Question

The capacitance of an isolated sphere of radius \( r_1 \) is increased by 5 times when enclosed by an earthed concentric sphere of radius \( r_2 \). The ratio \( \frac{r_1}{r_2} \) is:

• A. \( \frac{4}{5} \)
• B. \( \frac{5}{4} \)
• C. \( \frac{5}{1} \)
• D. \( \frac{1}{5} \)

Hint:

For concentric spherical capacitors, the capacitance increases when the outer sphere is grounded. Use \( C_2 = \frac{4 \pi \varepsilon_0 r_1 r_2}{r_2 - r_1} \) to find the ratio.

Answer 1

The Correct Option is A

Step 1: Apply Capacitance Formula for Concentric Spheres The capacitance of an isolated sphere is: \[ C_1 = 4 \pi \varepsilon_0 r_1 \] When enclosed by a conducting sphere at radius \( r_2 \), the new capacitance is: \[ C_2 = \frac{4 \pi \varepsilon_0 r_1 r_2}{r_2 - r_1} \] Given that \( C_2 = 5C_1 \), we get: \[ \frac{r_1 r_2}{r_2 - r_1} = 5 r_1 \] Step 2: Solve for \( \frac{r_1}{r_2} \) \[ \frac{r_1}{r_2} = \frac{4}{5} \] Thus, the correct answer is \( \frac{4}{5} \).

Answer 2

Given:
- An isolated conducting sphere of radius \( r_1 \) has capacitance \( C_1 \).
- When enclosed by an earthed concentric sphere of radius \( r_2 \), its capacitance increases by 5 times, so new capacitance \( C_2 = 5 C_1 \).

We need to find the ratio \( \frac{r_1}{r_2} \).

Step 1: Capacitance of an isolated sphere:
\[ C_1 = 4 \pi \varepsilon_0 r_1 \] where \( \varepsilon_0 \) is the permittivity of free space.

Step 2: Capacitance of the inner sphere enclosed by an earthed concentric spherical shell:
\[ C_2 = 4 \pi \varepsilon_0 \frac{r_1 r_2}{r_2 - r_1} \]

Step 3: Using the relation \( C_2 = 5 C_1 \), substitute:
\[ 4 \pi \varepsilon_0 \frac{r_1 r_2}{r_2 - r_1} = 5 \times 4 \pi \varepsilon_0 r_1 \]

Cancel \( 4 \pi \varepsilon_0 r_1 \) from both sides:
\[ \frac{r_2}{r_2 - r_1} = 5 \]

Step 4: Rearranging:
\[ r_2 = 5 (r_2 - r_1) \Rightarrow r_2 = 5 r_2 - 5 r_1 \] \[ 5 r_1 = 5 r_2 - r_2 = 4 r_2 \] \[ \frac{r_1}{r_2} = \frac{4}{5} \]

Therefore, the ratio \( \frac{r_1}{r_2} \) is:
\[ \boxed{\frac{4}{5}} \]