The spherical dielectric shell of inner radius a and outer radius b has the polarization vector P = k/r² r̂ + cosθθ̂ within the dielectric. a) Find the bound charge densities. b) Find the E field in the r ≤ a, a ≤ r ≤ b, and r > b regions.
Added by Albert G.
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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.]
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A sphere of radius $R$ carries a polarization $$\mathbf{P}(\mathbf{r})=k \mathbf{r},$$ where $k$ is a constant and $\mathbf{r}$ is the vector from the center. (a) Calculate the bound charges $\sigma_{b}$ and $\rho_{b}$. (b) Find the field inside and outside the sphere.
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