The electric potential at the center of two concentric half rings of radii \(R_1\) and \(R_2\), having same linear charge density \(λ\) is (\(ε_0\) = permittivity of free space)
Show Hint
Each half ring is at the same distance from the centre, so its potential is kQ/R with Q = lambda times pi R. Add the two.
Step 1: Integrate over the arc
Take a small arc $dl=R\,d\theta$ with charge $dq=\lambda R\,d\theta$. It lies at distance $R$ from O.
Step 2: Potential
$dV=\frac{dq}{4\pi\epsilon_0R}=\frac{\lambda\,d\theta}{4\pi\epsilon_0}$. Integrate over $\theta$ from $0$ to $\pi$: $V=\frac{\lambda\pi}{4\pi\epsilon_0}=\frac{\lambda}{4\epsilon_0}$.
Step 3: Two rings
The inner and outer rings give the same, so $V=2\cdot\frac{\lambda}{4\epsilon_0}=\frac{\lambda}{2\epsilon_0}$, option (B).
Final Answer:
The potential at the centre is $\frac{\lambda}{2\epsilon_0}$.
\[ \boxed{\dfrac{\lambda}{2\epsilon_0}} \]
