A spherical cavity of radius R is cut in an infinite dielectric which has dielectric constant ε.

A spherical cavity of radius R is cut in an infinite...

Key Concepts

Dielectric Constant

This is a measure of how a material polarizes in response to an applied electric field, thereby reducing the effective field within the medium compared to the vacuum. In problems like this, the dielectric constant (often denoted by ?) plays a crucial role in determining how the field propagates in the material and how boundary conditions at interfaces affect the overall field configuration.

Boundary Conditions at Dielectric Interfaces

When an interface exists between different media, such as between an infinite dielectric and a cavity, the electric fields on either side must satisfy specific continuity conditions. These conditions ensure that the normal component of the electric displacement and the tangential component of the electric field remain continuous across the boundary, which is key to correctly determining the field distribution inside the cavity.

Conductors in Electric Fields

Conductors respond to external electric fields by redistributing their free charges such that the internal electric field is zero. When a small conducting sphere is placed in an external field, it develops an induced surface charge distribution. This redistribution gives rise to an induced dipole moment, which is central to the solution of the problem.

Induced Dipole Moment

The induced dipole moment is a vector quantity that represents the separation of positive and negative charges in a system subjected to an external electric field. In the context of a conducting sphere in an external field, the sphere polarizes by accumulating surface charges, and the resulting dipole moment depends on the geometry of the sphere, the applied field, and the surrounding medium. Calculating this moment involves combining the effects of the external field with the modifications due to the dielectric environment.

Polarization in Dielectrics

Polarization refers to the alignment of bound charges within a dielectric material in response to an external electric field. This induced polarization creates additional electric fields that may either augment or reduce the applied field, depending on the geometry and material properties. Understanding polarization is essential when considering how an external electric field interacts with a composite system including both a dielectric medium and embedded conductive objects.

Transcript

00:02 When working with a linear dialectic, there are two things to keep in mind.

00:08 First is that there is a direct relationship between the d field and the electric field, the displacement field, and the electric field.

00:19 They are related through the dielectric constant epsilon, which is a combination of relative permittivity and the permittivity of free space.

00:32 And that relative permittivity is for the material just where it is polar, dielectric, i should say.

00:44 The other thing to keep in mind is that d is impervious to the polarization, and so it depends just on free charge.

00:59 So one of the important pictures to get is that there is no electric field in a linear dielectric, unless there is a group of free charges around creating a d field.

01:15 So our goal is to find the potential inside a dielectric that has an embedded free charge density, uniform embedded free charge density.

01:31 And what we can do is we can use that free charge density to find the d field, both inside and outside the sphere.

01:41 And then we may use the usual apparatus to find our potential.

01:49 So because of the spherical symmetry, we can use galses law to determine the d field.

02:00 So the flux of d through a spherical surface is going to be equal to the free charge enclosed i'll try to summarize that free charge and closed.

02:18 And outside the sphere, not too surprisingly, we're going to draw a spherical surface outside, and we may find the d field out there, and we may find it on the inside.

02:40 They're supposed to be spherical, a little wobbly there, but hopefully that gives the idea.

02:50 So outside, d times 4 pi r squared, is going to equal to the entire pre -charge, which is the density times the full volume of the sphere.

03:19 And inside, the amount of enclosed charge is going to depend on the radius of the gaussian surface, or a third's pie, little r cubed.

03:38 And so the d inside, just as we're used to finding for the electric field of a uniform ball is going to grow with radius.

03:52 And not too surprisingly, the electric field outside is just going to look like that of a point charge.

04:05 And inside, it's going to look like the electric field of a uniform distribution, with an important difference being the relative permittivity in the denominator.

04:27 Okay, now we want to find the potential of this sphere at its center, so v at zero.

04:44 Like usual, it's easier to start somewhere outside plus infinity and go in one.

04:52 Words from there.

04:56 Okay, so we can do our integral from infinity, and we can quickly recover a result for the surface...

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