Question

Unique distribution of charge on a conductor

If i place some charge on a conductor then it will distribute itself in such a way that electric field everywhere inside is zero. My text book says that only one kind of such charge distribution is possible. Or if i place charges outside the conductor then charges will be induced on the surface of a conductor to make field inside zero and again such charge distribution is unique. Intuitively it all seems correct but i am trying to figure out right arguments for this.

One such argument is that suppose i place some charge on conductor then i can solve poisson's equation to find potential at the surface(at boundary). Charge distribution outside the conductor is known. Then uniqueness theorem says that only one such potential function is possible which satisfies the given/known boundary conditions and poisson's equation outside. Since a particular potential function corresponds to a particular charge distribution the charge distribution must also be unique. This argument seems correct to me but at the same time something seems missing. Am i correct?

Answer 1

In electrostatic equilibrium, charges distribute themselves on a conductor in such a way that the electric field inside is zero. The distribution of charges on the surface is unique. The uniqueness can be argued using the uniqueness theorem, which states that there is only one potential function that satisfies the given boundary conditions and Poisson's equation outside the conductor.

Step-by-step explanation:

When a conductor is in electrostatic equilibrium, the charge distributes itself in such a way that the electric field inside the conductor is zero. This is because any electric field inside the conductor would cause the charges to move, resulting in a nonstatic condition, which contradicts the assumption of electrostatic equilibrium. The distribution of charges on the surface of the conductor is unique, meaning that only one kind of charge distribution is possible under these conditions.

This uniqueness can be argued using the uniqueness theorem, which states that there is only one potential function that satisfies the given boundary conditions and Poisson's equation outside the conductor. Since a particular potential function corresponds to a particular charge distribution, the charge distribution must also be unique.