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Multipole Moments of the Electric Field

The **multipole moments** of a charge distribution \(\rho\left(\mathbf{r}\right)\) arise naturally if we try to solve **Poisson's equation** for the electrostatic potential, \(\Delta\phi\left(\mathbf{r}\right)= -\rho\left(\mathbf{r}\right)/\varepsilon_{0}\):

**The Green's function** of the Laplace operator obeying \(\Delta_{\mathbf{r}}G\left(\mathbf{r},\mathbf{r}^{\prime}\right) = \delta\left(\mathbf{r}-\mathbf{r}^{\prime}\right)\), is given by \(G\left(\mathbf{r},\mathbf{r}^{\prime}\right) = -1/4\pi\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\), which provides us the well-known \[\phi\left(\mathbf{r}\right)=\frac{1}{4\pi\varepsilon_{0}}\int\frac{\rho\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}dV\ .\]The Green's function can now be **expanded** in a Taylor series around \(\mathbf{r}^{\prime}=0\) yielding (see also in the solution of "The Electric Field of a Dipole")\[\begin{eqnarray*}\phi\left(\mathbf{r}\right) & = & \frac{1}{4\pi\varepsilon_{0}}\left\{\int\frac{\rho\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}\right|}dV^{\prime}+\int\frac{\mathbf{r}\cdot\mathbf{r}^{\prime}\rho\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}\right|^{3}}dV^{\prime}+\dots\right\}\\&=& \frac{1}{4\pi\varepsilon_{0}}\left\{\frac{Q}{r}+\frac{\mathbf{p}\cdot\mathbf{r}}{r^{3}}+\dots\right\}\end{eqnarray*}\]with \(Q=\int\rho\left(\mathbf{r}^{\prime}\right)dV^{\prime}\) as the total **charge**, the **dipole moment** \(\mathbf{p}=\int\mathbf{r}^{\prime}\rho\left(\mathbf{r}^{\prime}\right)dV^{\prime}\) and \(r=\left|\mathbf{r}\right|\). You can see that the moments we already know are just the beginning of a **fundamental representation** of the electrostatic potential. In the following problems we shall make ourselfs familier with the multipole moments. Enjoy!

The **quadrupole moments** of a homogeneously charged ellipsoid are calculated. The result will help us to understand how and why the quadrupole moments of an **ellipsoidal charge distribution** changes if it gets into an **excited state**.

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We will calculate the first three multipole moments of a **deformed charged sphere**. Such a charge distribution is often used to **model nuclei** and understand their stability. The **rotational symmetry** of the problem will ease our computations significantly.

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The electric field of a dipole can be seen as the result of two charges approaching each other. Learn in this problem how to use the **Taylor expansion** of \(1/\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\) to calculate this field. Find out how this series expansion yields **multipole moments**, a very powerful description of the electric field.

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In **rotational symmetry**, the multipole moments of a charge distribution are described by **just one component** per order \(l\). This is a drastic difference to the usual \(2l+1\) independent components. Get to know this **simplification**s with two examples and **derive** it for any charge distribution with rotational symmetry!

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