The Electrostatic Field at the Surface of a Charged Conductor

Introduction:
When a conductor is charged, the charges distribute themselves in such a way that the electric field inside the conductor is zero. At the surface of the conductor, the electric field must be perpendicular to every point. This is due to the following reasons:

1. Electrostatic Equilibrium:
A charged conductor is in a state of electrostatic equilibrium, meaning there is no net movement of charges within the conductor. In this state, the electric field inside the conductor is zero. If the electric field were not perpendicular to the surface, it would indicate the presence of a component of the electric field along the surface, leading to a non-zero electric field inside the conductor. This would violate the condition of electrostatic equilibrium.

2. Gauss's Law:
Gauss's law states that the electric flux through a closed surface is proportional to the total charge enclosed by that surface. For a charged conductor, the electric field lines must be perpendicular to the surface at every point to ensure that the electric flux passing through the surface is zero. If the electric field were not perpendicular, there would be a non-zero flux passing through the surface, which contradicts Gauss's law.

3. Charge Redistribution:
When charges are placed on a conductor, they redistribute themselves in such a way that the electric field inside the conductor is zero. This redistribution of charges ensures that the electric field lines are perpendicular to the surface. If the field lines were not perpendicular, there would be a non-zero electric field inside the conductor, indicating that charges are still moving and redistributing. This would violate the condition of electrostatic equilibrium.

Conclusion:
The electrostatic field at the surface of a charged conductor must be perpendicular to every point on it to satisfy the conditions of electrostatic equilibrium, Gauss's law, and the redistribution of charges. This ensures that the electric field inside the conductor is zero and that the charges are in a stable equilibrium state.