Maxwell's Equations and Associated Laws in Various Forms

Name

Vector Calculus

Tensor Analysis

Spacetime Algebra

Lorenz gauge condition

\(\nabla\cdot\mathbf{A}+\frac{1}{c^2}\frac{\partial\varphi}{\partial t}=0\)

\(\partial_\mu A^\mu=0\)

\(\square\cdot A=0\)

Electromagnetic potential

\(\square^2\mathbf{A}=\mu_0\mathbf{J}\)

\(\square^2\varphi=\rho/\varepsilon_0\)

\(\partial_\mu\partial^\mu A^\nu=\mu_0J^\nu\)

\(\square^2 A=\mu_0J\)

Electromagnetic fields

\(\mathbf{E}=-\nabla\varphi-\frac{\partial\mathbf{A}}{\partial t}\)

\(\mathbf{B}=\nabla\times\mathbf{A}\)

\(F^{\mu\nu}=2\partial^{[\mu}A^{\nu]}\)

\(F=\square\wedge A\)

Maxwell's equations

\(\nabla\cdot\mathbf{D}=\rho\)

\(\nabla\times\mathbf{H}-\frac{\partial\mathbf{D}}{\partial t}=\mathbf{J}\)

\(\nabla\cdot\mathbf{B}=0\)

\(\nabla\times\mathbf{E}+\frac{\partial\mathbf{B}}{\partial t}=0\)

\(\partial_\mu F^{\mu\nu}=\mu_0 J^\nu\)

\(\partial_{[\mu}F_{\nu\sigma]}=0\)

\(\square F=\mu_0J\)

Continuity equation

\(\nabla\cdot\mathbf{J}+\frac{\partial\rho}{\partial t}=0\)

\(\partial_\mu J^\mu=0\)

\(\square\cdot J=0\)

Ohm's law

\(\mathbf{J}=\sigma(\mathbf{E}+\mathbf{v}\times\mathbf{B})\cosh\zeta\)

\(J^\mu=\sigma F^{\mu\nu}U_\nu\)

\(J=\sigma F\cdot U\)

Definitions

Vectors

\(\mathbf{A}\) is the magnetic vector potential
\(\varphi\) is the scalar electric potential
\(\mathbf{J}\) is the current density
\(\rho\) is the charge density
\(\mathbf{E}\) is the electric field
\(\mathbf{H}\) is the magnetic field
\(\mathbf{D}\) is the electric displacement
\(\mathbf{B}\) is the magnetic flux density
\(\sigma\) is the bulk conductivity
\(\cosh\zeta\) is the Lorentz factor
\(\nabla=\mathbf{x}\frac{\partial}{\partial x}+\mathbf{y}\frac{\partial}{\partial y}+\mathbf{z}\frac{\partial}{\partial z}\) is the del operator
\(\square^2=\frac{\partial^2}{\partial t^2}-\nabla^2\) is the d'Alembertian operator

Tensors

\(A^\mu=\left(\frac{\varphi}{c},A_x,A_y,A_z\right)\) is the four-potential
\(J^\mu=\left(c\rho,J_x,J_y,J_z\right)\) is the four-current density
\(F^{\mu\nu}\) is the Faraday tensor field
\(U^\nu=\left(c,v_x,v_y,v_z\right)\cosh\zeta\) is the four-velocity
\(\partial_\mu=\left(\frac{1}{c}\frac{\partial}{\partial t},\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right)\) is the spacetime gradient operator

Spacetime Algebra

\(A=A^\mu\gamma_\mu\) is the four-vector potential
\(J=J^\mu\gamma_\mu\) is the four-vector current density
\(F\) is the Faraday bivector field
\(U=U^\mu\gamma_\mu\) is the four-vector velocity
\(\square=\gamma^\mu\partial_\mu\) is the spacetime gradient operator
Geometric product: \(AB=A\cdot B+A\wedge B\)

For more information about the relativistic form and application of these equations, the reader is referred to my book, Relativistic Field Theory for Microwave Engineers :

Now available at BookBaby and Amazon.com.