Mathematical Methods of Physics/The multipole expansion

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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.

Introduction[edit]

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 expansion[edit]

Thus, the potential can be written as

We write this as , where,

and so on.

Monopole[edit]

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

Dipole[edit]

We can write

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

Quadrupole[edit]

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