A spherical capacitor consists of two concentric conducting shells. The inner shell has radius $R_1=2,\text{cm}$ and the outer shell has radius $R_2=8,\text{cm}$, and the space between them is filled with a dielectric of relative permittivity $\kappa =3$. (a) Find the capacitance of the capacitor. (b) If the capacitor is charged to a potential difference of $250,\text{V}$, calculate the energy stored in it. Use ${\epsilon }_0=8.85\times 10^{-12},{\text{F m}}^{-1}$.

Step-by-step solution

1. Convert the radii to metres: $R_1=0.02,\text{m}$, $R_2=0.08,\text{m}$.
2. Use the formula for a spherical capacitor with a dielectric: $C=4pivarepsilo{n}_0\kappa \frac{R_1{R}_2}{R_2-R_1}$.
3. Compute the numerator: $4pivarepsilo{n}_0\kappa R_1{R}_2=4\pi (8.85\times 10^{-12})(3)(0.02)(0.08)$.
4. Evaluate: $4pivarepsilo{n}_0\kappa R_1{R}_2\approx 5.33\times 10^{-13},\text{F}$.
5. Divide by $(R_2-R_1)=0.06,\text{m}$: $Capprox8.89\times 10^{-12},\text{F}$.
6. Use the energy formula for a capacitor: $U=\frac{1}{2}C{V}^2$.
7. Substitute $C=8.89\times 10^{-12},\text{F}$ and $V=250,\text{V}$.
8. Compute $U=0.5\times 8.89\times 10^{-12}\times (250{)}^2\approx 2.78\times 10^{-7},\text{J}$.

Concept tested: Spherical capacitor with dielectric

Published 16 September 2026 · How this solution was prepared and checked