Question

The electric potential at the center of two concentric half rings of radii $R_1$ and $R_2$, having same linear charge density $\lambda$ is :

• A. $\frac{2 \lambda}{\epsilon_0}$
• B. $\frac{\lambda}{2 \epsilon_0}$
• C. $\frac{\lambda}{4 \epsilon_0}$
• D. $\frac{\lambda}{\epsilon_0}$

Answer 1

The Correct Option is B

The potential at the center due to a half ring is given by:

\[ V = \frac{\lambda \cdot \pi R}{4 \pi \epsilon_0 R} = \frac{\lambda}{4 \epsilon_0} \]

For two concentric half rings:

\[ V_{\text{total}} = \frac{\lambda}{4 \epsilon_0} + \frac{\lambda}{4 \epsilon_0} = \frac{\lambda}{2 \epsilon_0} \]

Thus, the correct answer is \( \frac{\lambda}{2 \epsilon_0} \).

Answer 2

The correct answer is (B) : $\frac{\lambda}{2 \epsilon_0}$
Potential at centre
$V=\frac{\left(\lambda \cdot \pi R_2\right)}{4\pi {\epsilon }_0{R}_2}+\frac{\left(\lambda \cdot \pi R_1\right)}{4\pi {\epsilon }_0{R}_1}$
$=\frac{\lambda }{2{\epsilon }_0}$