Final answer:
The charge on a conducting sphere at a constant potential V₀ in equilibrium is distributed uniformly on its surface, with an electric field outside the sphere that behaves as if the charge were localized at the center and no electric field or charge inside the sphere.
Step-by-step explanation:
When considering a conducting sphere of center 0 and radius R in equilibrium, placed at a constant potential V₀, the charge distribution displays a unique behavior. At equilibrium, the excess charge on a conductor redistributes itself to minimize the total potential energy of the system, leading to all excess charge residing on the outer surface of the sphere. This results in a uniform surface charge density on the sphere. Inside the sphere, there is no electric field, and thus no charge is present.
Using Gauss's law, we can determine that the electric field both inside and outside the sphere is symmetric and emanates radially from the surface. Since the sphere is at constant potential V₀, and we can say that outside the sphere, at a distance r from the center, the electric field would be identical to that of a point charge located at the center of the sphere. Significant to note is that in the region r > R, the system's potential and electric field are influenced only by the total charge Q on the sphere, as if the charge were concentrated at the center. in summary, the charge distributes uniformly on the surface of the sphere only, with no electric field within the conductor. This is because the sphere itself forms an equipotential surface, meaning that every point on its surface has the same potential V₀.