Q amount of electric charge is present on the surface having radius R. Then electrical potential energy of this system is :

Options

A. `(kQ)/R`
B. `(kQ)^2/(R^2)`
C. `(kQ^2)/(2R)`
D. `(kQ^2)/(2R^2)`

Answer

To find the electrical potential energy of a system with charge \( Q \) distributed on the surface of a sphere with radius \( R \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the System**: We have a spherical surface with radius \( R \) that carries a total charge \( Q \). This charge is uniformly distributed over the surface.

2. **Identify the Capacitance**: The capacitance \( C \) of a spherical capacitor (or a sphere with charge) is given by the formula:
\[ C = 4 \pi \epsilon_0 R \]
where \( \epsilon_0 \) is the permittivity of free space.

3. **Calculate the Electric Potential \( V \)**: The electric potential \( V \) at the surface of the sphere due to the charge \( Q \) is given by:
\[ V = \frac{KQ}{R} \]
where \( K \) is Coulomb's constant, \( K = \frac{1}{4 \pi \epsilon_0} \). Thus, we can express \( V \) as:
\[ V = \frac{Q}{4 \pi \epsilon_0 R} \]

4. **Use the Formula for Potential Energy**: The electrical potential energy \( U \) stored in a capacitor is given by:
\[ U = \frac{Q^2}{2C} \]

5. **Substitute the Value of Capacitance**: Substitute the expression for capacitance \( C \) into the potential energy formula:
\[ U = \frac{Q^2}{2 \cdot (4 \pi \epsilon_0 R)} \]

6. **Simplify the Expression**: This simplifies to:
\[ U = \frac{Q^2}{8 \pi \epsilon_0 R} \]

7. **Final Result**: Thus, the electrical potential energy of the system is:
\[ U = \frac{K Q^2}{2R} \]
where \( K = \frac{1}{4 \pi \epsilon_0} \).

### Final Answer:
\[ U = \frac{K Q^2}{2R} \]