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When one solves Laplace's equation $$0~=~\nabla^2\Phi~=~\frac{1}{r^2}\frac{\partial}{\partial r}r^2 \frac{\partial \Phi}{\partial r} - \frac{1}{r^2}L^2\Phi $$in spherical coordinates in the bulk (away from a central region with sources), it separates in an angular problem on the $2$-sphere $S^2$ and a radial problem.

Since the rotation group $SO(3)$ acts on the $2$-sphere $S^2$, the angular solutions are representations of the Lie group $SO(3)$, namely linear combinations of the spherical harmonics. All finite-dimensional irreducible representations $V_{\ell}$ of the Lie group $SO(3)$ are characterized by an integer spin $\ell\in\mathbb{N}_0$, which are related to the $2^{\ell}$-pole term. Here the Casimir $L^2$ has eigenvalue $\ell(\ell+1)$. The irrep $V_{\ell}\equiv\underline{\bf 2\ell\!+\!1}$ has dimension $2\ell\!+\!1$. E.g. $\ell=0$ is a monopole, $\ell=1$ is a dipole, $\ell=2$ is a quadrupole, and so forth.

The main point is that potentials for any number of charges can be classified according to above scheme. The corresponding radial $2^{\ell}$-pole solution falls off as $ r^{-(\ell+1)}$. See also related Phys.SE posts here and here.

When one solves Laplace's equation $$0~=~\nabla^2\Phi~=~\frac{1}{r^2}\frac{\partial}{\partial r}r^2 \frac{\partial \Phi}{\partial r} - \frac{1}{r^2}L^2\Phi $$in spherical coordinates in the bulk (away from a central region with sources), it separates in an angular problem on the $2$-sphere $S^2$ and a radial problem.

Since the rotation group $SO(3)$ acts on the $2$-sphere $S^2$, the angular solutions are representations of the Lie group $SO(3)$, namely linear combinations of the spherical harmonics. All finite-dimensional irreducible representations $V_{\ell}$ of the Lie group $SO(3)$ are characterized by an integer spin $\ell\in\mathbb{N}_0$, which are related to the $2^{\ell}$-pole term. Here the Casimir $L^2$ has eigenvalue $\ell(\ell+1)$. The irrep $V_{\ell}\equiv\underline{\bf 2\ell\!+\!1}$ has dimension $2\ell\!+\!1$. E.g. $\ell=0$ is a monopole, $\ell=1$ is a dipole, $\ell=2$ is a quadrupole, and so forth.

The main point is that potentials for any number of charges can be classified according to above scheme. The corresponding radial $2^{\ell}$-pole solution falls off as $ r^{-(\ell+1)}$. See also related Phys.SE posts here and here.

When one solves Laplace's equation $$0~=~\nabla^2\Phi~=~\frac{\partial^2\Phi}{\partial r^2} + \frac{1}{r^2}L^2\Phi $$in spherical coordinates in the bulk (away from a central region with sources), it separates in an angular problem on the $2$-sphere $S^2$ and a radial problem.

Since the rotation group $SO(3)$ acts on the $2$-sphere $S^2$, the angular solutions are representations of the Lie group $SO(3)$, namely linear combinations of the spherical harmonics. All finite-dimensional irreducible representations $V_{\ell}$ of the Lie group $SO(3)$ are characterized by an integer spin $\ell\in\mathbb{N}_0$, which are related to the $2^{\ell}$-pole term. Here the Casimir $L^2$ has eigenvalue $\ell(\ell+1)$. The irrep $V_{\ell}\equiv\underline{\bf 2\ell\!+\!1}$ has dimension $2\ell\!+\!1$. E.g. $\ell=0$ is a monopole, $\ell=1$ is a dipole, $\ell=2$ is a quadrupole, and so forth.

The main point is that potentials for any number of charges can be classified according to above scheme. The corresponding radial $2^{\ell}$-pole solution falls off as $ r^{-(\ell+1)}$. See also related Phys.SE posts here and here.

