A uniformly charged solid sphere of radius `R` has potential `V_(0)` (measured with respect to `oo`) on its surface. For this sphere the equipotential surfaces with potentials `(3V_(0))/(2) , (5 V_(0))/(4), (3V_(0))/(4)` and `(V_(0))/(4)` have radius `R_(1), R_(2), R_(3)` and `R_(4)` respecatively. Then

Options

A. `R_(1)=0` and `R_(2) lt (R_(4)-R_(3))`
B. `2R lt R_(4)`
C. `R_(1)=0` and `R_(2) gt (R_(4)-R_(3))`
D. `R_(1) ne 0` and `(R_(2)-R_(1)) gt (R_(4)-R_(3))`

Answer

C, D

Explanation

For `r lt R " " V=(kQ)/(2R)(3-r^(2)/R^(2))`
& for `r gt R" "V=(kQ)/r`
at `r=R" "V_(0)=(kQ)/R`
at `r=0" "V_(0)=3/2(kQ)/R=3/2 V_(0)`
`:. R_(1)=0`
for `V=5/4 V_(0) (kQ)/(2R)=(3-r^(2)/R^(2))=5/4.(kQ)/R`
`R_(2)=R/sqrt(2)`
for `V=3/4 V_(0)" "(kQ)/R_(3)=3/4 (kQ)/R`
`R_(3)=(4R)/3`
for `V=V_(0)/4" "(kQ)/R_(4)=(kQ)/(4R_(0))`
`R_(4)=4R_(0)`