A thick spherical shell (inner radius a, outer radius b ) is...

A thick spherical shell (inner radius $a$, outer radius $b$ ) is made of dielectric material with a "frozen-in" polarization $$\mathbf{P}(\mathbf{r})=\frac{k}{r} \hat{\mathbf{r}},$$ where $k$ is a constant and $r$ is the distance from the center (Fig. 4.18). (There is no free charge in the problem.) Find the electric field in all three regions by two different methods: (a) Locate all the bound charge, and use Gauss's law (Eq. 2.13 ) to calculate the field it produces. (b) Use Eq. 4.23 to find $\mathbf{D}$, and then get $\mathbf{E}$ from Eq. 4.21 . [Notice that the second method is much faster, and avoids any explicit reference to the bound charges.]

A thick spherical shell (inner radius $a$, outer radius $b$ ) is made of dielectric material with a "frozen-in" polarization $$\mathbf{P}(\mathbf{r})=\frac{k}{r} \hat{\mathbf{r}},$$ where $k$ is a constant and $r$ is the distance from the center (Fig. 4.18). (There is no free charge in the problem.) Find the electric field in all three regions by two different methods:
(a) Locate all the bound charge, and use Gauss's law (Eq. 2.13 ) to calculate the field it produces.
(b) Use Eq. 4.23 to find $\mathbf{D}$, and then get $\mathbf{E}$ from Eq. 4.21 . [Notice that the second method is much faster, and avoids any explicit reference to the bound charges.]

Key Concepts

Polarization in Dielectrics

Polarization refers to the dipole moment per unit volume of a dielectric material. It describes how the material’s molecules align in an electric field (or may have a fixed, 'frozen-in' orientation), creating microscopic dipoles that contribute to the material’s response to electric fields. In problems like the spherical shell with a given spatially varying polarization, this concept is central to understanding how bound charge distributions are generated inside materials without any free charge present.

Bound Charge

Bound charges arise from the non-uniformity of a dielectric’s polarization. There are two types: volume bound charge density, determined by the negative divergence of the polarization field (-?·P), and surface bound charge density, obtained from the polarization’s normal component (P · n) at the material boundaries. These charges are not free to move but affect the electric field and are essential when applying Gauss’s law in media with polarization.

Gauss's Law

Gauss's law relates the electric flux through a closed surface to the charge enclosed by that surface. In the context of dielectrics, Gauss's law is used to compute the electric field produced by bound charges resulting from the polarization of the material. The law is particularly powerful in systems with high symmetry, such as the spherical shell, where it simplifies the calculation of the field in different regions by considering the net enclosed charge.

Electric Displacement Field (D)

The electric displacement field D combines the effects of the free charges and the bound charges due to the polarization in a dielectric material. It is defined as D = ??E + P, serving as an intermediary that can simplify electrostatics problems in media. In regions with no free charges, D's divergence vanishes, offering an alternative route to determine the electric field E, thus bypassing the need to explicitly compute the bound charges. This method can provide a more direct solution in cases with complex polarization profiles.

Transcript

00:01 So here we have a polarized spherical shell.

00:06 And what is good is that the polarization is radially symmetric.

00:12 And this suggests that we can use bound charges and solve for the electric field in different regions using galses law.

00:21 But before we do that, we're actually going to use the easy way.

00:26 So there is this relationship between deities.

00:33 Which is called the displacement field, and the electric field, epsilon not electric field plus polarization.

00:44 And what is true about d is it is impervious to bound charge.

00:52 So d does not depend, or i should say it depends only on free charge.

01:03 So typically if you had capacitors with pieces of metal, onto which charges could fly on and off, or point charges somehow embedded in the material, you would have a d field.

01:19 But because there is no free charge, in this case, none here, we can assume that the d is zero everywhere.

01:33 And that's not an assumption.

01:35 It is a truth that you can take to the bank.

01:40 So d is equal to zero everywhere.

01:46 And that means outside this sphere in its cavity, as well as in the polarized material shown in red.

01:56 And that means that the electric field is minus p over epsilon not everywhere.

02:06 And so this means that outside the polarized sphere where r is greater than b, the electric field collapses to zero, just like the d field, and r less than a, the electric field is equal to zero.

02:32 And in between where you have the material, the electric field is minus p over epsilon, not, which we can write as minus k over epsilon, r, and in the radial direction, if we want to be sneaky about it.

02:56 I think we'd say precise about it.

02:59 So that was the easy way.

03:01 And let's see if we can verify this using the second way.

03:06 We are going to find the surface bound charge, and there are two surfaces.

03:17 On the inside surface, we're grabbing onto the negative charge.

03:23 Or we can think about it as p pointing outwards and the normal pointing inwards.

03:32 So we have minus k, a squared.

03:46 A squared just minus k over a, at the end.

03:51 Inner surface, and it is equal to plus k times b at the outer surface.

04:07 So it is like a sphere that's been sapped with some negative charge on the inside, positive on the outside.

04:17 And the bound density charge density is minus the divergence of p, which we will use in spherical coordinates.

04:43 And there's just an r component.

04:47 And let's see, there's a k in there.

04:51 So the, let's see, the derivative just gives us the derivative of k times r.

05:00 We'll pull the k out.

05:05 And so we just have minus k over r squared.

05:13 Now to find the electric field, we are going to be using galses law.

05:21 So a reminder about galses law.

05:25 Spheres, it's pretty easy...

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