On the surface of earth, the gravitational field is $E_{g}$ and gravitational potential is V(R = radius of earth).
Column-I Column-II
i. At a height h = R, value of $E_{g}$ p. decreases by a factor $\frac{1}{4}$
ii. At a depth d = $\frac{R}{2}, value of E_{g}$ q. decreases by a factor $\frac{1}{2}$
iii. At a height h = R, value of V r. increases by a factor $\frac{11}{8}$
iv. At a depth d = $\frac{R}{2},$ value of V s. increases by a factor 2
t. decreases by a factor $\frac{11}{8}$
Match the given columns and select the correct option from the codes given below.
Codes

Moderate

Solution

$E_{h} = \frac{GM}{{(R + h)}^{2}} = \frac{GM}{4 R^{2}} = \frac{E}{4}$
$V_{h} = - \frac{GM}{2 R} = - \frac{V}{2}$
$E_{d} = E (1 - \frac{d}{R}) = \frac{E}{2}$
$V_{d} = - \frac{GM}{2 R^{3}} (3 R^{3} - r^{2})$
= $- \frac{GM}{2 R^{3}} (3 R^{2} - \frac{R^{2}}{4}) = - \frac{11 GM}{8 R}$

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