Physlib

Physlib.Electromagnetism.ThreeDimension.Basic

Three-Dimensional Electromagnetism

This directory provides a three-dimensional, vector-calculus-facing layer for electromagnetism.

The backend theory is formulated in a tensorial and dimension-general way. Here we re-express the relevant constructions in the familiar language of scalar and vector potentials, electric and magnetic fields, and spatial derivatives.

Fields from potentials

In this section we rewrite the electric and magnetic fields associated to an electromagnetic potential in terms of the scalar and vector potentials, using the standard vector-calculus expressions.

2 declarations

theorem

E=−∇ϕ−∂tA\mathbf{E} = -\nabla \phi - \partial_t \mathbf{A}

The electric field E\mathbf{E} is expressed in terms of the scalar potential ϕ\phi and the vector potential A\mathbf{A} as E(t,x)=−∇ϕ(t,x)−∂∂tA(t,x)\mathbf{E}(t, \mathbf{x}) = -\nabla \phi(t, \mathbf{x}) - \frac{\partial}{\partial t} \mathbf{A}(t, \mathbf{x}), where ∇\nabla denotes the spatial gradient and ∂∂t\frac{\partial}{\partial t} denotes the partial derivative with respect to time tt.

theorem

B=∇×A\mathbf{B} = \nabla \times \mathbf{A}

In three-dimensional electromagnetism, the magnetic field B\mathbf{B} at time tt and position x\mathbf{x} is defined as the curl of the vector potential A(t,x)\mathbf{A}(t, \mathbf{x}). That is, B(t,x)=∇×A(t,x)\mathbf{B}(t, \mathbf{x}) = \nabla \times \mathbf{A}(t, \mathbf{x}).