Question

A uniform sphere has radius ' \(R\) ' and mass ' \(M\) '. The magnitude of gravitational field at distances ' \(r_1\) ' and ' \(r_2\) ' from the centre are ' \(E_1\) ' and ' \(E_2\) ' respectively. The ratio \(E_1 : E_2\) is (\(r_1 > R\) and \(r_2 < R\))

• A. \(\frac{R^2}{r_1^2 r_2}\)
• B. \(\frac{R^3}{r_1 r_2}\)
• C. \(\frac{R^3}{r_1^2 r_2}\)
• D. \(\frac{R^3}{r_1 r_2^2}\)

Hint:

Outside a uniform sphere, the gravitational field follows the inverse square law; inside, it is directly proportional to the distance from the center.

Answer 1

The Correct Option is C

Step 1: Field outside the sphere ($r_1 > R$)
$E_1 = \frac{GM}{r_1^2}$.

Step 2: Field inside the sphere ($r_2 < R$)
$E_2 = \frac{GMr_2}{R^3}$.

Step 3: Calculation of Ratio
$\frac{E_1}{E_2} = \frac{GM/r_1^2}{GMr_2/R^3} = \frac{1}{r_1^2} \cdot \frac{R^3}{r_2} = \frac{R^3}{r_1^2 r_2}$.
Final Answer: (C)