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\section{Multipoles\label{Multipoles}}

Because 
\[
\frac{1}{\left| \vec{r}-\vec{r}~^{\prime }\right| }=\frac{1}{r}%
\sum_{0}^{\infty }\left( \frac{r^{\prime }}{r}\right) ^{l}P_{l}\left( \cos
\theta \right) 
\]
the potential can be written as: 
\[
V=\frac{Q}{4\pi \varepsilon }\sum_{n}\frac{k_{n}}{r^{n}} 
\]

For the lowest-order terms this results in:

\begin{itemize}
\item  Monopole: $l=0$, $k_{0}=\int \rho dV$

\item  Dipole: $l=1$, $k_{1}=\int r\cos (\theta )\rho dV$

\item  Quadrupole: $l=2$, $k_{2}=\frac{1}{2}\sum%
\limits_{i}(3z_{i}^{2}-r_{i}^{2})$
\end{itemize}

\begin{enumerate}
\item  The electric dipole: dipole moment: $\vec{p}=Ql\vec{e}_{{}}$, where $%
\vec{e}_{{}}$ goes from $\oplus $ to $\ominus $, and $\vec{F}=(\vec{p}\cdot
\nabla )\vec{E}_{\mathrm{ext}}$, and $W=-\vec{p}\cdot \vec{E}_{\mathrm{out}}$%
.\newline
Electric field: $\vec{E}\approx \dfrac{Q}{4\pi \varepsilon r^{3}}\left( 
\dfrac{3\vec{p}\cdot \vec{r}}{r^{2}}-\vec{p}\right) $.~~ The torque is: $%
\vec{\tau}=\vec{p}\times \vec{E}_{\mathrm{out}}$

\item  The magnetic dipole: dipole moment: if $r\gg \sqrt{A}$: $\vec{\mu}=%
\vec{I}\times (A\vec{e}_{\perp })$, $\vec{F}=(\vec{\mu}\cdot \nabla )\vec{B}%
_{\mathrm{out}}$, $|\mu |=\dfrac{mv_{\perp }^{2}}{2B}$, $W=-\vec{\mu}\times 
\vec{B}_{\mathrm{out}}$\newline
Magnetic field: $\vec{B}=\dfrac{-\mu }{4\pi r^{3}}\left( \dfrac{3\mu \cdot 
\vec{r}}{r^{2}}-\vec{\mu}\right) $.~~ The moment is: $\vec{\tau}=\vec{\mu}%
\times \vec{B}_{\mathrm{out}}$
\end{enumerate}

\subsection{\ Related Reference Topics}

{\small \hyperref{Calculus}{}{}{../Mathematics/CAL_contents.tex}}

{\small \hyperref{Differential Equations}{}{}{../Mathematics/ODE_contents.tex%
}}

{\small \hyperref{Matrix Algebra}{}{}{../Mathematics/MA_contents.tex}}

{\small \hyperref{Table of Integrals}{}{}{%
../Mathematics/TAB_integrals.tex#table of integrals}}

{\small \hyperref{Table of Derivatives}{}{}{%
../Mathematics/TAB_derivatives.tex}}

{\small \hyperref{Table of Physical Constants}{}{}{PhysicalConstants.tex}}

{\small \hyperref{Table of SI Units}{}{}{SIUnits.tex}}

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Numbers, Functions, and Units}{}{}{../../Help/DM2.tex}}

{\small \hyperref{Calculus}{}{}{../../Help/DM7.tex}}

{\small \hyperref{Matrix Algebra}{}{}{../../Help/DM8.tex}}

{\small \hyperref{Vector Calculus}{}{}{../../Help/DM9.tex}}

\noindent \hrulefill

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