Two ball with charges `5muC and 10muC` are at a distance of 1 m from each other. In order to reduce the distance between them to 0.5 m , the amount of work to be performed is

Options

A. 45 J
B. `0.45xx10^(-6)J`
C. `1.2xx10^(-4)J`
D. 0.45 J

Answer

To solve the problem of calculating the work done to reduce the distance between two charged balls, we will follow these steps:

### Step 1: Understand the Concept of Electrostatic Potential Energy
The electrostatic potential energy (U) between two point charges \( Q_1 \) and \( Q_2 \) separated by a distance \( r \) is given by the formula:

\[ U = \frac{k \cdot Q_1 \cdot Q_2}{r} \]

where \( k \) is Coulomb's constant, approximately \( 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \).

### Step 2: Calculate Initial Potential Energy (U_initial)
Given:
- \( Q_1 = 5 \, \mu C = 5 \times 10^{-6} \, C \)
- \( Q_2 = 10 \, \mu C = 10 \times 10^{-6} \, C \)
- Initial distance \( r_1 = 1 \, m \)

Using the formula for potential energy:

\[ U_{\text{initial}} = \frac{k \cdot Q_1 \cdot Q_2}{r_1} = \frac{9 \times 10^9 \cdot (5 \times 10^{-6}) \cdot (10 \times 10^{-6})}{1} \]

Calculating this gives:

\[ U_{\text{initial}} = 9 \times 10^9 \cdot 50 \times 10^{-12} = 9 \times 10^9 \cdot 5 \times 10^{-11} = 0.45 \, J \]

### Step 3: Calculate Final Potential Energy (U_final)
Now, we need to calculate the potential energy when the distance is reduced to \( r_2 = 0.5 \, m \):

\[ U_{\text{final}} = \frac{k \cdot Q_1 \cdot Q_2}{r_2} = \frac{9 \times 10^9 \cdot (5 \times 10^{-6}) \cdot (10 \times 10^{-6})}{0.5} \]

Calculating this gives:

\[ U_{\text{final}} = \frac{9 \times 10^9 \cdot 50 \times 10^{-12}}{0.5} = 9 \times 10^9 \cdot 100 \times 10^{-12} = 0.9 \, J \]

### Step 4: Calculate Work Done (W)
The work done by an external agent to bring the charges closer is the change in potential energy:

\[ W = U_{\text{final}} - U_{\text{initial}} = 0.9 \, J - 0.45 \, J = 0.45 \, J \]

### Final Answer
The amount of work to be performed to reduce the distance between the two charges from 1 m to 0.5 m is:

\[ \boxed{0.45 \, J} \]