Question

A capacitor is formed from two concentric spherical conducting shells separated by vacuum. The inner sphere has radius 12.5 cm, and the outer sphere has radius 14.8 cm. A potential difference of 120 V is applied to the capacitor. (a) What is the energy density at r = 12.6 cm, just outside the inner sphere? (b) What is the energy density at r = 14.7 cm, just inside the outer sphere? (c) For a parallel-plate capacitor the energy density is uniform in the region between the plates, except near the edges of the plates. Is this also true for a spherical capacitor?

Short answer

(a)The energy density at$r=12.6$ just outside the inner sphere is:$1.63\times {10}^{-4}J.m^{-}3$

(b) The energy density at$r=12.6$ just outside the inner sphere is$8.8\times {10}^{-5}J.m^{-3}$

(c) No, the energy density is not uniformly distributed in the region between the plates except the near the edges of the plate.

Step-by-step solution

Calculating the density

Given,

$$\begin{gathered} r_b=14.8cm \\ {r}_a=12.5cm \\ V=120V \end{gathered}$$

Now, to get density we need to find the charge for hollow spheres, and the potential difference between the surface is given by:

$V_{ab}=kQ\left(\frac{1}{r_a}-\frac{1}{r_b}\right)$

Substituting the values, we get:

$$\begin{gathered} Q=\frac{V_{ab}}{k\left({\displaystyle \frac{1}{r_a}}-{\displaystyle \frac{1}{r_b}}\right)}=\frac{120}{9\times {10}^9\left({\displaystyle \frac{1}{0.125}}-{\displaystyle \frac{1}{0.148}}\right)} \\ =10.72\times {10}^{-9}C \end{gathered}$$

Now, the density is given by:

$u=\frac{{\mathit{Q}}^{\mathit{2}}}{\mathit{32}{\mathit{\pi }}^{\mathit{2}}{\mathit{\epsilon }}_{\mathit{0}}{\mathit{r}}^{\mathit{4}}}$

Substituting the values, we get:

$$\begin{gathered} u=\frac{{\left(10.72\times {10}^{-9}\right)}^2}{32{\mathrm{\pi }}^2\times 8.85\times {10}^{-12}\times {\left(0.126\right)}^4} \\ =1.63\times {10}^{-4}J.m^{-3} \end{gathered}$$

Therefore, the density is$1.63\times {10}^{-4}J.m^{-3}$

Energy distribution

We, know that density is given by:

$$\begin{gathered} u=\frac{{\left(10.27\times {10}^{-9}\right)}^2}{32{\mathrm{\pi }}^2\times 8.85\times {10}^{-12}\times {\left(0.147\right)}^4} \\ =8.8\times {10}^{-5}J.m^{-3} \end{gathered}$$

Therefore, we can conclude that the energy density is not uniformly distributed, but decreases while being close to the outer shell.