Four charges `10^(-8), -2 xx 10^(8), +3xx10^(-8) and 2xx10^(-8)` coulomb are placed at the four corners of a square of side 1m the potential at the centre of the square is

Options

A. zero
B. 360 volt
C. 180 volt
D. `360sqrt(2)` volt

Explanation

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

### Step 1: Identify the Charges and Their Positions
We have four charges located at the corners of a square of side 1 meter:
- \( Q_1 = 10^{-8} \, \text{C} \) (at corner A)
- \( Q_2 = -2 \times 10^{-8} \, \text{C} \) (at corner B)
- \( Q_3 = 3 \times 10^{-8} \, \text{C} \) (at corner C)
- \( Q_4 = 2 \times 10^{-8} \, \text{C} \) (at corner D)

### Step 2: Calculate the Distance from Each Charge to the Center
The center of the square is at a distance \( r \) from each corner. For a square of side \( a = 1 \, \text{m} \), the distance from a corner to the center is given by:
\[ r = \frac{a}{\sqrt{2}} = \frac{1}{\sqrt{2}} \, \text{m} \]

### Step 3: Calculate the Potential Due to Each Charge
The electric potential \( V \) due to a point charge \( Q \) at a distance \( r \) is given by the formula:
\[ V = \frac{1}{4 \pi \epsilon_0} \cdot \frac{Q}{r} \]
where \( \epsilon_0 \approx 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \).

### Step 4: Calculate the Total Potential at the Center
Since potential is a scalar quantity, we can simply add the potentials due to each charge:
\[ V_{\text{total}} = V_1 + V_2 + V_3 + V_4 \]
Substituting the values:
\[ V_{\text{total}} = \frac{1}{4 \pi \epsilon_0} \left( \frac{Q_1}{r} + \frac{Q_2}{r} + \frac{Q_3}{r} + \frac{Q_4}{r} \right) \]
Factoring out \( \frac{1}{r} \):
\[ V_{\text{total}} = \frac{1}{4 \pi \epsilon_0 r} \left( Q_1 + Q_2 + Q_3 + Q_4 \right) \]

### Step 5: Substitute the Charges and Calculate
Now substituting the values of the charges:
\[ Q_1 + Q_2 + Q_3 + Q_4 = 10^{-8} - 2 \times 10^{-8} + 3 \times 10^{-8} + 2 \times 10^{-8} \]
Calculating this:
\[ = 10^{-8} - 2 \times 10^{-8} + 3 \times 10^{-8} + 2 \times 10^{-8} = 4 \times 10^{-8} \, \text{C} \]

### Step 6: Final Calculation
Now substituting \( r = \frac{1}{\sqrt{2}} \):
\[ V_{\text{total}} = \frac{1}{4 \pi \epsilon_0 \left( \frac{1}{\sqrt{2}} \right)} \cdot 4 \times 10^{-8} \]
\[ = \frac{4 \sqrt{2}}{4 \pi \epsilon_0} \times 10^{-8} \]
Using \( \epsilon_0 \approx 8.85 \times 10^{-12} \):
\[ V_{\text{total}} = \frac{4 \sqrt{2}}{4 \pi \times 8.85 \times 10^{-12}} \times 10^{-8} \]
Calculating this gives us the final potential at the center of the square.