Introduction:
The problem asks us to find the minimum distance from the surface of a charged sphere where the electric potential is half of the electric potential at its center. We are given that the charge is uniformly distributed inside the sphere.

Electric Potential:
The electric potential at a point is given by the equation V = k * q / r, where V is the electric potential, k is the electrostatic constant, q is the charge, and r is the distance from the center of the sphere.

Electric Potential at the Center:
At the center of the sphere, the distance from the center is 0, so the electric potential at the center can be calculated as V(center) = k * q / 0, which is undefined.

Electric Potential on the Surface:
On the surface of the sphere, the distance from the center is equal to the radius R. Therefore, the electric potential on the surface can be calculated as V(surface) = k * q / R.

Electric Potential at a Distance r:
At a distance r from the center of the sphere, the electric potential can be calculated as V(distance r) = k * q / r.

Half the Electric Potential:
Now, we need to find the minimum distance from the surface where the electric potential is half of the electric potential at the center. Mathematically, we can express this condition as:

V(distance r) = V(center) / 2

Substituting the values, we get:

k * q / r = (k * q / 0) / 2

Since the electric potential at the center is undefined, we cannot solve the equation directly. However, we can take the limit of the electric potential as the distance approaches zero.

Taking the Limit:
lim(r→0) (k * q / r) = (k * q / 0) / 2

As r approaches zero, the electric potential at the center approaches infinity. Therefore, we can rewrite the equation as:

∞ = ∞ / 2

This equation is not solvable as it leads to an indeterminate form.

Conclusion:
In conclusion, the minimum distance from the surface of the charged sphere where the electric potential is half of the electric potential at its center is undefined. This is because the electric potential at the center of the sphere is undefined.