Physics 304
Problem Set 6
Due: Wednesday, Oct. 19
(1) Consider the 4-charge quadrupole as in figure 3.27 of your text, where the 4 point charges are on
corners of a square with sides a.
(a) Find the first two nonzero terms in the multipole expansion for V(r,θ) for the case that the square is
centered at the origin, the θ = 0 observation direction is perpendicular to the square, and for nonzero
angles the observer is rotated by an angle θ about the long diagonal containing the two negative charges.
(Thus the observer sees the negative charges fixed while the positive charges change position.)
(b) Considering in this part only the quadrupole term in the expansion, find the contribution to V(r,θ) for
the cases that (i) the rotation is around the other long diagonal of the square, and (ii) the rotation is around
an axis that cuts the square into two rectangles (i.e. 45° rotated from the long diagonal direction).
[With a bit more effort the quadrupole potential can be computed for a general observation direction
given by (θ, φ). The orientation dependence is more complicated than that of the dipole, and as stated in
class can be parameterized by using a tensor, rather than the vector that represents the dipole charge
distribution.]
(2) Griffiths problem 3.40 (page 156). Also: Find the dipole moments of the 3 charge distributions.
(3) Griffiths problem 4.4 (page 163).
(4) Griffiths problem 4.6 (page165).
(5) (a) Consider a cylindrical insulating dielectric rod, length L and radius So, extending from z = 0 to L.
!
The polarization points along the z axis of the cylinder and is given by, P = (az 3 + b)ẑ , where a and b
are constants. Find the volume and surface bound charge densities for this rod, and show explicitly that
the total bound charge is equal to zero, as it must be for the charge due to a collection
of dipoles.
!
(b) Similarly, consider a sphere of radius R where the polarization is given by, P = (! / r)r̂ , with ! a
constant. Find the bound surface charge, and also the bound charge density within the volume of the
cylinder, and the total integrated bound charge in the bulk and on the surface.
(6) Consider a long cylindrical dielectric shell, with inner radius s = a, and outer radius s = b. The
!
polarization vector is given by P = ksŝ within the shell ( a < s < b ), but zero in the hollow inside region
and also zero outside. Assume that there are no free charges or any other applied fields in the vicinity.
(a) Find the bound surface and volume charges for this case.
(b) Using the integral form of Gauss’ law, find the electric field due to the distribution of charges
determined for this cylinder in part (a). Do this for the regions s < a , a < s < b , and b < s .
!
!
D
D
(c) Alternatively, find the displacement
vector
for
this
situation,
using
(4.23)
in
the
text.
Using
!
!
thus obtained, and the expression for P that was given, determine E directly using (4.21) in the text.