Mathematical Methods of Physics/The multipole expansion

< Mathematical Methods of Physics

Tensors are useful in all physical situations that involve complicated dependence on directions. Here, we consider one such example, the multipole expansion of the potential of a charge distribution.

Contents

IntroductionEdit

Consider an arbitrary charge distribution . We wish to find the electrostatic potential due to this charge distribution at a given point . We assume that this point is at a large distance from the charge distribution, that is if varies over the charge distribution, then

Now, the coulomb potential for a charge distribution is given by

Here, , where

Thus, using the fact that is much larger than , we can write , and using the binomial expansion,

(we neglect the third and higher order terms).

The multipole expansionEdit

Thus, the potential can be written as

We write this as , where,

and so on.

MonopoleEdit

Observe that is a scalar, (actually the total charge in the distribution) and is called the electric monopole

DipoleEdit

We can write

The vector is called the electric dipole. And its magnitude is called the dipole moment of the charge distribution.

QuadrupoleEdit

Let and be expressed in Cartesian coordinates as and . Then,

We define a dyad to be the tensor given by

Define the Quadrupole tensor as

Then, we can write as the tensor contraction