Question

A metal wire is bent in the shape of a circle of 10 cm radius. It is given a charge of 200 \(\mu\)C which spreads on it uniformly. The electric potential at its center is

• A. \(3 \times 10^{6}\) V
• B. \(6 \times 10^{6}\) V
• C. \(9 \times 10^{6}\) V
• D. \(18 \times 10^{6}\) V

Hint:

Every part of the ring is at distance R from the centre, so \(V = kQ/R\).

Answer 1

The Correct Option is D

Step 1: Method: Add Potentials of Small Elements:
Split the ring into small charges \(dq\). Each is at distance R from the centre. The potential from one element is \(dV = k\,dq/R\).

Step 2: Integrate:
Because R is the same for all elements, \[ V = \int \frac{k\,dq}{R} = \frac{k}{R}\int dq = \frac{kQ}{R} \] The field at the centre is zero by symmetry, but the potential is not zero, since potential is a scalar.

Step 3: Put in Numbers:
\[ V = \frac{(9\times10^{9})(200\times10^{-6})}{10\times10^{-2}} = \frac{1.8\times10^{6}}{0.1} = 1.8\times10^{7}\ \text{V} \]

Step 4: Express in the Option Format:
\(1.8\times10^{7} = 18\times10^{6}\) V, which is the fourth printed option.

Final Answer:
\[\boxed{18\times10^{6}\ \text{V}}\]