Four charge `q_(1)=2xx10^(-8)C,q_(2)=-2xx10^(-8)C,q_(3)=-3xx10^(-8)C,` and `q_(4)=6xx10^(-8)C` are palced at four corners of a square of side `sqrt(2)`m. What is the potential at the centre of the square?

Options

A. 270V
B. 300V
C. Zero
D. 100V

Answer

To find the electric potential at the center of a square with four charges placed at its corners, we can follow these steps:

### Step 1: Identify the Charges and Their Positions
We have four charges:
- \( q_1 = 2 \times 10^{-8} \, C \)
- \( q_2 = -2 \times 10^{-8} \, C \)
- \( q_3 = -3 \times 10^{-8} \, C \)
- \( q_4 = 6 \times 10^{-8} \, C \)

These charges are placed at the corners of a square with a side length of \( \sqrt{2} \, m \).

### Step 2: Calculate the Distance from Each Charge to the Center
The center of the square is equidistant from all four corners. The distance \( R \) from each corner to the center can be calculated using the formula for the diagonal of a square:
\[ R = \frac{d}{2} = \frac{\sqrt{2^2 + 2^2}}{2} = \frac{\sqrt{4}}{2} = 1 \, m \]
where \( d = \sqrt{2^2 + 2^2} = 2 \).

### Step 3: Calculate the Total Charge
The total charge \( Q_{total} \) is the algebraic sum of all the charges:
\[ Q_{total} = q_1 + q_2 + q_3 + q_4 \]
Substituting the values:
\[ Q_{total} = (2 \times 10^{-8}) + (-2 \times 10^{-8}) + (-3 \times 10^{-8}) + (6 \times 10^{-8}) \]
Calculating this gives:
\[ Q_{total} = 2 - 2 - 3 + 6 = 3 \times 10^{-8} \, C \]

### Step 4: Calculate the Electric Potential at the Center
The electric potential \( V \) at a point due to a point charge is given by:
\[ V = k \frac{Q}{R} \]
where \( k = 9 \times 10^9 \, N \cdot m^2/C^2 \) is Coulomb's constant.

Since all charges are at the same distance \( R = 1 \, m \), the total potential at the center can be calculated as:
\[ V = k \frac{Q_{total}}{R} \]
Substituting the values:
\[ V = 9 \times 10^9 \frac{3 \times 10^{-8}}{1} \]
Calculating this gives:
\[ V = 9 \times 3 \times 10^1 = 27 \times 10^1 = 270 \, V \]

### Final Answer
The potential at the center of the square is \( 270 \, V \).

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