When one solves Laplace's equation $$0~=~\nabla^2\Phi~=~\frac{1}{r^2}\frac{\partial}{\partial r}r^2 \frac{\partial \Phi}{\partial r} - \frac{1}{r^2}L^2\Phi $$in spherical coordinates in the bulk (away from a central region with sources), it separates in an angular problem on the $2$-sphere $S^2$ and a radial problem.

Since the rotation group $SO(3)$ acts on the $2$-sphere $S^2$, the angular solutions are representations of the Lie group $SO(3)$, namely linear combinations of the spherical harmonics. All finite-dimensional irreducible representations $V_{\ell}$ of the Lie group $SO(3)$ are characterized by an integer spin $\ell\in\mathbb{N}_0$, which are related to the $2^{\ell}$-pole term. Here the Casimir $L^2$ has eigenvalue $\ell(\ell+1)$. The irrep $V_{\ell}\equiv\underline{\bf 2\ell\!+\!1}$ has dimension $2\ell\!+\!1$. E.g. $\ell=0$ is a monopole, $\ell=1$ is a dipole, $\ell=2$ is a quadrupole, and so forth.

The main point is that potentials for any number of charges can be classified according to above scheme. The corresponding radial $2^{\ell}$-pole solution falls off as $ r^{-(\ell+1)}$. See also related Phys.SE posts here and here.

When one solves Laplace's equation $$0~=~\nabla^2\Phi~=~\frac{\partial^2\Phi}{\partial r^2} + \frac{1}{r^2}L^2\Phi $$in spherical coordinates in the bulk (away from a central region with sources), it separates in an angular problem on the $2$-sphere $S^2$ and a radial problem.

Since the rotation group $SO(3)$ acts on the $2$-sphere $S^2$, the angular solutions are representations of the Lie group $SO(3)$, namely linear combinations of the spherical harmonics. All finite-dimensional irreducible representations $V_{\ell}$ of the Lie group $SO(3)$ are characterized by an integer spin $\ell\in\mathbb{N}_0$, which are related to the $2^{\ell}$-pole term. Here the Casimir $L^2$ has eigenvalue $\ell(\ell+1)$. The irrep $V_{\ell}\equiv\underline{\bf 2\ell\!+\!1}$ has dimension $2\ell\!+\!1$. E.g. $\ell=0$ is a monopole, $\ell=1$ is a dipole, and so forth.

The main point is that potentials for any number of charges can be classified according to above scheme. The corresponding radial $2^{\ell}$-pole solution falls off as $ r^{-(\ell+1)}$.

When one solves Laplace's equation $$0~=~\nabla^2\Phi~=~\frac{\partial^2\Phi}{\partial r^2} + \frac{1}{r^2}L^2\Phi $$in spherical coordinates in the bulk (away from a central region with sources), it separates in an angular problem on the $2$-sphere $S^2$ and a radial problem.

Since the rotation group $SO(3)$ acts on the $2$-sphere $S^2$, the angular solutions are representations of the Lie group $SO(3)$, namely linear combinations of the spherical harmonics. All finite-dimensional irreducible representations $V_{\ell}$ of the Lie group $SO(3)$ are characterized by an integer spin $\ell\in\mathbb{N}_0$, which are related to the $2^{\ell}$-pole term. Here the Casimir $L^2$ has eigenvalue $\ell(\ell+1)$. The irrep $V_{\ell}\equiv\underline{\bf 2\ell\!+\!1}$ has dimension $2\ell\!+\!1$. E.g. $\ell=0$ is a monopole, $\ell=1$ is a dipole, $\ell=2$ is a quadrupole, and so forth.

The main point is that potentials for any number of charges can be classified according to above scheme. The corresponding radial $2^{\ell}$-pole solution falls off as $ r^{-(\ell+1)}$. See also related Phys.SE posts here and here.