Question

An arc of radius r, lying in the first quadrant is shown in the figure. The linear charge density on the arc is λ. Calculate the magnitude and direction of electric field intensity at the point of origin.

Answer 1

Let a circular arc of linear charge density is λ, a elementary Length dl cut from circular arc ,
now, dq = λdl
we know, $\theta=\frac{l}{r}$
so, dq = λrdθ

from coulomb's law,
dE = K.dq/r² = K λrdθ/r² = K λdθ/r
and then, E = ∫dEcosθ
E = $\int\limits^{\pi/4}_{-\pi/4}\frac{K\lambda}{r}cos\theta.d\theta$

= $\frac{K\lambda}{r}[sin\theta]^{\pi/4}_{-\pi/4}$

=$\frac{\sqrt{2}K\lambda}{r}$