Question

A ring of radius $R$ is charged uniformly with a charge $+Q$ . The electric field at any point on its axis at a distance $r$ from the circumference of the ring will be

• A. $\frac{KQ}{r}$
• B. $\frac{KQ}{r^2}$
• C. $\frac{KQ}{r^3}(r^2-R^2{)}^{\frac{1}{2}}$
• D. $\frac{KQr}{R^3}$

Answer 1

The correct option is C $\frac{KQ}{r^3}(r^2-R^2{)}^{\frac{1}{2}}$


We know that, electric field at any point on the axis is given by,
$E=\frac{KQx}{(R^2+x^2{)}^{\frac{3}{2}}}$
and from diagram, we get
$x=\sqrt{r^2-R^2}$
Therefore,
$E=\frac{KQ\sqrt{r^2-R^2}}{(R^2+r^2-R^2{)}^{\frac{3}{2}}}$
$=\frac{KQ}{r^3}(r^2-R^2{)}^{\frac{1}{2}}$