A cylindrical capacitor has has two co-axial cylinders of length 20 cm and radii 2r and r, inner cylinder is gliven a charge 10 `mu`C and and outer cylinder a charge of - 10 `mu`C. the potential difference between the two cylinders will be

Options

A. `(0.1 " In" 2 )/(4 pi epsilon_(0)) `m V
B. `( " In" 2 )/(4 pi epsilon_(0)) `m V
C. `(10 " In" 2 )/(4 pi epsilon_(0)) `m V
D. `(0.01 " In" 2 )/(4 pi epsilon_(0)) `m V

Answer

To find the potential difference between the two cylinders of a cylindrical capacitor, we can follow these steps:

### Step 1: Identify the given parameters
- Length of the cylindrical capacitor, \( L = 20 \, \text{cm} = 0.2 \, \text{m} \)
- Charge on the inner cylinder, \( Q = 10 \, \mu C = 10 \times 10^{-6} \, C \)
- Radius of the inner cylinder, \( r = R \)
- Radius of the outer cylinder, \( R_2 = 2R \)

### Step 2: Write the formula for capacitance of a cylindrical capacitor
The capacitance \( C \) of a cylindrical capacitor is given by the formula:
\[ C = \frac{2 \pi \epsilon_0 L}{\ln\left(\frac{R_2}{R_1}\right)} \]
where \( R_1 \) is the radius of the inner cylinder and \( R_2 \) is the radius of the outer cylinder.

### Step 3: Substitute the values into the capacitance formula
Here, \( R_1 = R \) and \( R_2 = 2R \). Thus, we can substitute these values into the capacitance formula:
\[ C = \frac{2 \pi \epsilon_0 (0.2)}{\ln\left(\frac{2R}{R}\right)} = \frac{2 \pi \epsilon_0 (0.2)}{\ln(2)} \]

### Step 4: Calculate the potential difference
The potential difference \( V \) between the two cylinders is given by the formula:
\[ V = \frac{Q}{C} \]
Substituting the expression for \( C \):
\[ V = \frac{Q}{\frac{2 \pi \epsilon_0 (0.2)}{\ln(2)}} = \frac{Q \cdot \ln(2)}{2 \pi \epsilon_0 (0.2)} \]
Now substituting \( Q = 10 \times 10^{-6} \, C \):
\[ V = \frac{(10 \times 10^{-6}) \cdot \ln(2)}{2 \pi \epsilon_0 (0.2)} \]

### Step 5: Simplify the expression
To simplify further:
\[ V = \frac{10 \times 10^{-6} \cdot \ln(2)}{0.4 \pi \epsilon_0} \]
\[ V = \frac{10^{-5} \cdot \ln(2)}{0.4 \pi \epsilon_0} \]

### Step 6: Final expression for potential difference
Thus, the potential difference \( V \) can be expressed as:
\[ V = \frac{0.1 \cdot \ln(2)}{4 \pi \epsilon_0} \, \text{volts} \]

### Conclusion
The potential difference between the two cylinders of the cylindrical capacitor is:
\[ V = \frac{0.1 \ln(2)}{4 \pi \epsilon_0} \, \text{volts} \]