Question

Charge $Q$ is uniformly distributed over a ring of radius $R$. The height $h$, on the axis of the ring at which electric field is maximum

• A. $\frac{R}{\sqrt{2}}$
• B. $\frac{R}{2}$
• C. R
• D. None of these

Answer 1

The correct option is A

$\frac{R}{\sqrt{2}}$

Step 1. Given Data,

Charge $=Q$

Radius $=R$

Height $=h$

Electric Field$=E$

We have to calculate the Height $h$, on the axis of the ring at which the Electric Field be maximum,

Step 2. Calculate the Height $h$, on the axis of the ring at which the Electric Field be maximum,

Electric Field (along) axis of circular coil $E$ $=\frac{kQh}{{\left(h^2+R^2\right)}^{\frac{3}{2}}}$ (where, $k$ is constant)

For max value of $E,\left(\frac{dE}{dh}\right)=0$

Now, differentiating $E$ with respect to $h$ and equating it to $0$

$$\begin{gathered} \frac{dE}{dh}=\left[\frac{R^2-2h^2}{{\left(h^2+R^2\right)}^{\frac{3}{2}}}\right] \\ \Rightarrow 0=R^2-2h^2 \\ \Rightarrow h=\pm \frac{R}{\sqrt{2}} \end{gathered}$$

Hence, the correct option is $\text{'}A\text{'}$.