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Gravitational Field Due to a Ring

Two concentri...

Question

Two concentric spherical shells have masses $M_{1}$ and $M_{2}$ and radii $R_{1}$ and $R_{2}$ $(R_{1} < R_{2})$ . What is the force exerted by this system on a particle of mass $m$ if it is at a distance $\frac{(R_{1} + R_{2})}{2}$ from the centre?

A

\frac{4 G (M_{1} + M_{2}) m}{(R_{1} + R_{2})^{2}}

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B

\frac{4 G M_{2} m}{(R_{1} + R_{2})^{2}}

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C

\frac{4 G M_{1} m}{(R_{1} + R_{2})^{2}}

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D

\frac{4 G (M_{1} - M_{2}) m}{(R_{1} + R_{2})^{2}}

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Solution

The correct option is C $\frac{4 G M_{1} m}{(R_{1} + R_{2})^{2}}$

Field due to a spherical shell at an inside point is zero. Hence the field at the point $P$ will be only due to $M_{1}$ .

So, the gravitational field at point $P$ ,

$E_{g} = \frac{4 G M_{1}}{(R_{1} + R_{2})^{2}}$

Thus, the force on mass $m$ is given by:

$F = m E_{g}$

Substituting the value of $E_{g}$ ,

$∴ F = \frac{4 G M_{1} m}{(R_{1} + R_{2})^{2}}$

Hence, option (c) is correct.

Why this question: It is absolutely important to be well versed with the field due to a solid sphere and a hollow sphere both inside and outside. The concept is extremely important from a JEE standpoint.

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