A system consists of two charges `4 mu C and -3 muC` with no external field placed at `(-5 cm, 0,0) and (+5 cm, 0,0)` respectively. The amount of work required to separated the two charges infinitely away from each other is

Options

A. `-1.1J`
B. ` 2 J `
C. `2.5J`
D. `3J`

Explanation

To find the amount of work required to separate the two charges infinitely away from each other, we can use the formula for the potential energy (U) of a system of two point charges:

\[ U = \frac{1}{4 \pi \epsilon_0} \frac{Q_1 Q_2}{R} \]

Where:
- \( U \) is the potential energy,
- \( Q_1 \) and \( Q_2 \) are the magnitudes of the two charges,
- \( R \) is the distance between the charges,
- \( \epsilon_0 \) is the permittivity of free space, approximately \( 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \).

### Step 1: Identify the charges and their positions
The charges are given as:
- \( Q_1 = 4 \, \mu C = 4 \times 10^{-6} \, C \) located at \( (-5 \, cm, 0, 0) \)
- \( Q_2 = -3 \, \mu C = -3 \times 10^{-6} \, C \) located at \( (5 \, cm, 0, 0) \)

### Step 2: Calculate the distance between the charges
The distance \( R \) between the two charges can be calculated as:
\[ R = |x_2 - x_1| = |5 \, cm - (-5 \, cm)| = |5 + 5| = 10 \, cm = 0.1 \, m \]

### Step 3: Substitute the values into the potential energy formula
Now, substituting the values into the formula for potential energy:
\[ U = \frac{1}{4 \pi \epsilon_0} \frac{Q_1 Q_2}{R} \]
Using \( \epsilon_0 \approx 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \):
\[ U = \frac{1}{4 \pi (8.85 \times 10^{-12})} \frac{(4 \times 10^{-6})(-3 \times 10^{-6})}{0.1} \]

### Step 4: Calculate the numerical value
Calculating the numerator:
\[ Q_1 Q_2 = (4 \times 10^{-6})(-3 \times 10^{-6}) = -12 \times 10^{-12} \, C^2 \]
Now substituting into the potential energy formula:
\[ U = \frac{1}{4 \pi (8.85 \times 10^{-12})} \frac{-12 \times 10^{-12}}{0.1} \]
Calculating the denominator:
\[ 4 \pi (8.85 \times 10^{-12}) \approx 1.112 \times 10^{-10} \]
Thus,
\[ U = \frac{-12 \times 10^{-12}}{1.112 \times 10^{-10} \times 0.1} = \frac{-12 \times 10^{-12}}{1.112 \times 10^{-11}} \approx -1.08 \, J \]

### Step 5: Final answer
The amount of work required to separate the two charges infinitely away from each other is approximately:
\[ U \approx -1.08 \, J \approx -1.1 \, J \]

### Conclusion
The answer to the question is:
\[ \text{Option A: } -1.1 \, J \]