Question

Electric field at center of semicircular ring shown in figure is :

• A. $-\frac{2kq}{{\pi \epsilon }_0{r}^2}i$
• B. $-\frac{4kq}{{2\pi \epsilon }_0{r}^2}i$
• C. $-\frac{2\sqrt{2}kq}{{2\pi \epsilon }_0{r}^2}i$
• D. None of these

Answer 1

The correct option is A $-\frac{2kq}{{\pi \epsilon }_0{r}^2}i$

$dE=-\frac{K\left(q\frac{d\theta }{\pi {ϵ}_0}\right)\cos \theta }{r^2\pi /2}$

$E_x=\frac{-K_q}{{ϵ}_0\pi r^2}{\int }_{-\pi /2}^{\pi /2}\cos \theta d\theta$

$E_x=\frac{-K_q}{\pi r^2{ϵ}_0}{\sin \theta |}_{-\pi /2}^{\pi /2}$

$E_x=\frac{-K_q}{r^2{ϵ}_0}(1+1)$

$=\frac{2K_q}{r^2{ϵ}_0}$

$\overline{E}=-\frac{2K_q}{r^2{ϵ}_0}\hat{i}$