If `10 mu C` charge exists at center of a square ABCD and `2 mu C ` point charge is moved form A to B. find the work done .

Options

A. 20 J
B. 5 J
C. 0
D. 8 J

Explanation

To solve the problem of finding the work done when a `2 µC` charge is moved from point A to point B in the presence of a `10 µC` charge at the center of a square ABCD, we can follow these steps:

### Step 1: Understand the Configuration
We have a square ABCD with a `10 µC` charge located at the center. The points A and B are at the corners of the square. We need to calculate the electric potential at points A and B due to the `10 µC` charge.

### Step 2: Calculate the Distance from the Center to the Corners
The distance from the center of the square to any corner (A or B) can be calculated using the Pythagorean theorem. If the side length of the square is `a`, then the distance `r` from the center to a corner is:
\[ r = \frac{a}{\sqrt{2}} \]

### Step 3: Calculate the Electric Potential at Point A
The electric potential \( V_A \) at point A due to the `10 µC` charge is given by the formula:
\[ V_A = \frac{k \cdot Q}{r} \]
Where:
- \( k \) is Coulomb's constant (\( 9 \times 10^9 \, \text{Nm}^2/\text{C}^2 \))
- \( Q = 10 \, \mu C = 10 \times 10^{-6} \, C \)
- \( r = \frac{a}{\sqrt{2}} \)

Substituting the values:
\[ V_A = \frac{9 \times 10^9 \cdot 10 \times 10^{-6}}{\frac{a}{\sqrt{2}}} = \frac{9 \times 10^9 \cdot 10 \times 10^{-6} \cdot \sqrt{2}}{a} \]

### Step 4: Calculate the Electric Potential at Point B
Using the same reasoning as for point A, the electric potential \( V_B \) at point B is also:
\[ V_B = \frac{k \cdot Q}{r} \]
Since point B is also at the same distance from the center as point A, we find:
\[ V_B = V_A \]

### Step 5: Calculate the Work Done
The work done \( W \) in moving the charge \( q \) from point A to point B is given by:
\[ W = q \cdot (V_B - V_A) \]
Substituting the values:
\[ W = 2 \times 10^{-6} \cdot (V_B - V_A) \]
Since \( V_B = V_A \):
\[ W = 2 \times 10^{-6} \cdot (0) = 0 \]

### Conclusion
The work done in moving the `2 µC` charge from point A to point B is:
\[ W = 0 \, \text{J} \]