I'm looking for the analytical expressions of the magnetic and induced electric fields inside a rotating uniformly magnetized sphere.

If the sphere isn't rotating, the electric field is 0 everywhere, and the magnetic field can be expressed as a function of the sphere's global magnetic moment : \begin{align} \vec{B}_{\text{int}} &= \frac{\mu_0 \, \vec{\mu}}{2 \pi R^3}, \tag{1} \\[12pt] \vec{E}_{\text{int}} &= 0. \tag{2} \end{align} Then consider the same sphere uniformly rotating around the $z$ axis. The magnetic moment $\vec{\mu}(t)$ has an inclination $\alpha$ relative to the rotation axis and is rotating around that axis : \begin{equation}\tag{3} \vec{\mu}(t) = \mu \, \sin{\alpha} \, \big(\cos{(\omega \, t)} \, \hat{x} + \sin{(\omega \, t)} \, \hat{y} \big) + \mu \, \cos{\alpha} \, \hat{z}. \end{equation} Take note that the sphere is neutral (the electric charge density is 0 everywhere). The currents on the surface can be described by this density vector : \begin{equation}\tag{4} \vec{J}(t, \vec{r}) = \frac{3}{4 \pi R^4} \, \vec{\mu}(t) \times \vec{r} \; \delta(r - R), \end{equation} such that \begin{equation}\tag{5} \vec{\mu}(t) \equiv \frac{1}{2} \int \vec{r} \times \vec{J}(t, \vec{r}) \, d^3 x. \end{equation} So what are the vectors $\vec{E}(t,\vec{r})$ and $\vec{B}(t,\vec{r})$ ?

Please, don't give any answers in terms of "macroscopic" fields $\vec{D}$ and $\vec{H}$. And I'm only interested in the fields inside the sphere (I know the solution for the outside, if the sphere is very small). I'm expecting that, like most radiation fields, the electromagnetic field inside the sphere should depend on some retarded time, something like $t - r / c$.

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    $\begingroup$ Maybe you can make use of the Maxwell equations in curved spacetime, to first calculate the electric field without rotation, and then change coordinate system to the rotated one? $\endgroup$ – Lukas Berns Aug 8 '17 at 6:26

By brute force, I may have found the solution, but I'm not sure it's the proper one. There are some pieces that are puzzling. The dots below are the time derivatives, and the magnetic moment must be evaluated at the retarded time : $\vec{\boldsymbol{\mu}} \equiv \vec{\boldsymbol{\mu}}(t - r / c)$ \begin{align} \vec{\boldsymbol{B}}(t, \vec{\boldsymbol{r}}) &= \frac{\mu_0}{4 \pi R^3} \Big( 2 \, \vec{\boldsymbol{\mu}} + 2 \, \frac{r}{c} \, \dot{\vec{\boldsymbol{\mu}}} + \frac{1}{c^2} \, (\ddot{\vec{\boldsymbol{\mu}}} \times \vec{\boldsymbol{r}}) \times \vec{\boldsymbol{r}} \Big), \tag{1} \\[12pt] \vec{\boldsymbol{E}}(t, \vec{\boldsymbol{r}}) &= -\, \frac{1}{4 \pi \varepsilon_0 R^3} \Big( \frac{1}{c^2} \, \dot{\vec{\boldsymbol{\mu}}} \times \vec{\boldsymbol{r}} + \frac{r}{c^3} \, \ddot{\vec{\boldsymbol{\mu}}} \times \vec{\boldsymbol{r}} \Big). \tag{2} \end{align} These expressions reduce to the constant solution (1) and (2) of my question, when the magnetic moment is a constant. Some tedious calculations show that Maxwell's equations are satisfied by these vectors, if we introduce a current density in the bulk of the sphere (I wasn't able to find a solution without a current inside the volume) : \begin{align} \vec{\boldsymbol{\nabla}} \cdot \vec{\boldsymbol{B}} &= 0, \tag{3} \\[12pt] \vec{\boldsymbol{\nabla}} \cdot \vec{\boldsymbol{E}} &= 0, \tag{4} \\[12pt] \vec{\boldsymbol{\nabla}} \times \vec{\boldsymbol{E}} &= -\, \frac{\partial \, \vec{\boldsymbol{B}}}{\partial \, t}, \tag{5} \\[12pt] \vec{\boldsymbol{\nabla}} \times \vec{\boldsymbol{B}} &= \mu_0 \, \vec{\boldsymbol{J}} + \frac{1}{c^2} \, \frac{\partial \, \vec{\boldsymbol{E}}}{\partial \, t}, \tag{6} \end{align} where \begin{equation}\tag{7} \vec{\boldsymbol{J}}(t, \vec{\boldsymbol{r}}) = \frac{3}{2 \pi R^3 \, c^2} \, \ddot{\vec{\boldsymbol{\mu}}} \times \vec{\boldsymbol{r}} \equiv \vec{\boldsymbol{\nabla}} \times \vec{\boldsymbol{K}}(t, \vec{\boldsymbol{r}}), \end{equation} and \begin{equation}\tag{8} \vec{\boldsymbol{K}}(t, \vec{\boldsymbol{r}}) = \frac{3}{2 \pi R^3} \Big( \, \vec{\boldsymbol{\mu}} + \frac{r}{c} \, \dot{\vec{\boldsymbol{\mu}}} \, \Big). \end{equation} I'm puzzled by the current density (7) and the vector (8) which may be interpreted as a magnetization density, but it's missing a factor of 2 at the denominator. Even when $\dot{\vec{\boldsymbol{\mu}}} = 0$, it's not exactly the magnetization in the bulk, which should be $\vec{\boldsymbol{M}} = 3 \, \vec{\boldsymbol{\mu}} / 4 \pi R^3$. However, when $\vec{\boldsymbol{\mu}}$ is a constant, so is $\vec{\boldsymbol{K}}$ and $\vec{\boldsymbol{J}} = 0$ (in the bulk), as it should for a uniformly magnetized sphere with a constant and uniform magnetic field (vector (1) of the question above).

Notice that this solution doesn't specify the form of the function $\vec{\boldsymbol{\mu}}(t - r / c)$ which is still arbitrary (not necessarily a rotation around a fixed axis). It can be a sphere with increasing uniform magnetization, which appears to produce some radiation inside the sphere.

Any opinions on this "solution" ? What should be the proper interpretation of the current density (7) ?

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  • $\begingroup$ I calculated the power output of the EM field (1)-(2), from the Poynting vector evaluated at $r = R$, for a rotating magnetic moment, and it's exactly the same as the classical well-known emission formula for a rotating dipole. So this is a good test for the "internal" solution. $\endgroup$ – Cham Aug 8 '17 at 14:56

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