# UY1: Capacitance Of Spherical Capacitor

Two concentric spherical conducting shells are separated by vacuum. The inner shell has total charge +Q and outer radius $r_{a}$, and outer shell has charge -Q and inner radius $r_{b}$. Find the capacitance of the spherical capacitor.

Consider a sphere with radius r between the two spheres and concentric with them as Gaussian surface. From Gauss’s Law,

\begin{aligned} EA &= \frac{q}{\epsilon_{0}} \\ E \times 4 \pi r^{2} &= \frac{Q}{\epsilon_{0}} \\ E &= \frac{Q}{4 \pi \epsilon_{0} r^{2}} \end{aligned}

To find V, we use integration on E:

\begin{aligned} E &= \frac{Q}{4 \pi \epsilon_{0} r^{2}} \\ – \frac{dV}{dr} &= \frac{Q}{4 \pi \epsilon_{0} r^{2}} \\ – \int\limits_{0}^{V} dV’ &= \int\limits_{0}^{r} \frac{Q}{4 \pi \epsilon_{0} r^{2}} \, dr’ \\ V &= \frac{Q}{4 \pi \epsilon_{0} r} \end{aligned}

Hence,

\begin{aligned} V_{ab} &= V_{a} – V_{b} \\ &= \frac{Q}{4 \pi \epsilon_{0} r_{a}} – \frac{Q}{4 \pi \epsilon_{0} r_{b}} \\ &= \frac{Q}{4 \pi \epsilon_{0}} \left( \frac{r_{b} – r_{a}}{r_{a}r_{b}} \right) \end{aligned}

The capacitance is then:

\begin{aligned} C &= \frac{Q}{V_{ab}} \\ &= 4 \pi \epsilon_{0} \frac{r_{a} r_{b}}{r_{b} – r_{a}} \end{aligned}

Next: Capacitance Of A Cylindrical Capacitor

Previous: Energy Stored In Capacitors

Back To Electromagnetism (UY1)

##### Mini Physics

As the Administrator of Mini Physics, I possess a BSc. (Hons) in Physics. I am committed to ensuring the accuracy and quality of the content on this site. If you encounter any inaccuracies or have suggestions for enhancements, I encourage you to contact us. Your support and feedback are invaluable to us. If you appreciate the resources available on this site, kindly consider recommending Mini Physics to your friends. Together, we can foster a community passionate about Physics and continuous learning.

This site uses Akismet to reduce spam. Learn how your comment data is processed.