Electrostatics MOC

Electrostatic Potential

Electric potential is a scalar field associated with an Electric field. It is a form of a scalar potential, and is therefore only possible since ⃗𝐄 is irrotational for static charge distributions.

Which can be converted back into an ⃗𝐄-field as follows

Note in order to define a field 𝑉 :ℝ3 →ℝ we have to pick a reference point O where the potential is zero. This is because voltage is a quantity of difference, not an absolute quantity. Conventionally this is taken to be infinitely far from the charge distribution, where ⃗𝐄 goes to zero.

Intuitively, regions of positive charge are potential hills, regions of negative charge are potential valleys, and the electric field points down the slope.

Electric potential is scalar additive, hence it obeys the Principle of Superposition.¹

For the more general (dynamic) case, see Electric and magnetic potentials.

Poisson's equation

Due to Gauß's law, an electric potential must satisfy Poisson's equation

where ∇2 =⃗∇ ⋅⃗∇ is the Laplacian. It follows that the potential satisfies Laplace's equation in all regions empty of charge.

From a charge distribution

It follows easily from the Electric potential of a point charge that given the volume Charge density 𝜌(⃗𝐫′), it follows

where ℝ3 represents all space, and by convention 𝑉 →0 as 𝑟 →∞.

Boundary conditions

At any boundary, the derivative of the potential normal to the boundary is discontinuous², with

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