In total, 10 charges (5 of them are `+q` each and other 5 are `-q` each) are placed randomly on the circumference of a circle. The radius of the circle is R. The electric potential at the center of this circle due to these charges will be

Options

A. `0`
B. `10 Kq//R`
C. `5 Kq//R`
D. Cannot be calculated unless the positions of the charges on the circle are specified.

Answer

To find the electric potential at the center of a circle due to charges placed on its circumference, we can follow these steps:

### Step-by-Step Solution

1. **Understanding Electric Potential**:
The electric potential \( V \) at a point due to a point charge \( Q \) is given by the formula:
\[ V = \frac{kQ}{r} \]
where \( k \) is Coulomb's constant, \( Q \) is the charge, and \( r \) is the distance from the charge to the point where the potential is being calculated.

2. **Identifying Charges**:
In this problem, we have 10 charges: 5 positive charges of \( +q \) and 5 negative charges of \( -q \). All these charges are placed on the circumference of a circle with radius \( R \).

3. **Calculating Potential Due to Positive Charges**:
The potential at the center of the circle due to one positive charge \( +q \) is:
\[ V_{+} = \frac{k(+q)}{R} \]
Since there are 5 positive charges, the total potential due to all positive charges is:
\[ V_{+ \text{ total}} = 5 \cdot \frac{kq}{R} = \frac{5kq}{R} \]

4. **Calculating Potential Due to Negative Charges**:
Similarly, the potential at the center due to one negative charge \( -q \) is:
\[ V_{-} = \frac{k(-q)}{R} \]
For 5 negative charges, the total potential due to all negative charges is:
\[ V_{- \text{ total}} = 5 \cdot \frac{k(-q)}{R} = -\frac{5kq}{R} \]

5. **Finding the Net Potential at the Center**:
The net potential \( V_{\text{net}} \) at the center of the circle is the sum of the potentials due to positive and negative charges:
\[ V_{\text{net}} = V_{+ \text{ total}} + V_{- \text{ total}} = \frac{5kq}{R} - \frac{5kq}{R} = 0 \]

6. **Conclusion**:
The electric potential at the center of the circle due to the 10 charges is:
\[ V_{\text{net}} = 0 \]

### Final Answer
The electric potential at the center of the circle is \( 0 \).