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:
$\frac{E_{1}}{E_{2}} = \frac{r_{1}}{r_{2}}$ , if $r_{1} < R$ and $r_{2} < R$
$\frac{E_{1}}{E_{2}} = \frac{r_{2}^{2}}{r_{1}^{2}}$ , if $r_{1} > R$ and $r_{2} > R$
$\frac{E_{1}}{E_{2}} = \frac{r_{1}^{3}}{r_{2}^{3}}$ , if $r_{1} < R$ and $r_{2} < R$
$\frac{E_{1}}{E_{2}} = \frac{r_{1}^{2}}{r_{2}^{2}}$ , if $r_{1} < R$ and $r_{2} < R$

Question

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:
$\frac{E_{1}}{E_{2}} = \frac{r_{1}}{r_{2}}$ , if $r_{1} < R$ and $r_{2} < R$
$\frac{E_{1}}{E_{2}} = \frac{r_{2}^{2}}{r_{1}^{2}}$ , if $r_{1} > R$ and $r_{2} > R$
$\frac{E_{1}}{E_{2}} = \frac{r_{1}^{3}}{r_{2}^{3}}$ , if $r_{1} < R$ and $r_{2} < R$
$\frac{E_{1}}{E_{2}} = \frac{r_{1}^{2}}{r_{2}^{2}}$ , if $r_{1} < R$ and $r_{2} < R$

Toppr Admin · Accepted Answer

-If r-x2264-R- then-E-GMR3-r-x27F9-E-x221D-r-x27F9-E1E2-r1r2 if r1-lt-R and-r2-lt-R-If r-x2265-R- then-E-GMr2-x27F9-E-x221D-1r2-x27F9-E1E2-r22r21 if r1-gt-R and r2-gt-R-

The magnitudes of the gravitational field at distances r1 and r2 from the centre of a uniform sphere of radius R and mass M are E1 and E2 respectively. Then:E1E2=r1r2, if r1<R and r2<RE1E2=r22r21, if r1>R and r2>RE1E2=r31r32, if r1<R and r2<RE1E2=r21r22, if r1<R and r2<R

If r≤R, then E=GMR3(r)⟹E∝r ⟹E1E2=r1r2 if r1<R and r2<R If r≥R, then E=GMr2⟹E∝1r2 ⟹E1E2=r22r21 if r1>R and r2>R