Calculate the electric potential at the center of a square of side `sqrt(2) m`, having charges `100 muc, -50 muC, 20 muc and -60 muC` at the four corners of the square.

Options

A. 21.6 N
B. 64.8 N
C. 0
D. 43.2 N

Answer

To calculate the electric potential at the center of a square with side length \(\sqrt{2}\) m and charges \(100 \, \mu C\), \(-50 \, \mu C\), \(20 \, \mu C\), and \(-60 \, \mu C\) located at the corners, we will follow these steps:

### Step 1: Determine the distance from each charge to the center of the square.

The diagonal \(D\) of the square can be calculated using the Pythagorean theorem:
\[ D = \sqrt{(\text{side})^2 + (\text{side})^2} = \sqrt{(\sqrt{2})^2 + (\sqrt{2})^2} = \sqrt{2 + 2} = \sqrt{4} = 2 \, \text{m} \]
The distance from the center to each corner (which is half the diagonal) is:
\[ r = \frac{D}{2} = \frac{2}{2} = 1 \, \text{m} \]

### Step 2: 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 \(k = 9 \times 10^9 \, \text{N m}^2/\text{C}^2\) is Coulomb's constant, \(Q\) is the charge, and \(r\) is the distance from the charge to the point.

### Step 3: Calculate the total electric potential at the center.

The total potential at the center due to all four charges is the algebraic sum of the potentials due to each charge:
\[ V_{\text{total}} = V_1 + V_2 + V_3 + V_4 \]

Calculating each potential:
- For \(Q_1 = 100 \, \mu C = 100 \times 10^{-6} \, C\):
\[ V_1 = k \frac{100 \times 10^{-6}}{1} = 9 \times 10^9 \times 100 \times 10^{-6} = 9 \times 10^4 \, \text{V} \]

- For \(Q_2 = -50 \, \mu C = -50 \times 10^{-6} \, C\):
\[ V_2 = k \frac{-50 \times 10^{-6}}{1} = 9 \times 10^9 \times -50 \times 10^{-6} = -4.5 \times 10^4 \, \text{V} \]

- For \(Q_3 = 20 \, \mu C = 20 \times 10^{-6} \, C\):
\[ V_3 = k \frac{20 \times 10^{-6}}{1} = 9 \times 10^9 \times 20 \times 10^{-6} = 1.8 \times 10^4 \, \text{V} \]

- For \(Q_4 = -60 \, \mu C = -60 \times 10^{-6} \, C\):
\[ V_4 = k \frac{-60 \times 10^{-6}}{1} = 9 \times 10^9 \times -60 \times 10^{-6} = -5.4 \times 10^4 \, \text{V} \]

### Step 4: Sum the potentials.

Now, we sum all the potentials:
\[ V_{\text{total}} = 9 \times 10^4 + (-4.5 \times 10^4) + 1.8 \times 10^4 + (-5.4 \times 10^4) \]
Calculating this step by step:
\[ V_{\text{total}} = 9 \times 10^4 - 4.5 \times 10^4 + 1.8 \times 10^4 - 5.4 \times 10^4 \]
\[ = (9 - 4.5 + 1.8 - 5.4) \times 10^4 \]
\[ = (9 - 4.5 - 5.4 + 1.8) \times 10^4 = (1.8) \times 10^4 \, \text{V} \]

### Final Answer:
The electric potential at the center of the square is:
\[ V_{\text{total}} = 1.8 \times 10^4 \, \text{V} \]