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Question

A solid sphere of mass M and radius R has a spherical cavity of radius R2 such that the centre of cavity is at a distance R2 from the centre of the sphere. The force of attraction that this sphere would exert on a particle of mass m which lies at a distance d(>R) from the centre of the solid sphere on the straight line joining the centre of the sphere and the center of cavity is

A
GMmd2
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B
GMmd2⎢ ⎢ ⎢ ⎢ ⎢118(1R2d)2⎥ ⎥ ⎥ ⎥ ⎥
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C
GMmd2⎢ ⎢ ⎢ ⎢ ⎢1+18(1R2d)2⎥ ⎥ ⎥ ⎥ ⎥
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D
GMm8d2
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Solution

The correct option is B GMmd2⎢ ⎢ ⎢ ⎢ ⎢118(1R2d)2⎥ ⎥ ⎥ ⎥ ⎥
Given that,
Radius of solid sphere =R
Mass of the solid sphere =M

The density of the solid sphere, ρ=M43πR3


Radius of the spherical cavity =R2.

Let M be the mass of the spherical cavity.

M=M43πR3×43π(R2)3

M=M8

Let the gravitational field at P due to the complete sphere is −−Ecom.

Let us assume that the solid sphere is not there and only removed sphere of radius R2 is there.

Now the gravitational field due to this spherical cavity at P is Erem.

Then, the net gravitational field at P is

Enet=−−Ecom−−Erem

Enet=GMd2GM(dR/2)2

Enet=GMd2GM8(dR/2)2

Enet=GMd2GM8d2(1R2d)2

Enet=GMd2⎢ ⎢ ⎢ ⎢ ⎢118(1R2d)2⎥ ⎥ ⎥ ⎥ ⎥

Force on particle m,

F=mEnet=GMmd2⎢ ⎢ ⎢ ⎢ ⎢118(1R2d)2⎥ ⎥ ⎥ ⎥ ⎥

Hence, option (b) is the correct answer.
Why this question: To enhance the understanding of electric field at any point due to the combination of solid and hollow bodies.

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