Four charges `+3muC, -1muC, +5muC and -7muC` are arranged on the circumference of a circle of radius 0.5m. The potential at the centre is

Options

A. zero
B. `18 xx 10^(4)V`
C. `-18 xx 10^(4)V`
D. `18xx 10^(-4)V`

Explanation

To find the electric potential at the center of a circle where four charges are arranged on its circumference, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Charges and Their Values:**
We have four charges:
- Charge 1: \( +3 \mu C = +3 \times 10^{-6} C \)
- Charge 2: \( -1 \mu C = -1 \times 10^{-6} C \)
- Charge 3: \( +5 \mu C = +5 \times 10^{-6} C \)
- Charge 4: \( -7 \mu C = -7 \times 10^{-6} C \)

2. **Determine the Radius of the Circle:**
The radius of the circle is given as \( r = 0.5 \, m \).

3. **Formula for Electric Potential:**
The electric potential \( V \) due to a point charge \( Q \) at a distance \( r \) is given by:
\[ V = \frac{kQ}{r} \]
where \( k \) is Coulomb's constant, approximately \( 8.99 \times 10^9 \, N m^2/C^2 \).

4. **Calculate the Potential at the Center Due to Each Charge:**
Since all charges are at the same distance \( r \) from the center, we can calculate the potential due to each charge and then sum them up.

- Potential due to Charge 1:
\[ V_1 = \frac{k \cdot 3 \times 10^{-6}}{0.5} \]
- Potential due to Charge 2:
\[ V_2 = \frac{k \cdot (-1) \times 10^{-6}}{0.5} \]
- Potential due to Charge 3:
\[ V_3 = \frac{k \cdot 5 \times 10^{-6}}{0.5} \]
- Potential due to Charge 4:
\[ V_4 = \frac{k \cdot (-7) \times 10^{-6}}{0.5} \]

5. **Sum the Potentials:**
The total potential at the center \( V \) is the sum of the potentials due to all four charges:
\[ V = V_1 + V_2 + V_3 + V_4 \]
Substituting the values:
\[ V = \frac{k \cdot 3 \times 10^{-6}}{0.5} + \frac{k \cdot (-1) \times 10^{-6}}{0.5} + \frac{k \cdot 5 \times 10^{-6}}{0.5} + \frac{k \cdot (-7) \times 10^{-6}}{0.5} \]

6. **Combine the Charges:**
Combine the coefficients of \( k \):
\[ V = k \left( \frac{3 - 1 + 5 - 7}{0.5} \times 10^{-6} \right) \]
Simplifying the expression inside the parentheses:
\[ 3 - 1 + 5 - 7 = 0 \]
Thus:
\[ V = k \cdot 0 = 0 \]

### Final Result:
The potential at the center of the circle is \( V = 0 \, V \).