Question

[Here $g=GM/R_{2}$, where $M$ is the mass of the sphere]

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Solution

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Correct option is B)

Gravitational field at mass $m$ due to full solid sphere

$Eˉ_{1}=3ε_{0}prˉ =6ε_{0}pR .......[ε_{0}=4πG1 ]$

Gravitational field at mass $m$ due to cavity $(−p)$

$Eˉ_{2}=3ε_{0}R_{2}(−p)(R/2)_{3}rˉ .......[usingE=3ε_{0}r_{2}pa_{3} ]$

$−24ε_{0}R_{2}(−p)R_{3} =−24ε_{0}−pR $

Net gravitational field

$Eˉ=Eˉ_{1}+Eˉ_{2}=6ε_{0}pR −24ε_{0}pR =8ε_{0}pR $

Net force on $m→F=mEˉ=8ε_{0}mpR $

Here, $p=(4/3)πR_{3}M $ & $ε_{0}=4πG1 $

Then, $F=83mg $

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