4. Electromagnetism#
From 3-dimensional to 4-dimensional formalism. In this section the 4-dimensional formalism is introduced, and the 4-dimensional version of governing equations of electromagnetism that naturally suits special relativity theory is derived starting from the governing equations of classical electromagnetism.
Definition of physical quantities
4-potential vector, \(\mathbf{A} = \frac{\phi}{c} \mathbf{E}_0 + a^i \mathbf{E}_i\)
4-current vector, \(\mathbf{J} = c \rho \mathbf{E}_0 + j^i \mathbf{E}_i\)
EM field tensor,
\[\begin{split}\begin{aligned} \mathbf{F} & = - \mathbf{E}_0 \otimes \mathbf{E}_i \dfrac{e_i}{c} + \dfrac{e_i}{c} \mathbf{E}_i \otimes \mathbf{E}_0 + \varepsilon_{ijk} b_j \mathbf{E}_i \otimes \mathbf{E}_k \\ \mathbf{D} & = - \mathbf{E}_0 \otimes \mathbf{E}_i c d_i + d_i \mathbf{E}_i \otimes \mathbf{E}_0 + \varepsilon_{ijk} h_j \mathbf{E}_i \otimes \mathbf{E}_k \\ \end{aligned}\end{split}\]or
\[\begin{split}\begin{aligned} \left[F^{\alpha \beta}\right] = \begin{bmatrix} 0 & - \mathbf{e}^T / c \\ \mathbf{e}/c & \mathbf{b}_{\times} \end{bmatrix} \qquad , \qquad \left[D^{\alpha \beta}\right] = \begin{bmatrix} 0 & - c \mathbf{d}^T \\ c \mathbf{d} & \mathbf{h}_{\times} \end{bmatrix} \\ \end{aligned}\end{split}\]4-momentum-energy density tensor, \(\mathbf{T}\)
Definitions, relations and equations
Lorentz’s gauge, \(\nabla \cdot \mathbf{A} = 0\)
EM field tensor, \(\mathbf{F} = \nabla \mathbf{A} - \left( \nabla \mathbf{A} \right)^T\)
4-current continuity equation, \(\nabla \cdot \mathbf{J} = 0\)
Maxwell’s equations
\[\begin{split}\begin{aligned} \mathbf{\nabla} \cdot \mathbf{D} & = \mathbf{J}^f \\ \nabla \cdot \left( \symbf{\epsilon} : \mathbf{F} \right) & = \mathbf{0} \end{aligned}\end{split}\]Constitutive equations
\[\mathbf{D} = \dfrac{1}{\mu_0} \mathbf{F} + \mathbf{P}\]Wave equation for the pontential (what about wave equations for the EM field?)
\[\nabla \cdot \nabla \mathbf{A} = \mu_0 \mathbf{J}\]Energy balance equation
\[\nabla \cdot \mathbf{T} = - \mathbf{F} \cdot \mathbf{J}\]Equation of motion of a point charge in a EM field,
\[m \mathbf{X}'' = q \mathbf{F} \cdot \mathbf{X}'\]
Einstein’s special relativity and Lorentz’s transformations for the EM quantities. After the equations are derived, Lorentz’s transformations are used to discuss special relativity. Low-speed relations, used in classical electromagnetism for systems with charateristic speed \(v \ll c\), are derived.
Lagrangian approach to electromagnetism in special relativity. Weak form of the equations are derived, and the Lagrangian approach to electromagnetism in special relativity is discussed: both field equations and dynamical equations of charges moving in an electromagnetic field are re-derived with a Lagrangian approach.