- #1

Nitacii

- 4

- 0

- Homework Statement
- Given the current time dependent density ##\mathbf{J}## find the EM field inside and outside a sphere.

- Relevant Equations
- ##\mathbf{J} = k(t) \delta(r-a) \sin \theta \hat{\boldsymbol{\phi}}##

So I tried to solve this using the Hertz potentials. I choose the magnetic one since this one corresponds to the magnetisation.

Before I start let me note that I denote a unit vector with a hat, while ##{x,y,z}## are the Cartesian coordinates and ##{r,\theta,\phi}## are the spherical coordinates.

First we need to relate the current to the magnetisation. Since for a surface current ##\mathbf{K}##, the magnetization satisfies the equation ##\mathbf{K} =\mathbf{M} \times \hat{\mathbf{n}}## we get

$$

\mathbf{M}|_{r =a} = k(t) \hat{\mathbf{z}}.

$$

The Hertz potential ##\boldsymbol{\Pi}_m## satisfies the equation

$$

\nabla^2 \boldsymbol{\Pi}_m - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \boldsymbol{\Pi}_m = -\mu_0 \mathbf{M}

$$

since the magnetiazion is non-zero only in the ##\hat{\mathbf{z}}## direction and utilizing the spherical symmetry of the problem I assumed that ##\boldsymbol{\Pi}_m = \Pi_z(r,t) \hat{\mathbf{z}}##, then the equation for the potential is

$$

\frac{1}{r} \frac{\partial^2}{\partial r^2} \left(r \Pi_z(r,t) \right) - \frac{1}{c^2} \frac{\partial^2}{\partial t^2 }\Pi_z(r,t) = -\mu_0 H(t) \delta (r-a).

$$

Now this is basically where I get stuck, I'm unable to solve this equations, I've tried combining the spherical Bessel functions so that the field is regular at the origin and zero at infinity (an outgoing wave), but I am not able to solve the equations above.

I assume when I would get the homogeneous equation for the potential I would then assume that the potential is continuous and impose the Maxwell equations by checking the boundary, note that ##\mathbf{A} = \nabla \times \boldsymbol{\Pi}## and ##\mathbf{B} = \nabla \times \nabla \times \boldsymbol{\Pi}_m##.

So the equations is basically what form of solutions satisfy the equation for the potential?

Edit: I've found an article which solves exactly the same problem avaiable on Arxiv, but they use Laplace transform and for me the derivation is quite unclear.

Before I start let me note that I denote a unit vector with a hat, while ##{x,y,z}## are the Cartesian coordinates and ##{r,\theta,\phi}## are the spherical coordinates.

First we need to relate the current to the magnetisation. Since for a surface current ##\mathbf{K}##, the magnetization satisfies the equation ##\mathbf{K} =\mathbf{M} \times \hat{\mathbf{n}}## we get

$$

\mathbf{M}|_{r =a} = k(t) \hat{\mathbf{z}}.

$$

The Hertz potential ##\boldsymbol{\Pi}_m## satisfies the equation

$$

\nabla^2 \boldsymbol{\Pi}_m - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \boldsymbol{\Pi}_m = -\mu_0 \mathbf{M}

$$

since the magnetiazion is non-zero only in the ##\hat{\mathbf{z}}## direction and utilizing the spherical symmetry of the problem I assumed that ##\boldsymbol{\Pi}_m = \Pi_z(r,t) \hat{\mathbf{z}}##, then the equation for the potential is

$$

\frac{1}{r} \frac{\partial^2}{\partial r^2} \left(r \Pi_z(r,t) \right) - \frac{1}{c^2} \frac{\partial^2}{\partial t^2 }\Pi_z(r,t) = -\mu_0 H(t) \delta (r-a).

$$

Now this is basically where I get stuck, I'm unable to solve this equations, I've tried combining the spherical Bessel functions so that the field is regular at the origin and zero at infinity (an outgoing wave), but I am not able to solve the equations above.

I assume when I would get the homogeneous equation for the potential I would then assume that the potential is continuous and impose the Maxwell equations by checking the boundary, note that ##\mathbf{A} = \nabla \times \boldsymbol{\Pi}## and ##\mathbf{B} = \nabla \times \nabla \times \boldsymbol{\Pi}_m##.

So the equations is basically what form of solutions satisfy the equation for the potential?

Edit: I've found an article which solves exactly the same problem avaiable on Arxiv, but they use Laplace transform and for me the derivation is quite unclear.

Last edited: