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.


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.


Observe that is a scalar, (actually the total charge in the distribution) and is called the electric monopole. This term indicates point charge electrical potential.


We can write

The vector is called the electric dipole. And its magnitude is called the dipole moment of the charge distribution. This terms indicates the linear charge distribution geometry of a dipole electrical potential.


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 this term indicates the three dimensional distribution of a quadruple electrical potential.