# Boundary condition for electrostatics problem - found issue?

• vogtster
In summary, the conversation discusses the problem of finding a scalar potential in a system with different dielectric constants. The main question is whether the assumption of different dielectric constants leads to an inconsistency in the system. The conversation suggests that an iterative solution may not converge and that GMRES may be the best option for finding a reasonable solution.
vogtster
Hey everyone

Just a picture of my configuration.

The assumption here is $$\epsilon_a,\epsilon_b,\epsilon_c$$ are different from one another. Really the interest of this problem is to find the scalar potential $$\phi$$, such that $$\nabla^2 \phi = 0$$.

So now my question, about jump conditions,
Surface at $$y=0$$ has tangent $$\vec{E}$$ continous, thus
\begin{align}
-\hat{x} \cdot \nabla \phi_a = -\hat{x} \cdot \nabla \phi_b \\
-\hat{x} \cdot \nabla \phi_a = -\hat{x} \cdot \nabla \phi_c \\
\end{align}

However if we look at $$x=0$$ then normal $$\vec{D}$$ is continuous thus

\begin{align}
-\epsilon_b \hat{x} \cdot \nabla \phi_b = - \epsilon_c \hat{x} \cdot \nabla \phi_c \\
\end{align}

From our relation above this implies that $$\epsilon_b=\epsilon_c$$, which we made no such assumption. So this looks like a contradiction to me.

Can someone tell me where I have gone wrong?

Thank you!

Last edited:
Mathematically, there is also the solution ##\hat{x} \cdot \nabla \phi_a = 0##.

There might a smooth solution that has a violation of the boundary conditions in an arbitrarily small region where the three materials cross, and the actual fields then will depend on the non-exact material structure there.

mfb said:
Mathematically, there is also the solution ##\hat{x} \cdot \nabla \phi_a = 0##.

There might a smooth solution that has a violation of the boundary conditions in an arbitrarily small region where the three materials cross, and the actual fields then will depend on the non-exact material structure there.

Hey mfb,

Thanks for the response. I guess I'm in the game of solving these problems by numerical methods. I suppose my worry right now is if I implement this with $$\epsilon_b \neq \epsilon_c$$, then there is an underlying inconsistency in the system. You know how I can get around this?

You can check if an iterative solution converges to something stable.

mfb said:
You can check if an iterative solution converges to something stable.

Hi mfb,

I do not believe it would converge, or at the very best converge slowly, the underlying assumption would be that the matrix is well conditioned. Let's assume we did some sort of finite differencing and obtained an $$Ax=b$$ system. This inconsistency in the equations, will cause $$A^{-1}$$ not to exist analytically. Thus numerically, $$A$$ will be ill-conditioned, so iterative linear methods will converge slowly in order to find a $$x$$ such that $$Ax=b$$. Do you think GMRES, would be the best hope to find something reasonable?

Last edited:
I would put it in a program and see what happens.

## 1. What are boundary conditions in electrostatics problems?

Boundary conditions are constraints or specifications that are applied to the surfaces or boundaries of a system in an electrostatics problem. They determine the behavior of electric fields and potentials at these boundaries.

## 2. How do boundary conditions affect the solution of an electrostatics problem?

Boundary conditions play a crucial role in determining the unique solution to an electrostatics problem. They provide the necessary information for solving the equations that describe the behavior of electric fields and potentials in a given system.

## 3. What is the significance of boundary conditions in electrostatics experiments?

Boundary conditions are essential in electrostatics experiments as they help in setting up the experimental conditions accurately. They ensure that the electric fields and potentials at the boundaries of the experimental setup are known, which is necessary for obtaining accurate results.

## 4. What are the types of boundary conditions used in electrostatics problems?

The two main types of boundary conditions used in electrostatics problems are Dirichlet boundary conditions and Neumann boundary conditions. Dirichlet boundary conditions specify the value of the electric potential at a given boundary, while Neumann boundary conditions specify the normal derivative of the potential at the boundary.

## 5. How do you handle boundary conditions for complex geometries in electrostatics problems?

For complex geometries, boundary conditions can be handled using numerical methods such as finite element analysis or boundary element methods. These methods allow for the accurate simulation of electric fields and potentials in complex systems with varying boundary conditions.

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