Mathematical Methods of Physics/The multipole expansion

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. This term indicates point charge electrical potential.

Dipole edit

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.

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