A solid sphere of radius `R` has total charge `2Q` and volume charge density `p=kr` where `r` is distance from centre. Now charges `Q` and `-Q` are placed diametrically opposite at distance `2a` where a is distance form centre of sphere such that net force on charge `Q` is zero then relation between a and `R` is

Options

A. `a=2^(-1//4)R`
B. `a=R/sqrt3`
C. `a=8^(-1//4)R`
D. `a=(3R)/2^(1//4)`

Answer

C

Explanation

Consider force on A due to B
`|F|=(Q^2)/(4piin_(0)(2a)^(2))=(Q^2)/(16piin_(0)a^(2))`

Since A is in equilibrium, force on it due to changed sphere, `F_(s)=F` (magnitude)
So, electric field at distance a from centre
`E=(Q)/(16piin_(0)a^(2))`
If `rgtR` , result for electric field is `E=(2Q)/(4piin_(0)r^(2))`
So, assuming `agtR," "E=(Q)/(16piin_(0)a^(2))=(2Q)/(4piin_(0)r^(2))" "rArr" "r=2sqrt(2)a" "rArr" "a=(v)/(2sqrt2)`
But the result was only valid for `rgtR" "rArr" "` The point is inside the sphere and `altR`
Let electric field inside the sphere at distance r from centre be E, then by gauss law
`E(4pir^(2))=(1)/(in_0)overset(r)underset(0)(int)kx(4pix^2)dx" "rArr" "E=(kr^2)/(4in_0)`
Also `overset(R)underset(0)(int)kx(4pix^2)dx=2Q" "rArr" "k=(2Q)/(piR^4)" "rArr" "E=(Qr^2)/(2piin_(0)R^(4))=(Qa^2)/(2piin_(0)R^(4))`
Comparing with `E=(Q)/(16piin_(0)a^(2))" "rArr" "a=8^(-1//4)R`