1 \(\dfrac{4}{3} G M \hat{i}\)
2 \(\dfrac{1}{3} G M \hat{i}\)
3 \(\dfrac{1}{4} G M \hat{i}\)
4 \(\dfrac{2}{3} G M \hat{i}\)
Explanation:
Net gravitational field intensity at the origin,
\(E_{\text {net }}=E_{1}+E_{2}+E_{3}+\ldots\)
\( = \frac{{GM}}{{{{(1)}^2}}}\widehat i + \frac{{GM}}{{{{(2)}^2}}}\widehat i + \frac{{GM}}{{{{(4)}^2}}}\widehat i + ...\)
\( = GM\hat i\left[ {1 + \frac{1}{4} + \frac{1}{{16}} + ......\infty } \right]\)
\(\left( {{\rm{Here}}\,a = 1\,{\rm{and}}\,r = \frac{1}{4}} \right)\)
So, \({E_{net}} = GM\left( {\frac{a}{{1 - r}}} \right)\widehat i = GM\widehat i\left( {\frac{1}{{1 - \frac{1}{4}}}} \right)\)
\( \Rightarrow \,\,\,\,\,\,{E_{net}} = \frac{4}{3}GM\widehat i\)