Question

Two spheres of radii $12\mathrm{cm}$ and $16\mathrm{cm}$ have equal charges.The ratio of their energies is :

Answer 1

Step 1: Given data

$$\begin{gathered} r_1=12\mathrm{cm} \\ {r}_2=16\mathrm{cm} \end{gathered}$$

Here,

$r_1$ is the radius of the first sphere.

$r_2$ is the radius of the second sphere.

Step 2: Formulation

Capacitance

1. Capacitance is defined as the ratio of change in electrical charge to the change in potential of the system.
2. The capacitance of a spherical object can be written as follows $\left(R\to \infty \right)$:

$C=4\pi {\epsilon }_{0}r$

Here, $C$ is the capacitance, $r$ is the radius of the sphere, and ${\epsilon }_0$ is the constant.

Hence, capacitance is directly proportional to the radius of the sphere.

3. The equation for the energy can be written as follows:

$$\begin{gathered} E=\frac{Q^2}{2C} \\ E=\frac{Q^2}{8\pi {\epsilon }_{0}r} \end{gathered}$$

Here, $E$ is the energy and $Q$ is the charge.

Hence, energy is inversely proportional to capacitance.

Step 3: Calculation

Since both the spheres have equal charges, the relation between the energy and the radius can be written by combining the equation as follows:

$$\begin{gathered} E\propto \frac{1}{r} \\ \frac{E_1}{E_2}=\frac{r_2}{r_1} \\ \frac{E_1}{E_2}=\frac{16\mathrm{cm}}{12\mathrm{cm}} \\ \frac{E_1}{E_2}=\frac{4}{3} \end{gathered}$$

Hence, the ratio of the energies of the spheres is $4:3$.