0≤r≤a2:ρ=ρ1
a2≤r≤a:ρ=ρ2
The system is equivalent to two solid spheres (i) radius a, density ρ2 (ii) radius a2, density (ρ1−ρ2).
∴ potential at O:Vo=14πεo.32.4πa33ρ2a+14πεo.32.4π3a38×(ρ1−ρ2)×2a
=a2ρ22εo+a28εo(ρ1−ρ2)=a28εo(ρ1+3ρ2)
Potential on the surface at P :
VP=14πεo.4πa33ρ2a+14πεo.4π3a38(ρ1−ρ2)a
=a2ρ23εo+a2(ρ1−ρ2)24εo=a224εo(8ρ2+ρ1−ρ2)
a224εo(ρ1+7ρ2)
Vo=2Vp, then, 2(ρ1+7ρ2)=(3ρ1+9ρ2)
i.e., 2ρ1+14ρ2=3ρ1+9ρ2
ρ=5ρ2