Answer:

Introduction:
In this problem, we need to find the electrostatic potential energy stored outside the spherical shell when a point charge Q is placed at the centre of an isolated conducting shell of radius R which contains charge Q distributed uniformly over it.

Explanation:
Let's consider a Gaussian surface, which is a sphere of radius r, outside the charged shell. The electric field at any point on the Gaussian surface due to the charge Q at the centre is given by Coulomb's Law as:

E = Q/(4πε₀r²)

Since the charge is uniformly distributed over the shell, the charge enclosed within the Gaussian surface is:

q = Qr²/R²

According to Gauss's Law, the electric field due to the uniformly charged shell at any point outside it is the same as the field due to a point charge Q located at the centre of the shell. Therefore, the electric field at any point on the Gaussian surface due to the uniformly charged shell is also given by Coulomb's Law as:

E' = q/(4πε₀r²) = Qr²/(4πε₀R²r²)

The total electric field E_total at any point on the Gaussian surface is the vector sum of the electric fields due to the point charge Q and the uniformly charged shell, given by:

E_total = E + E' = Q/(4πε₀r²) + Qr²/(4πε₀R²r²)

The electrostatic potential energy U stored outside the shell is given by:

U = ∫E_total·dS

where the integral is taken over the entire surface of the Gaussian sphere. Since the electric field is radial, the dot product reduces to:

U = ∫E_total·dS = ∫E_totaldS = ∫E_total(4πr²)dr

Substituting the expression for E_total and integrating over r from R to ∞, we get:

U = Q²/(8πε₀R)

Therefore, the electrostatic potential energy stored outside the spherical shell is given by U = Q²/(8πε₀R).