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Suppose that $$ \vec\nabla \cdot \vec J = 0. $$ What I know about the boundary condition is, that form the normal direction $$ J_{1n}=J_{2n} $$ and for the tangential direction $$ \dfrac{J_{1t}}{\sigma_1} = \dfrac{J_{2t}}{\sigma_2}. $$

But I read that $$ \langle n\cdot J\rangle=-\nabla_s \cdot C $$ where $\langle \cdot \rangle$ means the difference across the boundary, and the right hand side means the 2D surface divergence on surface current density $C$.

I would like to ask which situation will make this boundary condition?

Thank you!

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    $\begingroup$ "where means the difference in the normal direction is the 2-D surface divergence on surface current density." Please fix this sentence $\endgroup$ Sep 4, 2019 at 6:43
  • $\begingroup$ Sorry for the unclear expression. I fixed. $\endgroup$
    – Yui
    Sep 4, 2019 at 7:13
  • $\begingroup$ I’m not sure I understand the question. What is the relationship between the last three equations and which boundary condition are you referring to? $\endgroup$ Sep 4, 2019 at 7:15
  • $\begingroup$ For the first equation, I think that means the current density is continuous in normal direction, so the difference is zero. However, the third equation seems the current density is discontinuous, so I would like to know which situation will make this discontinuity. The second equation is just a supplement, not have the direct relation in this question. $\endgroup$
    – Yui
    Sep 4, 2019 at 7:24
  • $\begingroup$ Or maybe I misunderstand the first equation? Doesn't it mean the continuity? Thank you very much again! $\endgroup$
    – Yui
    Sep 4, 2019 at 7:25

1 Answer 1

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The first boundary condition you gave assumes that there is no surface current density in the boundary. This is often a perfectly reasonable assumption, unless there is a very thin and conductive layer at the boundary of the two media.

Your last equation assumes that the boundary can support a non-zero surface charge density. This means that the continuity equation no longer requires the normal components of the current density on either side of the boundary to be equal. If it happens that there is a net current flow into a small area of the boundary layer from the two sides of boundary, the same current must flow sideways within the boundary layer, meaning the surface current density must have a non-zero divergence. Your last equation is a mathematical statement of this.

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  • $\begingroup$ I have one further question. Suppose now the condition is that medium 1 is nearly a perfect conductor, medium 2 is nearly a insulator, and the boundary is conductive layer which allows ohmic diffusion. Even the surface current density could be set initially, it will diffuse later. Then normal components of current density will be continuous. But no matter it is in the initial time or later, it always allows that can have non-zero charge density in the boundary (if the ratio of permittivity and conductivity of both medium are not equal), these charges are just from moving state to stop. $\endgroup$
    – Yui
    Sep 5, 2019 at 9:55
  • $\begingroup$ But because the current can't penetrate to medium 2, as long as new current is generated by media 1, they can flow through the boundary layer by the existing charge density, which again showed that the required charge density in the boundary layer. Is this statement correct? Thank you! @Puk $\endgroup$
    – Yui
    Sep 5, 2019 at 9:55
  • $\begingroup$ From what I can understand, I think you have the right idea. Medium 1 can have a non-zero normal component of the current density if either the rate of change of the surface charge density or the divergence of the surface current density is non-zero. $\endgroup$
    – Puk
    Sep 5, 2019 at 10:16
  • $\begingroup$ Specifically, the continuity equation at the boundary reads (assuming medium 2 is not conducting) $\frac{\partial \rho_s}{\partial t} + J_n + \nabla_\parallel · \vec{K} = 0$, where $\rho_s$ is the surface charge density, $J_n$ is the normal component of the current density into medium 1, $\vec{K}$ is the surface current density and $\nabla_\parallel ·$ denotes the 2D divergence in the boundary. $\endgroup$
    – Puk
    Sep 5, 2019 at 10:17
  • $\begingroup$ Actually what I know about continuity equation is simply like this : maxwells-equations.com/equations/continuity.php. And I want to learn more about continuity equation on boundary condition and how to derive (like you mentioned above), could you give some reference? Thank you! @Puk $\endgroup$
    – Yui
    Sep 5, 2019 at 14:27

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