# 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

Consider an arbitrary charge distribution ${\displaystyle \rho (\mathbf {r} ')}$ . We wish to find the electrostatic potential due to this charge distribution at a given point ${\displaystyle \mathbf {r} }$ . We assume that this point is at a large distance from the charge distribution, that is if ${\displaystyle \mathbf {r} '}$  varies over the charge distribution, then ${\displaystyle \mathbf {r} >>\mathbf {r} '}$

Now, the coulomb potential for a charge distribution is given by ${\displaystyle V(\mathbf {r} )={\frac {1}{4\pi \epsilon _{0}}}\int _{V'}{\frac {\rho (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r'} |}}dV'}$

Here, ${\displaystyle |\mathbf {r} -\mathbf {r'} |=|r^{2}-2\mathbf {r} \cdot \mathbf {r} '+r'^{2}|^{\frac {1}{2}}=r\left|1-2{\frac {{\hat {\mathbf {r} }}\cdot \mathbf {r} '}{r}}+\left({\frac {r'}{r}}\right)^{2}\right|^{\frac {1}{2}}}$ , where ${\displaystyle {\hat {\mathbf {r} }}\triangleq \mathbf {r} /r}$

Thus, using the fact that ${\displaystyle \mathbf {r} }$  is much larger than ${\displaystyle \mathbf {r} '}$ , we can write ${\displaystyle {\frac {1}{|\mathbf {r} -\mathbf {r'} |}}={\frac {1}{r}}{\frac {1}{\left|1-2{\frac {{\hat {\mathbf {r} }}\cdot \mathbf {r} '}{r}}+\left({\frac {r'}{r}}\right)^{2}\right|^{\frac {1}{2}}}}}$ , and using the binomial expansion,

${\displaystyle {\frac {1}{\left|1-2{\frac {{\hat {\mathbf {r} }}\cdot \mathbf {r} '}{r}}+\left({\frac {r'}{r}}\right)^{2}\right|^{\frac {1}{2}}}}=1+{\frac {{\hat {\mathbf {r} }}\cdot \mathbf {r} '}{r}}+{\frac {1}{2r^{2}}}\left(3({\hat {\mathbf {r} }}\cdot \mathbf {r} '\right)^{2}-r'^{2})+O\left({\frac {r'}{r}}\right)^{3}}$  (we neglect the third and higher order terms).

## The multipole expansion

Thus, the potential can be written as ${\displaystyle V(\mathbf {r} )={\frac {1}{4\pi \epsilon _{0}r}}\int _{V'}\rho (\mathbf {r} ')\left(1+{\frac {{\hat {\mathbf {r} }}\cdot \mathbf {r} '}{r}}+{\frac {1}{2r^{2}}}\left(3({\hat {\mathbf {r} }}\cdot \mathbf {r} '\right)^{2}-r'^{2})+O\left({\frac {r'}{r}}\right)^{3}\right)dV'}$

We write this as ${\displaystyle V(\mathbf {r} )=V_{\text{mon}}(\mathbf {r} )+V_{\text{dip}}(\mathbf {r} )+V_{\text{quad}}(\mathbf {r} )+\ldots }$ , where,

${\displaystyle V_{\text{mon}}(\mathbf {r} )={\frac {1}{4\pi \epsilon _{0}r}}\int _{V'}\rho (\mathbf {r} ')dV'}$

${\displaystyle V_{\text{dip}}(\mathbf {r} )={\frac {1}{4\pi \epsilon _{0}r^{2}}}\int _{V'}\rho (\mathbf {r} ')\left({\hat {\mathbf {r} }}\cdot \mathbf {r} '\right)dV'}$

${\displaystyle V_{\text{quad}}(\mathbf {r} )={\frac {1}{8\pi \epsilon _{0}r^{3}}}\int _{V'}\rho (\mathbf {r} ')\left(3\left({\hat {\mathbf {r} }}\cdot \mathbf {r} '\right)^{2}-r'^{2}\right)dV'}$

and so on.

### Monopole

Observe that ${\displaystyle Q=\int _{V'}\rho (\mathbf {r} ')dV'}$  is a scalar, (actually the total charge in the distribution) and is called the electric monopole. This term indicates point charge electrical potential.

### Dipole

We can write ${\displaystyle V_{\text{dip}}(\mathbf {r} )={\frac {\hat {\mathbf {r} }}{4\pi \epsilon _{0}r^{2}}}\cdot \int _{V'}\rho (\mathbf {r} ')\mathbf {r} 'dV'}$

The vector ${\displaystyle \mathbf {p} =\int _{V'}\rho (\mathbf {r} ')\mathbf {r} 'dV'}$  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 ${\displaystyle {\hat {\mathbf {r} }}}$  and ${\displaystyle \mathbf {r} '}$  be expressed in Cartesian coordinates as ${\displaystyle (r_{1},r_{2},r_{3})}$  and ${\displaystyle (x_{1},x_{2},x_{3})}$ . Then, ${\displaystyle ({\hat {\mathbf {r} }}\cdot \mathbf {r} ')^{2}=(r_{i}x_{i})^{2}=r_{i}r_{j}x_{i}x_{j}}$
We define a dyad to be the tensor ${\displaystyle {\hat {\mathbf {r} }}{\hat {\mathbf {r} }}}$  given by ${\displaystyle \left({\hat {\mathbf {r} }}{\hat {\mathbf {r} }}\right)_{ij}=r_{i}r_{j}}$
Define the Quadrupole tensor as ${\displaystyle T=\int _{V'}\rho (\mathbf {r} ')\left(3(\mathbf {r} '\mathbf {r} ')-\mathbf {I} r'^{2}\right)dV'}$
Then, we can write ${\displaystyle V_{\text{qua}}}$  as the tensor contraction ${\displaystyle V_{\text{qua}}(\mathbf {r} )=-{\frac {{\hat {\mathbf {r} }}{\hat {\mathbf {r} }}}{4\pi \epsilon _{0}r^{3}}}::T}$  this term indicates the three dimensional distribution of a quadruple electrical potential.