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.
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.