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Electrostatics is a field of physics that deals with the study of forces, fields, and potentials arising from stationary or slow-moving electric charges. At the core of electrostatics is the concept that like charges repel each other, while opposite charges attract. These interactions are quantified using laws such as Coulomb's law, which calculates the electrostatic force between two point charges.

In electrostatics, the electric potential ( V ) plays a crucial role. It is a scalar quantity that represents the electric potential energy per unit charge at a point in space. The electric potential is a measure of the work done to move a unit positive charge from a reference point (usually infinity) to a particular point in the field without accelerating. This potential is related to the electric field, which is a vector quantity representing the force per unit charge. Understanding these foundational concepts is essential when analyzing scenarios involving conductive materials in electrostatic equilibrium, such as a hollow metal sphere.

A hollow metal sphere is an intriguing object in electrostatics because of its properties when placed in an electric field. The key point to understand is that a conductor in electrostatic equilibrium has a set of unique characteristics.

• First, the electric field inside a conductor is zero, which is why the hollow region inside the sphere lacks an electric field.
• The charges reside on the surface of the conductor, creating a uniform potential across both the interior and surface of the conductor.

This uniform potential means that if we know the potential on the surface, as given in the problem (+400 V), this potential is the same throughout the hollow interior, including the center.

Therefore, the potential at any point inside a hollow conductor such as this one remains constant, regardless of where you measure it (surface or center), provided that the conductor is isolated and remains undisturbed by external charges.

Electrostatic equilibrium is a state reached when electric charges are at rest and the electric field within a conductor is zero. In this condition, several important properties about conductors can be noted:

• The electric field inside the conductor is zero, eliminating the possibility of charge movement within the material itself.
• Any excess charge resides on the conductor's surface, not inside.
• The potential is constant throughout the conductor, meaning there is no potential difference (\(dV = 0\)) that would otherwise cause charge migration.

For a hollow conductive sphere, this results in a uniform electric potential, as seen in the problem. Thus, even at the center, the potential is dictated by the surface value (+400 V in this case) because of the absence of internal electric fields and charge redistribution capabilities.