Boundary condition for a charged surface

In summary, the conversation discusses the equation of continuity in the context of electrically conducting fluids and the accrual of charge at the interface. There is a need for a covariant derivative in the equation, but it is often quoted without derivation in papers. The books "Electrodynamics of Continuous Media" and "Electrodynamics of Moving Media" are recommended for further understanding, but access to them may be difficult.
  • #1
hunt_mat
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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:
[tex]
\frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf{J}=0
[/tex]

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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  • #2
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 [itex]\partial \mathbf D/\partial t[/itex] 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.
 
  • #3
I don't have access to these book either.
 
  • #5
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.
 
Last edited:

FAQ: Boundary condition for a charged surface

What is a boundary condition for a charged surface?

A boundary condition for a charged surface is a mathematical statement that describes how the electric potential, electric field, or electric charge is related at the interface between two different materials or regions.

Why is a boundary condition important for a charged surface?

Boundary conditions are important because they allow us to solve complex problems involving charged surfaces. They help us determine the behavior of electric charges and fields at the interface between two materials, and can provide insight into the physical properties of the materials themselves.

What are the different types of boundary conditions for a charged surface?

There are several types of boundary conditions for a charged surface, including the Dirichlet boundary condition, the Neumann boundary condition, and the mixed boundary condition. These conditions differ in how they specify the relationship between electric potential and electric field at the interface.

How do boundary conditions affect the behavior of a charged surface?

Boundary conditions play a crucial role in determining the behavior of a charged surface. They can affect the distribution of electric charges and fields at the interface, as well as the overall behavior of the surface in response to external influences such as applied electric fields or changes in material properties.

What factors should be considered when choosing a boundary condition for a charged surface?

When choosing a boundary condition for a charged surface, factors such as the geometry and materials involved, the desired accuracy of the solution, and any physical constraints or known symmetries should be taken into account. It is important to select a boundary condition that accurately reflects the real-world conditions and allows for a meaningful analysis of the problem at hand.

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