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,
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.
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.