For a given uniform spherical shell of mass `M` and radius `R` find gravitational field at a distancer r from centre in following two cases (a) `r ge R` (b) `r lt R` .

Explanation

`dE = (GdM)/(l^(2)) cos alpha " " ge R`
`dM = (M)/(4pi R^(2)) xx 2pi R sin theta R d theta`
`dM = (M)/(2) sin theta d theta`
`therefore dE = (Gm sin theta cos alpha d theta)/(2l^(2))`
Now `l^(2) = R^(2) + r^(2) - 2rR cos theta`....(1)
`R^(2) = l^(2) + r^(2) - 2lr cos alpha`...(2)
`:. cos alpha = (l^(2)+ r^(2) -R^(2))/(2lr)`
`cos theta = (R^(2) + r^(2) - l^(2))/(2rR)`
differentiating (1)
`:. 2 l d l = 2r R sin theta d theta`
`:. dE = (GM)/(2l^(2)) . (ldl)/(Rr). (l^(2) + r^(2) -R^(2))/((2lr)) rArr dE = (GM)/(4Rr^(2)) [1 + (r^(2)-R^(2))/(l^(2))]dl`
`:.E =intdE=(GM)/(4Rr^(2))[int_(r-R)^(r+R)dl+(r^(2)dl-R^(2))int_(r-R)^(r+R)(dl)/(l^(2))]`
`rArr E = (GM)/(r^(2)) r ge R`
If point is inside the shell limit changes to [(R -r) to R +r]
`E =0` where `r lt R`
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