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A half ring of radius $R$ has a charge of $\lambda$ per unit length. The electric force on $1\, C$ charged placed at the centre is
Zero
$\frac{k \lambda}{R}$
$\frac{2 k \lambda}{R}$
$\frac{k \pi \lambda}{R}$
Solution
As, $R$ be the radius of the ring. Consider a small strip of length $d l$ having charge $d q$ lying at an angle $\theta$.
$d l=R d \theta$
Charge on $d l=\lambda R d \theta$
Force at $1 C$ due to $d l$
$=\frac{k \lambda R d \theta}{R^{2}}=\frac{k \lambda}{R} d \theta=d F$
We need to consider only the component $d F \cos \theta,$ as the component $d F \sin \theta$ will cancel out because of the symmetrical element $d l$.
The total force on $1 C$ is
$F=\int_{-\pi / 2}^{\pi / 2} d F \cos \theta$
$=\frac{k \lambda}{R} \int_{-\pi / 2}^{\pi / 2} \cos \theta d \theta$
$=\frac{k \lambda}{R} \times 2=\frac{2 k \lambda}{R}$
