Physlib.Electromagnetism.ThreeDimension.Basic

2 declarations

theorem

$\mathbf{E} = -\nabla \phi - \partial_t \mathbf{A}$

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

theorem

$\mathbf{B} = \nabla \times \mathbf{A}$

#magneticField_eq_3D

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

Physlib.Electromagnetism.ThreeDimension.Basic

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

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

$\mathbf{E} = -\nabla \phi - \partial_t \mathbf{A}$

$\mathbf{B} = \nabla \times \mathbf{A}$