A metal sphere of radius r₁ carries a positive charge of amount Q. A concentric spherical metal shell with inner radius r₂ and outer radius r₃ surrounds the inner sphere and carries a total positive charge of amount 4Q, with some of this charge on the outer surface (at r₃) and some on the inner surface (at r₂).(a) How is the charge 4Q distribute on the two surfaces of the outer shell? Prove this! (b) What is the potential (relative to infinity) just outside r₃ halfway between r₂ and r₃ just inside r₂, just outside r₁ and at the center?

The charge of metal sphere is Q₁ = Q.

The radius of metal sphere is r₁.

The total positive charge on concentric shell is Q₂ + Q₃= 4Q

The net charge between metal sphere and inner surface of shell should be zero to maintain the polarity there.

The net charge between metal sphere and inner surface of the shell is zero, so Q₁ + Q₂ = 0

Here, Q₂ is the charge on the inner surface of shell.

Substitute all the values in the above equation.

Q + Q₂ = 0

Q₂ = - Q

The charge on the outer surface of concentric shell is given as: Q₂ + Q₃= 4Q

Here, Q₃ is the charge on the outer surface of shell.

Substitute all the values in the above equation.

$$\begin{gathered} -Q+Q_3=4Q \\ {Q}_3=5Q \end{gathered}$$

Therefore, the distributed on the inner surface and outer surface of the concentric shell are -Q and Q5 .