422 Appendix: Exercises...
which are equivalent to the homogeneous Maxwell equations, cast into the pertaining
three-dimensional form? Introduce a three-by-three field tensor Fijand formulate
their homogeneous Maxwell equations. How about the inhomogeneous Maxwell
equations in flatland?
The antisymmetric field tensor is defined by
Fik:=∂kΦi−∂iΦk, i,k= 1 , 2 , 3.In matrix notation, it reads
Fik:=⎛
⎜
⎝
0 B^1 cE 1−B (^01) cE 2
−^1 cE 1 −^1 cE 2 0
⎞
⎟
⎠.
From the definition of the field tensor follows
∂ 1 F 23 +∂ 2 F 31 +∂ 3 F 12 = 0.This Jacobi identity corresponds to
∂ 1 E 2 −∂ 2 E 1 =−
∂B
∂t,
The two-dimensionalD-field and the H-field tensor are combined in the tensor
Hik:=⎛
⎝
0 HcD 1
−H 0 cD 2
−cD 1 −cD 2 0⎞
⎠. (A.20)
The inhomogeneous Maxwell equations are equivalent to
∂kHik=Ji.The casei=3 corresponds to
∂ 1 D 1 +∂ 2 D 2 =ρ.The casesi=1 andi=2are
∂ 2 H=j 1 +∂D 1 /∂t, −∂ 1 H=j 2 +∂D 2 /∂t.Also in flatland, constitutive relations are needed to close the Maxwell equations.
In vacuum and in a linear medium, the linear relationHik ∼ Fikapplies. This
corresponds toDi∼Ei,i= 1 ,2 andH∼B.