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Conductors in electrostatic equilibrium are fascinating because they showcase a fundamental property of electric fields and potentials. When we say a conductor is in electrostatic equilibrium, we mean that there are no net motions of electric charge within it. This condition has several important implications.

Firstly, inside a conductor at electrostatic equilibrium, the electric field is zero. Why? Because if there were any electric fields inside, charges would move along these fields, disrupting equilibrium. This absence of an internal electric field ensures that the surface of the conductor is the only place where excess charges can be found.

Secondly, the electric potential remains constant throughout the conductor's volume. This means that every point inside the conductor, including on the surface, shares the same electric potential. This constant potential is a consequence of having no electric fields inside. Since electric potential is closely related to electric fields, and with fields being absent inside, the potential cannot vary. Therefore, when we calculate potential for points within a conductor, we know it is the same everywhere inside.

A hollow metal sphere is a remarkable example of conductors in electrostatic equilibrium. Imagine a metal shell with empty space inside, similar to a bubble. When this sphere is in electrostatic equilibrium, the principles of conductors apply here as well.

Inside this hollow part of the sphere, the electric field is zero, provided that the sphere contains no charges in the cavity. This means that within that empty space, any point is effectively shielded from external electric fields. It's quite amazing how the metal surface blocks out external electric influences.

Even though the sphere may carry an electric charge, all this charge stays on its outer surface. This exterior charge distribution allows charges to arrange themselves in such a way that they cancel their collective internal electric fields. This property allows for uniform potential across the whole structure, including the very center of the hollow. For someone at the center of the sphere, it would be as if they were in a magical charge-free zone, perfectly safe from external electric forces.

Charge distribution in conductors, specifically in hollow metal spheres, is a key concept in understanding electrostatics. When we talk about charge distribution, we refer to how charge is spread out over the surface of a conductor.

In the case of our hollow metal sphere, although the sphere may carry a certain amount of net charge, that charge is distributed uniformly over the outer surface of the sphere. This uniform distribution is crucial. It's driven by the mutual repulsion of like charges, which push themselves as far apart as possible while maintaining an equilibrium condition.

The charge on a hollow sphere's surface ensures that there are no excess charges within the hollow. Therefore, not only does the outer surface have charge, but also the interior surface does not feature any. This careful charge arrangement maintains uniform potential and zero electric field inside the sphere. This uniformity is essential for ensuring that any point within the hollow has the same potential value as the surface, safeguarding the principle of electrostatic equilibrium.

Uniform potential in the context of a conductor in electrostatic equilibrium is a direct result of zero internal electric fields. The term "uniform potential" means that the electric potential is the same at every location within the conductor.

For the hollow metal sphere from our example, the uniform potential is demonstrated by its constant value from the surface all the way to the center. The potential does not drop, rise, or change as you move closer to the center of the sphere. It remains equal everywhere.

This uniform potential ensures that there is no potential difference between any two points within the conductor. As a result, there are no forces to move charges around, reinforcing the condition of electrostatic equilibrium. For the sphere with a potential of 400 V, this value is maintained consistently both on the surface and inside. So, for any point you consider, whether at the core or on the crust, the potential remains 400 V, a beautiful illustration of nature’s symmetry in electrostatics.