Question

The capacitance of a spherical capacitor is \( 100 \) pF. If the spacing between the two spheres is \( 1 \) cm, then the radius of the inner sphere of the capacitor is:

• A. \( 9 \) cm
• B. \( 10 \) cm
• C. \( 19 \) cm
• D. \( 20 \) cm

Hint:

For spherical capacitors, approximate using: \[ C \approx 4\pi \epsilon_0 \frac{r_1 r_2}{r_2 - r_1} \] and solve for \( r_1 \).

Answer 1

The Correct Option is B

Step 1: Formula for Capacitance of a Spherical Capacitor The capacitance of a spherical capacitor is given by: \[ C = 4\pi \epsilon_0 \frac{r_1 r_2}{r_2 - r_1} \] where: - \( C = 100 \) pF, - \( r_2 - r_1 = 1 \) cm, - \( \epsilon_0 = 8.85 \times 10^{-12} \) F/m. Step 2: Solving for \( r_1 \) Approximating for large radius: \[ C = 4\pi \epsilon_0 \times r_1 \] \[ 100 \times 10^{-12} = 4\pi \times (8.85 \times 10^{-12}) r_1 \] \[ r_1 = \frac{100 \times 10^{-12}}{4\pi \times 8.85 \times 10^{-12}} \] \[ \approx 10 \text{ cm} \] Conclusion Thus, the correct answer is: \[ 10 \text{ cm} \]