A ring has a mass $M$ and radius $R$. The distance of the point on its geometric axis from its centre at which the gravitational field is strongest is

• A

  $\frac {R}{2}$
• B

  $\frac {R}{4}$
• C

  $\frac {R}{\sqrt 3}$
• Correct answer
  D

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

Solution

Gravitational field from the centre is given by $\mathrm{E}=\frac{\mathrm{GMx}}{\left(\mathrm{R}^2+\mathrm{x}^2\right)^{3 / 2}}$ The gravitational field is strongest. So $\begin{aligned} & \frac{d E}{d x}=0 \\ & 0=\frac{\left(R^2+x^2\right)^{3 / 2} G M-G M x \frac{3}{2}\left(R^2+x^2\right)^{1 / 2} 2 x}{\left(R^2+x^2\right)^3} \\ & \Rightarrow 0=G M\left(R^2+x^2\right)^{1 / 2}\left[R^2+x^2-3 x^2\right] \\ & \Rightarrow x=\frac{R}{\sqrt{2}} \end{aligned}$

Hence, the correct answer is option: 'D'.