Question

A semicircular ring of radius $0.5$ m is uniformly charged with a total charge of 4 coulomb The electric potential at the center of this ring is

Answer 1

consider a small element of length dl on the ring ..let @ is the angle be radious vector with base then angle substended

by this element at center is d@ ...

$\frac{arc}{radius}=@$

$dl=Rd@$ ..............1

let charge dencity = p then charge of this element is dq

$dq=p\left(dl\right)=pRd@$ ..............2

electric field due to this element at center dE (magnitude)= kdq/R²

$dE={\frac{Kdq}{R}}^{2}cos@$

(i) +${\frac{kdq}{R}}^{2}sin@(-j)$ with direction

$dE={\frac{kdq}{R}}^2[cos@i-sin@j]$

$dE=\frac{kp}{R}\left[cos@d@\right(i)-sin@d@(j\left)\right]$ lim from 0 to $\pi$

$E=\frac{kp}{R}[-2j]$

now we have p (charge per unit length) = Q/piR so

$E=\frac{2kQ}{pi{R}^2}(-j)$

putting all the values we get

$E=3.2\times {10}^{12}N/C$