Question

A ring of radius $r$ has charge $Q$. A length $dl$ is cut from it. Find the electric field at the centre.

• A. Zero
• B. $\frac{Qdl}{2\pi r^2{\epsilon }_0}$
• C. $\frac{Qdl}{2\pi r^3{\epsilon }_0}$
• D. $\frac{Qdl}{8{\pi }^2{\epsilon }_0{r}^3}$

Answer 1

Answer: (D)

$\frac{Qdl}{8{\pi }^2{\epsilon }_0{r}^3}$

The electric field due to full ring is zero at center when arc of length $dl$ is cut, the electric field becomes non zero, so
$E_{smallarc}=-E_{restofportion}$
If $\lambda$ be the line charge density, $Q=\lambda .2\pi r\Rightarrow \lambda =\frac{Q}{2\pi r}$
Charge on cut part is $dq=\lambda .dl=\frac{Qdl}{2\pi r}$
The electric field at center due to arc of length dl is $E_o=\frac{1}{4\pi {ϵ}_0}.\frac{dq}{r^2}=\frac{1}{4\pi {ϵ}_0}.\frac{Qdl}{2\pi r^3}=\frac{Qdl}{8{\pi }^2{ϵ}_0{r}^3}$