Tensors for Physics

7.5 Maxwell Equations in Differential Form 103

The current density and the charge density are the sources for the vector potential
and the scalar potential. Electromagnetic radiation is caused by accelerated charges,
then the time derivatives in (7.68) are essential. For a stationary situation, on the
other hand, the time derivatives vanish and the second of these equations reduces
to the Poisson equation, cf. Sect.7.3.2. The first of the equations (7.68), linking the
vector potential with the current density is mathematically equivalent to the Poisson
equation, when a stationary situation is considered. It determines the magnetic field
generated by a steady current, as formulated in theBiot-Savart relation.

7.5.5 Magnetic Field Tensors

In 3D, there exists a dual relation between an antisymmetric second rank tensor and
a vector, cf. Sects.3.3and4.1.3. This allows to replace the magnetic field vectorsB
andHby antisymmetric tensors. To see what this means, consider the homogeneous
Maxwell equationελσ τ∇σEτ=−∂∂tBλ. Multiplication of this equation byεμνλand
use ofεμνλελσ τ=δμσδντ−δμτδνσyields

∇μEν−∇νEμ=−

∂

∂t

εμνλBλ.

Both sides of this equation, which is equivalent to the first of the Maxwell equations
(7.57), are antisymmetric tensors. The right hand side can be expressed in terms of
the magnetic field tensorBμν, which is related to the vector fieldBλ,by

Bμν≡εμνλBλ. (7.69)

In matrix notation, this relation is equivalent to

Bμν:=

⎛

⎝

0 B 3 −B 2

−B 3 0 B 1

B 2 −B 1 0

⎞

⎠. (7.70)

The reciprocal relation between the vector and the antisymmetric tensor is

Bλ=

1

2

ελμνBμν. (7.71)

The field tensor is linked with the vector potentialAvia

Bμν=∇μAν−∇νAμ. (7.72)

The Lorentz ForceF,viz.

Fμ=eEμ+eεμλνBλvν,