Conducting sphere of radius `R_(1)` is covered by concentric sphere of radius `R_(2)`. Capacity of this combination is proportional to

Options

A. `(R_(2)-R_(1))/(R_(1)R_(2))`
B. `(R_(2)+R_(1))/(R_(1)R_(2))`
C. `(R_(1)R_(2))/(R_(1)+R_(2))`
D. `(R_(1)R_(2))/(R_(2)-R_(1))`

Explanation

To solve the problem of finding the capacity of a conducting sphere of radius \( R_1 \) covered by a concentric sphere of radius \( R_2 \), we will follow these steps:

### Step 1: Understanding the System
We have a conducting sphere of radius \( R_1 \) and a concentric outer sphere of radius \( R_2 \). The inner sphere will have a charge \( +Q \), and the outer sphere will have a charge \( -Q \) due to the nature of capacitors.

### Step 2: Determine the Electric Potential
The electric potential \( V \) at a distance \( r \) from a charged sphere is given by:
\[ V = \frac{1}{4 \pi \epsilon_0} \frac{Q}{r} \]
For the inner sphere at radius \( R_1 \), the potential \( V_1 \) is:
\[ V_1 = \frac{1}{4 \pi \epsilon_0} \frac{Q}{R_1} \]
For the outer sphere at radius \( R_2 \), the potential \( V_2 \) is:
\[ V_2 = -\frac{1}{4 \pi \epsilon_0} \frac{Q}{R_2} \]

### Step 3: Calculate the Potential Difference
The potential difference \( \Delta V \) between the two spheres is given by:
\[ \Delta V = V_1 - V_2 = \frac{1}{4 \pi \epsilon_0} \left( \frac{Q}{R_1} + \frac{Q}{R_2} \right) \]
This simplifies to:
\[ \Delta V = \frac{Q}{4 \pi \epsilon_0} \left( \frac{1}{R_1} + \frac{1}{R_2} \right) \]

### Step 4: Relate Charge and Capacitance
The capacitance \( C \) of the system is defined by the relationship:
\[ Q = C \Delta V \]
Substituting for \( \Delta V \):
\[ C = \frac{Q}{\Delta V} = \frac{Q}{\frac{Q}{4 \pi \epsilon_0} \left( \frac{1}{R_1} + \frac{1}{R_2} \right)} \]
This simplifies to:
\[ C = 4 \pi \epsilon_0 \left( \frac{1}{R_1} + \frac{1}{R_2} \right)^{-1} \]

### Step 5: Final Expression for Capacitance
Using the formula for the sum of reciprocals:
\[ \frac{1}{R_1} + \frac{1}{R_2} = \frac{R_1 + R_2}{R_1 R_2} \]
Thus, we can express \( C \) as:
\[ C = 4 \pi \epsilon_0 \frac{R_1 R_2}{R_1 + R_2} \]

### Conclusion
The capacity of the combination of the conducting sphere and the concentric sphere is proportional to:
\[ C \propto \frac{R_1 R_2}{R_1 + R_2} \]