Find the potential at the centre of a square of side ` sqrt(2)` m . Which carries at its four corners charges `q_(1) = 3 xx 10^(-6) C, q_(2) = - 3 xx 10^(-6) C, q_(3) = - 4 xx 10^(6)C, q_(4) = 7 xx 10^(-6)C`.

Options

A. `2.7xx 10^(4)V`
B. `1.5 xx 10^(3)V`
C. `3 xx 10^(2)V`
D. ` 5 xx 10^(3)V`

Explanation

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

### Step 1: Identify the configuration
We have a square of side length \( \sqrt{2} \) m with the following charges at the corners:
- \( q_1 = 3 \times 10^{-6} \, \text{C} \)
- \( q_2 = -3 \times 10^{-6} \, \text{C} \)
- \( q_3 = -4 \times 10^{-6} \, \text{C} \)
- \( q_4 = 7 \times 10^{-6} \, \text{C} \)

### Step 2: Calculate the distance from the center to a corner
The distance \( r \) from the center of the square to any corner can be calculated using the Pythagorean theorem. Since the side of the square is \( \sqrt{2} \), each half of the side (from the center to a corner) is:
\[ r = \frac{\sqrt{2}}{2} \sqrt{2} = 1 \, \text{m} \]

### Step 3: Use the formula for electric potential
The electric potential \( V \) at a point due to a point charge is given by:
\[ V = k \frac{q}{r} \]
where \( k \) is Coulomb's constant (\( k \approx 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \)), \( q \) is the charge, and \( r \) is the distance from the charge to the point of interest.

### Step 4: Calculate the total potential at the center
Since the potential is a scalar quantity, we can sum the potentials due to each charge:
\[ V_{\text{total}} = V_1 + V_2 + V_3 + V_4 = k \left( \frac{q_1}{r} + \frac{q_2}{r} + \frac{q_3}{r} + \frac{q_4}{r} \right) \]
Substituting \( r = 1 \, \text{m} \):
\[ V_{\text{total}} = k \left( q_1 + q_2 + q_3 + q_4 \right) \]

### Step 5: Substitute the values of the charges
Now substituting the values of the charges:
\[ V_{\text{total}} = k \left( 3 \times 10^{-6} - 3 \times 10^{-6} - 4 \times 10^{-6} + 7 \times 10^{-6} \right) \]
Calculating the sum:
\[ V_{\text{total}} = k \left( 3 - 3 - 4 + 7 \right) \times 10^{-6} = k \left( 3 \times 10^{-6} \right) \]

### Step 6: Calculate the potential
Now substituting \( k \):
\[ V_{\text{total}} = 8.99 \times 10^9 \times 3 \times 10^{-6} = 26.97 \times 10^3 \, \text{V} = 2.697 \times 10^4 \, \text{V} \]

### Final Result
The potential at the center of the square is approximately:
\[ V \approx 2.7 \times 10^4 \, \text{V} \]