# Boundary condition for a charged surface

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## Main Question or Discussion Point

Hi,

I am looking at a problem where I have two electrically conducting fluids where charge accrues on the interface, I know that one of the equations that I have to use comes straight from the usual boundary conditions for the normal component of the electric field, the other one apparent comes from integrating the continuity equation:
$$\frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf{J}=0$$

around a closed surface ending up with a covariant derivative, in all the papers I have seen the equation simply quoted and not derived, I am interested in it's derivation.

Can anyone suggest any papers or give be a few pointers?

Mat

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Jano L.
Gold Member
Hello Mat,

the equation of continuity is a mathematical formulation of the experience that electric charges moves continuously in space, the total charge of isolated system of bodies is constant. It is not derived from anything simpler.

Maxwell used this law to argue that there should be new term $\partial \mathbf D/\partial t$ in the equation stating the Ampere law.

The electromagnetism of continuous media is treated briefly in Landau&Lifgarbagez, Electrodynamics of continuous media, and there is also an important book

P. Penfield, H. A. Haus, Eletrodynamics of moving media, Cambridge, MIT, 1967

which I would like to get access to, but so far I didn't have luck.

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Jano L.
Gold Member

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Anyone else care to comment? The equation I understand involves a covariant derivative.

I understand that the model in question is the leaky dielectric model.

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