Question

A thread carrying a charge (uniform) $\lambda$ per unit length has configuration shown in figure.

Assuming a curvature radius $R$ to be considerably less than the length of thread. Find the magnitude of electric field strength at point $O$.

Answer 1

Net $\vec{E}$ due to semicircular arc of radius $R$
$E =\frac{2 \lambda}{4 \pi \varepsilon_{0} R }=\frac{\lambda}{2 \pi \varepsilon_{0} R } \hat{ j } (+y$ direction)
Net $y$ component of $\vec{E}$ due to semi infinite line charges $A B$ and $C D$ in $(-y)$ direction $= E ^{\prime}$
$E '=\frac{\lambda}{4 \pi \varepsilon_{0} R }+\frac{\lambda}{4 \pi \varepsilon R }=\frac{\lambda}{2 \pi \varepsilon_{0} R }(-\hat{ j })$
$X$ components of field of these line charges cancel each other.
$E'' =E+E'$
$=\frac{\lambda}{2 \pi \varepsilon_{0} R}(-\hat{j})+\frac{\lambda}{2 \pi \varepsilon_{0} R}(+\hat{j}) $
$=0$ (zero)