0 . Why?
b)

Show that the total force F acting on  can be expanded as a series
whose first two terms are



    
F qE   p E x 
 
x 0

where q and p are the charge and dipole moment of  .
c)
Show that the first term in the expansion of the torque on  is

  
N  p E 0 .
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