The capacitance of two concentric spherical shells of radii `R_(1)` and `R_(2) (R_(2) gt R_(1))` is

Options

A. `4pi epsi_(0)R_(2)`
B. `4pi epsi_(0) ((R_(2)-R_(1)))/(R_(1)R_(2))`
C. `4pi epsi_(0) (R_(1)R_(2))/((R_(2)-R_(1)))`
D. `4pi epsi_(0) R^(1)`

Answer

To find the capacitance of two concentric spherical shells with radii \( R_1 \) and \( R_2 \) (where \( R_2 > R_1 \)), we can follow these steps:

### Step 1: Understand the Setup
We have two concentric spherical shells. The inner shell has a radius \( R_1 \) and the outer shell has a radius \( R_2 \). The inner shell is given a charge \( +Q \) and the outer shell is given a charge \( -Q \).

### Step 2: Determine the Electric Field
The electric field \( E \) in the region between the two shells (i.e., for \( R_1 < r < R_2 \)) can be calculated using Gauss's law. The electric field due to the inner shell at a distance \( r \) from the center is given by:
\[ E = \frac{kQ}{r^2} \]
where \( k \) is Coulomb's constant, \( k = \frac{1}{4\pi\epsilon_0} \).

### Step 3: Calculate the Potential Difference
The potential difference \( V \) between the two shells is calculated by integrating the electric field from \( R_1 \) to \( R_2 \):
\[ V = -\int_{R_1}^{R_2} E \, dr = -\int_{R_1}^{R_2} \frac{kQ}{r^2} \, dr \]
Calculating this integral:
\[ V = -kQ \left[-\frac{1}{r}\right]_{R_1}^{R_2} = kQ \left(\frac{1}{R_1} - \frac{1}{R_2}\right) \]

### Step 4: Relate Charge, Capacitance, and Potential
The capacitance \( C \) is defined as the ratio of the charge \( Q \) to the potential difference \( V \):
\[ C = \frac{Q}{V} \]
Substituting the expression for \( V \):
\[ C = \frac{Q}{kQ \left(\frac{1}{R_1} - \frac{1}{R_2}\right)} = \frac{1}{k \left(\frac{1}{R_1} - \frac{1}{R_2}\right)} \]

### Step 5: Simplify the Expression
Using \( k = \frac{1}{4\pi\epsilon_0} \):
\[ C = \frac{1}{\frac{1}{4\pi\epsilon_0} \left(\frac{1}{R_1} - \frac{1}{R_2}\right)} = 4\pi\epsilon_0 \left(\frac{R_1 R_2}{R_2 - R_1}\right) \]

### Final Result
Thus, the capacitance \( C \) of the two concentric spherical shells is given by:
\[ C = \frac{4\pi\epsilon_0 R_1 R_2}{R_2 - R_1} \]