The capacity of a spherical conductor is

Options

A. `(R)/(4 pi epsilon_(0))`
B. `(4 pi epsilon_(0))/(R)`
C. `4 pi epsilon_(0)R`
D. `4 pi epsilon_(0)R^(2)`

Answer

To find the capacitance of a spherical conductor, we can follow these steps:

### Step 1: Understand the Definition of Capacitance
Capacitance (C) is defined as the ratio of charge (Q) stored on the conductor to the potential difference (V) across it. Mathematically, this is expressed as:
\[ C = \frac{Q}{V} \]

### Step 2: Determine the Expression for Potential (V)
For a spherical conductor of radius \( R \), the electric potential \( V \) at its surface due to a charge \( Q \) is given by:
\[ V = \frac{KQ}{R} \]
where \( K \) is Coulomb's constant, which can also be expressed as:
\[ K = \frac{1}{4 \pi \epsilon_0} \]
Thus, we can rewrite the potential as:
\[ V = \frac{Q}{4 \pi \epsilon_0 R} \]

### Step 3: Substitute the Expression for V into the Capacitance Formula
Now, substituting the expression for \( V \) into the capacitance formula:
\[ C = \frac{Q}{V} = \frac{Q}{\frac{Q}{4 \pi \epsilon_0 R}} \]

### Step 4: Simplify the Expression
When we simplify the above expression, we find:
\[ C = 4 \pi \epsilon_0 R \]

### Step 5: Conclusion
Thus, the capacitance of a spherical conductor is:
\[ C = 4 \pi \epsilon_0 R \]