- #1

euphoricrhino

- 22

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- TL;DR Summary
- "method of image charge in electrostatics" works because of uniqueness of Laplace equation solutions, what makes the "method of image current" work in magnetostatics?

Hi wise folks,

I am working through Jackson problems, and have just encountered problem 5.17:

It is pretty straightforward to show that the given image current distributions will satisfy the boundary conditions (both tangent and normal) at the ##z=0## plane. But my question is actually: "why is the method of image current even valid"?

Recall that the method of image charge works in electrostatics because of the uniqueness of Laplace equation solutions, in particular the Dirichlet boundary condition - once we have a virtual image charge producing the same potential on the conductor's boundary surface, we know immediately its effect is identical to the situation without image charge but only the conductor.

Even with conductors replaced by dielectrics, as in section 4.4, we can use the method of images since the solution is known to be uniquely expandable into legendre series given symmetry.

But can we say the same thing in magnetostatics? For the problem above, can we claim that once the "ansatz" image currents produce field that meet consistently at the boundary, these image currents will produce identical effect elsewhere (i.e., off the slab boundary)? Here we are not dealing with dirichlet boundary conditions unless ##\mu_r## is infinity or unity. For general ##\mu_r##, the boundary may not even be an equipotential surface (that is, the magnetic scalar potential). Without this condition, how is the method of image current even valid?

Please enlighten me.

Thanks!

I am working through Jackson problems, and have just encountered problem 5.17:

A current distribution ##\mathbf{J}(\mathbf{x})## exists in a medium of unit relative permeability adjacent to a semi-infinite slab of material having relative permeability ##\mu_r## and filling the halfspace ##z<0##.

Show that for ##z>0## the magnetic induction can be calculated by replacing the medium of permeability ##\mu_r## by an image current distribution ##\mathbf{J}^*##, with components

\begin{align*}

\left(\frac{\mu_r-1}{\mu_r+1}\right)J_x(x,y,-z) && \left(\frac{\mu_r-1}{\mu_r+1}\right)J_y(x,y,-z) && -\left(\frac{\mu_r-1}{\mu_r+1}\right)J_z(x,y,-z)

\end{align*}

and that for ##z<0## the magnetic induction appears to be due to a current distribution ##2\mu_r\mathbf{J}/(\mu_r+1)## in a medium of unit relative permeability.

It is pretty straightforward to show that the given image current distributions will satisfy the boundary conditions (both tangent and normal) at the ##z=0## plane. But my question is actually: "why is the method of image current even valid"?

Recall that the method of image charge works in electrostatics because of the uniqueness of Laplace equation solutions, in particular the Dirichlet boundary condition - once we have a virtual image charge producing the same potential on the conductor's boundary surface, we know immediately its effect is identical to the situation without image charge but only the conductor.

Even with conductors replaced by dielectrics, as in section 4.4, we can use the method of images since the solution is known to be uniquely expandable into legendre series given symmetry.

But can we say the same thing in magnetostatics? For the problem above, can we claim that once the "ansatz" image currents produce field that meet consistently at the boundary, these image currents will produce identical effect elsewhere (i.e., off the slab boundary)? Here we are not dealing with dirichlet boundary conditions unless ##\mu_r## is infinity or unity. For general ##\mu_r##, the boundary may not even be an equipotential surface (that is, the magnetic scalar potential). Without this condition, how is the method of image current even valid?

Please enlighten me.

Thanks!