Three long concentric conducting cylindrical shells have radii R, 2R and `2sqrt2 R` Inner and outer shells are connected to each other. The capacitance across middle and inner shells per unit length is:

Options

A. `(1/3in_0)/(ln2)`
B. `(6piin_0)/(ln2)`
C. `(piin_0)/(ln2)`
D. None

Answer

To find the capacitance across the middle and inner cylindrical shells per unit length, we can follow these steps:

### Step 1: Identify the Radii
We have three concentric cylindrical shells with the following radii:
- Inner shell (R1) = R
- Middle shell (R2) = 2R
- Outer shell (R3) = 2√2 R

### Step 2: Use the Capacitance Formula for Cylindrical Capacitors
The capacitance \( C \) per unit length for a cylindrical capacitor is given by the formula:
\[ C = \frac{2\pi \epsilon_0}{\ln\left(\frac{R_2}{R_1}\right)} \]
where \( R_1 \) is the inner radius and \( R_2 \) is the outer radius.

### Step 3: Calculate Capacitance \( C_1 \) between Inner and Middle Shells
For the capacitance \( C_1 \) between the inner shell (R) and the middle shell (2R):
- \( R_1 = R \)
- \( R_2 = 2R \)

Substituting these values into the formula:
\[ C_1 = \frac{2\pi \epsilon_0}{\ln\left(\frac{2R}{R}\right)} = \frac{2\pi \epsilon_0}{\ln(2)} \]

### Step 4: Calculate Capacitance \( C_2 \) between Middle and Outer Shells
For the capacitance \( C_2 \) between the middle shell (2R) and the outer shell (2√2 R):
- \( R_1 = 2R \)
- \( R_2 = 2\sqrt{2}R \)

Substituting these values into the formula:
\[ C_2 = \frac{2\pi \epsilon_0}{\ln\left(\frac{2\sqrt{2}R}{2R}\right)} = \frac{2\pi \epsilon_0}{\ln\left(\sqrt{2}\right)} = \frac{2\pi \epsilon_0}{\frac{1}{2}\ln(2)} = \frac{4\pi \epsilon_0}{\ln(2)} \]

### Step 5: Combine the Capacitances
Since the capacitors \( C_1 \) and \( C_2 \) are in parallel, the total capacitance \( C_{total} \) is given by:
\[ C_{total} = C_1 + C_2 \]
Substituting the values we calculated:
\[ C_{total} = \frac{2\pi \epsilon_0}{\ln(2)} + \frac{4\pi \epsilon_0}{\ln(2)} = \frac{6\pi \epsilon_0}{\ln(2)} \]

### Final Answer
Thus, the capacitance across the middle and inner shells per unit length is:
\[ C_{total} = \frac{6\pi \epsilon_0}{\ln(2)} \]

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