Multipole

Complex charge distributions can be approximated as a sum of different poles from far away enough

$V(\vec{r})=k\sum _{n=0}^{\infty }\frac{1}{r^{(n+1)}}\int (r^{\prime }{)}^n{P}_n(\cos \alpha )\rho ({\vec{r}}^{\prime })\mathrm{d}{\tau }^{\prime }$

• Voltage of a charge density
• $k$ is the Coloumb constant
• $r$ is the position vector magnitude
• $r^{\prime }$ is the source vector magnitude
• $P_n$ is the Legendre Polynomial
• $\alpha$ is the angle between the position vector and source vector
• ${\tau }^{\prime }$ is the volume element