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The way I initially justified the potential inside a charged conductor to be $\frac{kQ}{R}\ for\ \ 0\le r\ <R$ was that from Gaus's law, you could determine that when you were a distance $r$ away from a conducting sphere, $ V =\frac{kQ}{r}\ for\ \ R\le r\ $, and as you approached the surface of the sphere V tends to $\frac{kQ}{R}\ $. As E inside a conductor is always 0. there can be no further change in potential and so the potential inside a conductor is fixed at $V =\frac{kQ}{R}\ $.

However, if there were other charges around the conductor, the potential near the surface of the sphere would be larger than $\frac{kQ}{R}\ $ by the super position principle, also this potential would be different at different positions near the spheres surface. How would the potential inside the sphere change? It has to be fixed seeing as the electric field inside is still 0, but the potential at the surface of the sphere is now different at different points.

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. . . . but the potential at the surface of the [conducting] sphere is now different at different points.

Something has to happen if you in the realm of electrostatics.

Suppose that initially you had a conducting sphere which was positively charged.
That charge would be distributed uniformly across the surface of the conducting sphere.

Now if a positive charge is brought close to the conducting sphere it would also produce an electric field in the vicinity of the conducting sphere.

That electric field would make the surface charge on the conducting sphere move to ensure that the final state is such that the potential of the charged sphere is the same throughout.

Regions on the charged conducting sphere closer to the positively charge would suffer a reduction in surface charge density (become less positive) and regions on the other side of the conducting sphere would undergo an increase in the surface charge density (become more positive).

Think of a charge producing induced charges on a conductor which is in the vicinity of the charge.

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