A semi-circular arc of radius ’ a ’ is charged uniformly and the charge per unit length is λ. The electric field at the centre of this arc is ?

A semi-circular arc of radius ’ a ’ is charged uniformly and the charge per unit length is λ. The electric field at the centre of this arc is ?

(a) \( \frac{\lambda}{2\pi\varepsilon_0 a} \)

(b) \( \frac{\lambda}{2\pi\varepsilon_0 a^2} \)

(c) \( \frac{\lambda}{4\pi^2\varepsilon_0 a} \)

(d) \( \frac{\lambda^2}{2\pi\varepsilon_0 a} \)

Answer :  (a) \( \frac{\lambda}{2\pi\varepsilon_0 a} \)

\( E = \int_{\theta=0}^{\theta=\pi} dE \sin\theta \)

\( = \int_{\theta=0}^{\pi} \frac{k \, dq}{a^2} \sin\theta \)

\( = \frac{k \lambda}{a^2} \int dl \, \sin\theta \)

\( = \frac{k \lambda a}{a^2} \int d\theta \, \sin\theta \)

\( = \frac{k \lambda}{a} \left[ -\cos\theta \right]_{\theta=0}^{\pi} \)

\( = -\frac{k \lambda}{a} \left( -1 - 1 \right) \)

\( = \frac{2}{4 \pi \varepsilon_0} \cdot \frac{\lambda}{a} \)

\( = \frac{\lambda}{2\pi\varepsilon_0 a} \)