Electric potential generated by the accumulated charge on a grounded spherical conductor near a point charge

Question

Suppose that a positive point charge is placed near a grounded spherical conductor. According to my understanding:

1. the point charge would generate a positive, generally non-uniform electric potential at every point in the sphere;
2. the sphere would "absorb" a certain number of electrons from the earth and redistribute them on its surface in such a way that the electric potential generated by the point charge is nullified at every point in the sphere.

My question is: would it be wrong to assume that the accumulated charge on the surface of the sphere is uniformly distributed?

In other words, would it be wrong to say that the electric potential generated by the accumulated charge would be the same as that of a point charge $Q$, where $Q$ is the accumulated charge on the surface of the sphere?

Answer 1

$\dots$ the point charge would generate a positive, generally non-uniform electric potential at every point in the sphere $\dots$

Electrostatics and a conductor $\Rightarrow$ the potential of the surface of thesphere is the same everywhere.

$\dots$ would it be wrong to assume that the accumulated charge on the surface of the sphere is uniformly distributed?

Yes, it would be wrong.

Suppose the point charge is positive and the sphere was uncharged.
The work done in moving a unit positive charge from infinity passing the positive charge and then reaching the sphere would be different from the work done in moving the unit positive charge from infinity to the other side of the sphere (and positive point charge), ie the potentials would be different.

What actually happens is the charges on the sphere redistribute themselves so that there is a larger negative change density on the parts of the sphere close to the positive point charge.
This will mean that along whatever path a unit positive charge travels from infinity to the sphere the work done is always the same.

Some maths - Point Charge and a Grounded Sphere