# Boundary condition for dielectric sphere

• lonewolf219
In summary, the potential across the boundary of a dielectric sphere embedded in a dielectric material is continuous, allowing for the potential inside the sphere to be set equal to the potential outside at r=R. This is because the most that can be present at the boundary is a finite surface charge density, which results in a finite electric field. Despite the electric field pointing in opposite directions on either side of the boundary, the change in potential is infinitesimal. This was confirmed in a debate during a class discussion.
lonewolf219
Is the potential across the boundary continuous for a dielectric sphere embedded in a dielectric material, so that the potential inside the sphere can be set equal to the potential outside of it at r=R ?

Yes. And the reason is at the boundary the most you will get is a surface charge density ## \sigma_p ##. The electric field from an infinite sheet of surface charge, which is what ## \sigma_p ## will look like at very close range is finite, even though the field from the surface charge will point in opposite directions on opposite sides of the sheet of surface charge. Thereby, the change in potential as one traverses an infinitesimal distance across the boundary of surface charge is also infinitesimal. (The electric field sees a discontinuity as one crosses the boundary, but not the potential.)

## 1. What is a boundary condition for a dielectric sphere?

A boundary condition for a dielectric sphere refers to the relationship between the electric field and the charge distribution on the surface of the sphere. This boundary condition helps to determine the electric field inside and outside of the sphere.

## 2. How is the boundary condition for a dielectric sphere derived?

The boundary condition for a dielectric sphere is derived by applying Gauss's law to a spherical Gaussian surface, which encloses the sphere. This allows us to relate the electric field to the charge distribution on the surface of the sphere.

## 3. What are the boundary conditions for a dielectric sphere with a uniform charge distribution?

If the charge distribution on the surface of the dielectric sphere is uniform, then the boundary condition is given by the continuity of the electric field and the continuity of the normal component of the electric displacement vector at the interface between the dielectric and the surrounding medium.

## 4. How does the boundary condition for a dielectric sphere change with a non-uniform charge distribution?

If the charge distribution on the surface of the dielectric sphere is non-uniform, then the boundary condition becomes more complex. In addition to the continuity of the electric field and the normal component of the electric displacement vector, the boundary condition also includes a surface charge density term that accounts for the non-uniform distribution of charge.

## 5. Can the boundary condition for a dielectric sphere be applied to other shapes?

While the boundary condition for a dielectric sphere specifically refers to a spherical shape, similar principles can be applied to other shapes. For example, a boundary condition for a dielectric cylinder would involve integrating over a cylindrical Gaussian surface, while a boundary condition for a dielectric cube would involve integrating over a cubic Gaussian surface.

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