A sperical capacitor composed of two concentric metal spheres one having a radius twice as large as the other. The region in which the energy is stored has a volume of

Options

A. `1/4c^3/(pi^2epsilon_0^3)`
B. `1/(16) c^3/(pi^2epsilon_0^2)`
C. `1/(384) c^3/(pi^2epsilon_0^3)`
D. `(7c^3)/(384pi^2epsilon_0^2)`

Answer

To solve the problem of finding the volume of the region in which energy is stored in a spherical capacitor composed of two concentric metal spheres, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Radii of the Spheres**:
Let the radius of the inner sphere be \( R_1 = R \) and the radius of the outer sphere be \( R_2 = 2R \).

2. **Identify the Volume of the Region**:
The energy is stored in the space between the two spheres. The volume \( V \) of this region can be calculated as the difference between the volume of the outer sphere and the volume of the inner sphere.

3. **Calculate the Volume of the Outer Sphere**:
The volume \( V_2 \) of the outer sphere (radius \( R_2 \)) is given by the formula:
\[ V_2 = \frac{4}{3} \pi R_2^3 = \frac{4}{3} \pi (2R)^3 = \frac{4}{3} \pi (8R^3) = \frac{32}{3} \pi R^3 \]

4. **Calculate the Volume of the Inner Sphere**:
The volume \( V_1 \) of the inner sphere (radius \( R_1 \)) is given by the formula:
\[ V_1 = \frac{4}{3} \pi R_1^3 = \frac{4}{3} \pi R^3 \]

5. **Find the Volume of the Region Between the Spheres**:
The volume of the region where energy is stored is:
\[ V = V_2 - V_1 = \frac{32}{3} \pi R^3 - \frac{4}{3} \pi R^3 = \frac{32\pi R^3 - 4\pi R^3}{3} = \frac{28\pi R^3}{3} \]

6. **Final Expression for the Volume**:
Thus, the volume of the region in which the energy is stored in the spherical capacitor is:
\[ V = \frac{28}{3} \pi R^3 \]