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.

## 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