The capacitance of a spherical condenser is `1 mu F`. If the spacing between the two spheres is `1mm`, then the radius of the outer sphere is

Options

A. `30 cm`
B. `6 m`
C. `5cm`
D. `3 m`

Explanation

To find the radius of the outer sphere of a spherical condenser with a capacitance of \(1 \mu F\) and a spacing of \(1 mm\) between the two spheres, we can follow these steps:

### Step 1: Understand the formula for capacitance of a spherical capacitor
The capacitance \(C\) of a spherical capacitor is given by the formula:
\[ C = \frac{4 \pi \epsilon_0 A}{B - A} \]
where:
- \(A\) is the inner radius,
- \(B\) is the outer radius,
- \(\epsilon_0\) is the permittivity of free space, approximately \(8.85 \times 10^{-12} \, F/m\).

### Step 2: Define the variables
Let:
- \(A\) = inner radius,
- \(B\) = outer radius.

From the problem, we know that the spacing between the two spheres is \(1 mm\), which can be converted to meters:
\[ B - A = 1 \, mm = 1 \times 10^{-3} \, m \]

### Step 3: Substitute the given values into the capacitance formula
We know that \(C = 1 \mu F = 1 \times 10^{-6} F\). Now substituting into the capacitance formula:
\[ 1 \times 10^{-6} = \frac{4 \pi (8.85 \times 10^{-12}) A}{B - A} \]
Substituting \(B - A = 1 \times 10^{-3}\):
\[ 1 \times 10^{-6} = \frac{4 \pi (8.85 \times 10^{-12}) A}{1 \times 10^{-3}} \]

### Step 4: Simplify the equation
Rearranging the equation gives:
\[ 1 \times 10^{-6} \times 1 \times 10^{-3} = 4 \pi (8.85 \times 10^{-12}) A \]
\[ 1 \times 10^{-9} = 4 \pi (8.85 \times 10^{-12}) A \]

### Step 5: Solve for \(A\)
Now we can solve for \(A\):
\[ A = \frac{1 \times 10^{-9}}{4 \pi (8.85 \times 10^{-12})} \]
Calculating this gives:
\[ A \approx \frac{1 \times 10^{-9}}{1.112 \times 10^{-10}} \approx 8.99 \, m \]

### Step 6: Use the relationship \(B - A = 1 \times 10^{-3}\) to find \(B\)
From the relationship \(B - A = 1 \times 10^{-3}\):
\[ B = A + 1 \times 10^{-3} \]
Substituting the value of \(A\):
\[ B \approx 8.99 + 0.001 = 9.991 \, m \]

### Conclusion
Thus, the radius of the outer sphere \(B\) is approximately:
\[ B \approx 9.991 \, m \]