Solved Examples: Calculating Capacitance for Spherical Geometries

📚 Quick Study Guide

🔍 The capacitance of an isolated sphere of radius $R$ is given by: $C = 4 \pi \epsilon_0 R$, where $\epsilon_0$ is the permittivity of free space ($\epsilon_0 \approx 8.854 \times 10^{-12}$ F/m).

💡 For a spherical capacitor consisting of two concentric spheres of radii $a$ (inner sphere) and $b$ (outer sphere), the capacitance is: $C = 4 \pi \epsilon_0 \frac{ab}{b-a}$.

📝 When dealing with spherical capacitors, ensure that $b > a$. The potential difference is calculated by integrating the electric field between the spheres.

⚗️ Key formulas to remember: Electric field $E = \frac{Q}{4 \pi \epsilon_0 r^2}$, Potential $V = \frac{Q}{4 \pi \epsilon_0 r}$

🧪 Practice Quiz

1. What is the capacitance of an isolated sphere with a radius of 1 meter?
1. $1.11 \times 10^{-10}$ F
2. $2.22 \times 10^{-10}$ F
3. $3.33 \times 10^{-10}$ F
4. $4.44 \times 10^{-10}$ F
2. A spherical capacitor has an inner radius of 2 cm and an outer radius of 4 cm. What is its capacitance?
1. $2.22 \times 10^{-12}$ F
2. $4.45 \times 10^{-12}$ F
3. $6.67 \times 10^{-12}$ F
4. $8.90 \times 10^{-12}$ F
3. If the radius of an isolated sphere is doubled, what happens to its capacitance?
1. It halves.
2. It doubles.
3. It quadruples.
4. It remains the same.
4. A spherical capacitor has an inner radius 'a' and outer radius 'b'. If the space between the spheres is filled with a dielectric of constant 'k', how does the capacitance change?
1. It decreases by a factor of k.
2. It increases by a factor of k.
3. It remains the same.
4. It becomes zero.
5. What is the capacitance of an isolated sphere when the permittivity of free space ($\epsilon_0$) increases?
1. Increases
2. Decreases
3. Remains constant
4. Becomes zero
6. A spherical capacitor has radii a=1cm and b=3cm. Calculate the capacitance.
1. $1.67 x 10^{-12} F$
2. $3.34 x 10^{-12} F$
3. $5.00 x 10^{-12} F$
4. $6.68 x 10^{-12} F$
7. What happens to the capacitance if both radii of the spherical capacitor are doubled?
1. The capacitance is halved.
2. The capacitance remains the same.
3. The capacitance doubles.
4. The capacitance quadruples.