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The magnitudes of the gravitational field at distances $r_{1}$ and $r_{2}$ from the centre of a uniform sphere of radius $R$ and mass $M$ are $E_{1}$ and $E_{2}$ respectively. Then:

This question has multiple correct options

A

$\frac{E _{1}}{E _{2}} = \frac{r _{1}}{r _{2}}$ , if $r_{1} < R$ and $r_{2} < R$

B

$\frac{E _{1}}{E _{2}} = \frac{r _{2}^{2}}{r _{1}^{2}}$ , if $r_{1} > R$ and $r_{2} > R$

C

$\frac{E _{1}}{E _{2}} = \frac{r _{1}^{3}}{r _{2}^{3}}$ , if $r_{1} < R$ and $r_{2} < R$

D

$\frac{E _{1}}{E _{2}} = \frac{r _{1}^{2}}{r _{2}^{2}}$ , if $r_{1} < R$ and $r_{2} < R$

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Correct options are A) and B)

If $r \leq R$ , then $E = \frac{G M}{R ^{3}} (r) ⟹ E \propto r$

$⟹ \frac{E _{1}}{E _{2}} = \frac{r _{1}}{r _{2}}$ if $r_{1} < R$ and $r_{2} < R$

If $r \geq R$ , then $E = \frac{G M}{r ^{2}} ⟹ E \propto \frac{1}{r ^{2}}$

$⟹ \frac{E _{1}}{E _{2}} = \frac{r _{2}^{2}}{r _{1}^{2}}$ if $r_{1} > R$ and $r_{2} > R$

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