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 given charges at its corners, we can follow these steps:

### Step 1: Understand the Configuration
We have a square with a side length of \( \sqrt{2} \) m, and charges placed at each corner:
- \( q_1 = 3 \times 10^{-6} \) C
- \( q_2 = -3 \times 10^{-6} \) C
- \( q_3 = -4 \times 10^{-6} \) C
- \( q_4 = 7 \times 10^{-6} \) C

### Step 2: Determine the Distance from the Center to Each Charge
The distance from the center of the square to each corner can be calculated using the Pythagorean theorem. The diagonal of the square is \( \sqrt{2^2 + 2^2} = 2 \) m. The distance from the center to any corner (half the diagonal) is:
\[ R = \frac{2}{2} = 1 \text{ m} \]

### Step 3: Formula for Electric Potential
The electric potential \( V \) due to a point charge is given by:
\[ V = \frac{kQ}{R} \]
where \( k \) is Coulomb's constant (\( 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \)), \( Q \) is the charge, and \( R \) is the distance from the charge to the point where the potential is being calculated.

### Step 4: Calculate the Potential from Each Charge
Since the distance \( R \) is the same for all charges (1 m), we can calculate the potential at the center due to each charge:
- \( V_1 = \frac{kq_1}{R} = \frac{9 \times 10^9 \times 3 \times 10^{-6}}{1} = 27 \times 10^3 \, \text{V} \)
- \( V_2 = \frac{kq_2}{R} = \frac{9 \times 10^9 \times (-3) \times 10^{-6}}{1} = -27 \times 10^3 \, \text{V} \)
- \( V_3 = \frac{kq_3}{R} = \frac{9 \times 10^9 \times (-4) \times 10^{-6}}{1} = -36 \times 10^3 \, \text{V} \)
- \( V_4 = \frac{kq_4}{R} = \frac{9 \times 10^9 \times 7 \times 10^{-6}}{1} = 63 \times 10^3 \, \text{V} \)

### Step 5: Calculate the Total Potential
The total potential \( V \) at the center is the sum of the potentials due to all four charges:
\[ V = V_1 + V_2 + V_3 + V_4 \]
Substituting the values:
\[ V = 27 \times 10^3 - 27 \times 10^3 - 36 \times 10^3 + 63 \times 10^3 \]
\[ V = 0 - 36 \times 10^3 + 63 \times 10^3 \]
\[ V = 27 \times 10^3 \, \text{V} \]

### Step 6: Final Result
Thus, the potential at the center of the square is:
\[ V = 27 \times 10^3 \, \text{V} \quad \text{or} \quad 2.7 \times 10^4 \, \text{V} \]