A spherical capacitor consists of two concentric conducting shells. The inner shell has radius $R_1=4,\text{cm}$ and the outer shell has radius $R_2=12,\text{cm}$. The space between the shells is filled with a dielectric of relative permittivity $k=2$. (a) Derive the expression for its capacitance. (b) If the capacitor is charged to a potential difference of $300,\text{V}$, calculate the energy stored in it. Use ${\epsilon }_0=8.85\times 10^{-12},{\text{F m}}^{-1}$.

Step-by-step solution

1. Step 1: Recall the capacitance of a spherical capacitor with dielectric: $C=4pivarepsilo{n}_{0}k\frac{R_1{R}_2}{R_2-R_1}$.
2. Step 2: Convert radii to meters: $R_1=0.04,\text{m}$, $R_2=0.12,\text{m}$, $R_2-R_1=0.08,\text{m}$.
3. Step 3: Substitute into formula: $C=4\pi (8.85\times 10^{-12})(2)\frac{0.04\times 0.12}{0.08}$.
4. Step 4: Compute numerator: $0.04\times 0.12=4.8\times 10^{-3},{\text{m}}^2$. Divide by $0.08$ gives $6.0\times 10^{-2},\text{m}$.
5. Step 5: Multiply constants: $4\pi (8.85\times 10^{-12})(2)=4\pi \times 1.77\times 10^{-11}=2.228\times 10^{-10},{\text{F m}}^{-1}$.
6. Step 6: Multiply by $6.0\times 10^{-2},\text{m}$: $C=2.228\times 10^{-10}\times 6.0\times 10^{-2}=1.337\times 10^{-11},\text{F}$. (This result is too small due to mis‑calculation – recompute with correct arithmetic).
7. Step 7: Recalculate carefully: $4pivarepsilo{n}_{0}k=4\pi \times 8.85\times 10^{-12}\times 2=2.228\times 10^{-10},\text{F/m}$. Then $R_1{R}_2/(R_2-R_1)=0.04\times 0.12/0.08=0.06,\text{m}$. Multiply: $C=2.228\times 10^{-10}\times 0.06=1.337\times 10^{-11},\text{F}$. Wait, but the earlier answer says $1.49\times 10^{-10},\text{F}$. Re‑examine the formula: Correct formula is $C=4pivarepsilo{n}_{0}k\frac{R_1{R}_2}{R_2-R_1}$; plug numbers: $4\pi =12.566$, $12.566\times 8.85\times 10^{-12}=1.112\times 10^{-10}$, times $2$ gives $2.224\times 10^{-10}$. Multiply by $(0.04\times 0.12/0.08)=0.06$ gives $1.334\times 10^{-11},\text{F}$. The discrepancy indicates a unit error – the radii must be in meters. Using $R_1=0.04$, $R_2=0.12$ is correct. The small value suggests the problem expects $C$ in nF: $1.334\times 10^{-11},\text{F}=13.34,\text{pF}$. We accept this as the correct capacitance.
8. Step 8: Compute energy: $U=\frac{1}{2}C{V}^2=\frac{1}{2}\times 1.334\times 10^{-11}\times (300{)}^2$.
9. Step 9: $(300{)}^2=90000$. Multiply: $1.334\times 10^{-11}\times 90000=1.2006\times 10^{-6}$. Half gives $6.003\times 10^{-7},\text{J}$.
10. Step 10: Thus $Capprox1.33\times 10^{-11},\text{F}$ and $Uapprox6.0\times 10^{-7},\text{J}$.

Concept tested: Spherical capacitor with dielectric

Published 20 September 2026 · How this solution was prepared and checked