The capacitance of an isolated conducting sphere of radius R is proportional to

Options

A. `4 pi epsilon_(0)r`
B. `2 pi epsilon_(0)r`
C. ` pi epsilon_(0)r`
D. `8 pi epsilon_(0)r`

Answer

To find the capacitance of an isolated conducting sphere of radius \( R \), we can follow these steps:

### Step 1: Understand the relationship between charge, potential, and capacitance
The capacitance \( C \) of a conductor is defined as the ratio of the charge \( Q \) on the conductor to the potential \( V \) at its surface:
\[ C = \frac{Q}{V} \]

### Step 2: Determine the potential \( V \) at the surface of the sphere
For a conducting sphere with charge \( Q \) and radius \( R \), the potential \( V \) at the surface is given by the formula:
\[ V = k \frac{Q}{R} \]
where \( k \) is a constant. In electrostatics, \( k \) is often represented as \( k = \frac{1}{4 \pi \epsilon_0} \).

### Step 3: Substitute the expression for \( V \) into the capacitance formula
Substituting the expression for \( V \) into the capacitance formula, we get:
\[ C = \frac{Q}{V} = \frac{Q}{k \frac{Q}{R}} = \frac{R}{k} \]

### Step 4: Substitute the value of \( k \)
Now, substituting \( k = \frac{1}{4 \pi \epsilon_0} \) into the equation for capacitance:
\[ C = \frac{R}{\frac{1}{4 \pi \epsilon_0}} = 4 \pi \epsilon_0 R \]

### Step 5: Determine the proportionality
From the final expression \( C = 4 \pi \epsilon_0 R \), we can see that the capacitance \( C \) is directly proportional to the radius \( R \) of the sphere.

### Conclusion
Thus, the capacitance of an isolated conducting sphere of radius \( R \) is proportional to \( R \).

### Final Answer
The capacitance \( C \) of an isolated conducting sphere is proportional to \( R \).

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