Question

14.2 A conducting sphere of radius $r_1$ is concentrically surrounded by a conducting shell with an inner radius $r_2$ ($> r_1$). The sphere has a uniform free-charge density $\sigma_f$ on its surface, and the outer conducting shell is grounded. The region between the two conductors is filled a dielectric that has dielectric constant $\epsilon_r$. (a) Using appropriate boundary conditions for the potential, find $V(r)$ in the region between the two conductors. (b) From your answer in (a) find the electric field $E(r)$ in the same region. (c) Find the polarization $P(r)$ of the dielectric. (d) Calculate the capacitance between the two conductors. Is the result what you expected? Explain. Answer: (a) $V(r) = \frac{\sigma_f r_1^2}{\epsilon_0 \epsilon_r} \left[\frac{1}{r} - \frac{1}{r_2}\right]$, (b) $E(r) = \frac{\sigma_f r_1^2}{\epsilon_0 \epsilon_r r^2} \hat{r}$, (c) $P(r) = \sigma_f \frac{\epsilon_r - 1}{\epsilon_r} \frac{r_1^2}{r^2} \hat{r}$ (d) $C = \frac{4\pi \epsilon_0 \epsilon_r}{1/r_1 - 1/r_2}$

          14.2 A conducting sphere of radius $r_1$ is concentrically surrounded by a conducting
shell with an inner radius $r_2$ ($> r_1$). The sphere has a uniform free-charge density $\sigma_f$
on its surface, and the outer conducting shell is grounded. The region between the two
conductors is filled a dielectric that has dielectric constant $\epsilon_r$.
(a) Using appropriate boundary conditions for the potential, find $V(r)$ in the region
between the two conductors.
(b) From your answer in (a) find the electric field $E(r)$ in the same region.
(c) Find the polarization $P(r)$ of the dielectric.
(d) Calculate the capacitance between the two conductors. Is the result what you
expected? Explain.
Answer:
(a) $V(r) = \frac{\sigma_f r_1^2}{\epsilon_0 \epsilon_r} \left[\frac{1}{r} - \frac{1}{r_2}\right]$, (b) $E(r) = \frac{\sigma_f r_1^2}{\epsilon_0 \epsilon_r r^2} \hat{r}$, (c) $P(r) = \sigma_f \frac{\epsilon_r - 1}{\epsilon_r} \frac{r_1^2}{r^2} \hat{r}$
(d) $C = \frac{4\pi \epsilon_0 \epsilon_r}{1/r_1 - 1/r_2}$

14.2 A conducting sphere of radius r1 is concentrically surrounded by a conducting
shell with an inner radius r2 (> r1). The sphere has a uniform free-charge density
on its surface, and the outer conducting shell is grounded. The region between the two
conductors is filled a dielectric that has dielectric constant .
(a) Using appropriate boundary conditions for the potential, find V(r) in the region
between the two conductors.
(b) From your answer in (a) find the electric field E(r) in the same region.
(c) Find the polarization P(r) of the dielectric.
(d) Calculate the capacitance between the two conductors. Is the result what you
expected? Explain.
Answer:
(a) V(r) = ( r1^2)/(ϵ0 )[(1)/(r) - (1)/(r2)], (b) E(r) = ( r1^2)/(ϵ0 r^2)r̂, (c) P(r) = ( - 1)/()(r1^2)/(r^2)r̂
(d) C = (4πϵ0 )/(1/r1 - 1/r2)

Added by Bryan V.

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