Electricity and Magnetism · Problem 10.6

MAGNETIC FIELDS IN MATTER — Edward Purcell & David Morin

Physics — E&M · quadrupole-tensor

Problem

You should see the quadrupole tensor at least once in your life, so here it is. Calculate the general form of the quadrupole tensor by writing the

R

in Eq. (10.5) in Cartesian form as

R = \sqrt{(x_1 - x_1')^2 + (x_2 - x_2')^2 + (x_3 - x_3')^2},

and then performing a Taylor expansion as we did in Section 10.2. (It's a little cleaner to work with

x_1, x_2, x_3

instead of

x, y, z

.) Your goal is to write the

1/r^3

part of the potential in a form that has all of its dependence on the primed coordinates collected into a matrix. (By analogy, the

1/r^2

part of the potential in Eq. (10.12) has all of its dependence on the primed coordinates collected into the vector

\mathbf{p}

given by Eq. (10.13).)