Obtain an expression of capacitance of spherical capacitor.

To derive the expression for the capacitance of a spherical capacitor, we need to consider the geometry of the capacitor and the principles of electrostatics. A spherical capacitor consists of two concentric spherical conductors: an inner sphere of radius \( r_1 \) and an outer sphere of radius \( r_2 \). The capacitance can be understood as the ability of this configuration to store electric charge per unit voltage.

Understanding the Basics of Capacitance

Capacitance (\( C \)) is defined as the ratio of the charge (\( Q \)) stored on a conductor to the potential difference (\( V \)) between the conductors:

C = \frac{Q}{V}

Electric Field Between the Spheres

To find the capacitance, we first need to determine the electric field (\( E \)) in the region between the two spheres. According to Gauss's law, the electric field due to a charged sphere can be calculated using a Gaussian surface. For a point outside a charged sphere, the electric field behaves as if all the charge were concentrated at the center.

For our inner sphere with charge \( Q \), the electric field at a distance \( r \) (where \( r_1 < r < r_2 \)) is given by:

E = \frac{1}{4\pi \epsilon_0} \cdot \frac{Q}{r^2}

Here, \( \epsilon_0 \) is the permittivity of free space.

Calculating the Potential Difference

The potential difference (\( V \)) between the inner and outer spheres can be calculated by integrating the electric field from \( r_1 \) to \( r_2 \):

V = -\int_{r_1}^{r_2} E \, dr

Substituting the expression for \( E \), we have:

V = -\int_{r_1}^{r_2} \frac{1}{4\pi \epsilon_0} \cdot \frac{Q}{r^2} \, dr

Carrying out the integration gives:

V = \frac{Q}{4\pi \epsilon_0} \left( \frac{1}{r_1} - \frac{1}{r_2} \right)

Finding the Capacitance

Now that we have the expression for the potential difference, we can substitute \( V \) back into the capacitance formula:

C = \frac{Q}{V} = \frac{Q}{\frac{Q}{4\pi \epsilon_0} \left( \frac{1}{r_1} - \frac{1}{r_2} \right)}

After simplifying, we find:

C = 4\pi \epsilon_0 \left( \frac{r_1 r_2}{r_2 - r_1} \right)

Final Expression

Thus, the capacitance of a spherical capacitor is given by:

C = 4\pi \epsilon_0 \left( \frac{r_1 r_2}{r_2 - r_1} \right)

This formula illustrates how the capacitance depends on the radii of the spheres and the permittivity of the medium between them. The larger the radii and the smaller the distance between them, the greater the capacitance, allowing the capacitor to store more charge for a given voltage.