A charge of `5 mu C` is placed at the center of a square `ABCD` of side `10 cm`. Find the work done `("in" mu J)` in moving a charge of `1 mu C` from A to B.

Explanation

To solve the problem of finding the work done in moving a charge of \(1 \mu C\) from point A to point B in a square ABCD with a charge of \(5 \mu C\) placed at the center, we can follow these steps:

### Step 1: Understand the Configuration
We have a square ABCD with a side length of \(10 \, cm\) and a charge of \(5 \mu C\) located at the center (point O) of the square. The points A, B, C, and D are the vertices of the square.

### Step 2: Calculate the Distance from Center to Vertices
The distance from the center O to any vertex (A, B, C, or D) can be calculated using the Pythagorean theorem. The distance \(r\) from the center to a vertex is given by:
\[ r = \frac{10 \, cm}{2} \cdot \sqrt{2} = 5\sqrt{2} \, cm = 0.05\sqrt{2} \, m \]

### Step 3: Calculate the Electric Potential at Points A and B
The electric potential \(V\) at a distance \(r\) from a point charge \(Q\) is given by the formula:
\[ V = \frac{kQ}{r} \]
where \(k\) is Coulomb's constant (\(k \approx 9 \times 10^9 \, N m^2/C^2\)).

For the charge at the center:
\[ V_A = V_B = \frac{(9 \times 10^9) \cdot (5 \times 10^{-6})}{0.05\sqrt{2}} \, V \]
Since both points A and B are equidistant from the charge at the center, the potential at both points will be the same.

### Step 4: Determine the Potential Difference
The potential difference \(dV\) between points A and B is:
\[ dV = V_B - V_A = 0 \, V \]

### Step 5: Calculate the Work Done
The work done \(W\) in moving a charge \(q\) through a potential difference \(dV\) is given by:
\[ W = q \cdot dV \]
Substituting the values:
\[ W = (1 \times 10^{-6}) \cdot 0 = 0 \, J \]
Since \(1 \, J = 10^6 \, \mu J\), the work done in microjoules is:
\[ W = 0 \, \mu J \]

### Final Answer
The work done in moving a charge of \(1 \mu C\) from A to B is \(0 \, \mu J\).

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