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

Answer: 0.00

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