chapter  1
Pages 78

Early studies of the properties of electric and magnetic ﬁelds, their

interplay and the resulting emission of radiation, led to formation

of several fundamental laws by Carl F. Gauss (1777-1855), Michael

Faraday (1791-1867), Andre´ M. Ampe`re (1775-1836), and James

C. Maxwell (1831-1879). These were presented in a compact

diﬀerential formalism by Maxwell (1865), which are referred to as

Maxwell’s equations:

∇ · D(r, t) = ρ(r, t), (1.1) ∇ · B(r, t) = 0, (1.2) ∇ × E(r, t) = −∂B(r, t)

∂t , (1.3)

∇ × H(r, t) = J(r, t)+ ∂D(r, t) ∂t

. (1.4)

The electric E(r, t) and the magnetic ﬁelds B(r, t), are time dependent and can be speciﬁed at every point in space and time, in

which r(= x , y, z) denotes the position vector, t is the time, J = σE, the electric current density, q(= 1.6 × 10−19 C) the elementary charge, E and B, the electric ﬁeld strength and the magnetic induction, respectively, H = B/μ, the magnetic ﬁeld, D = E, the

electric displacement, ρ, the volume density of free charge, , μ, and

σ the respective dielectric permittivity, the magnetic permeability,

and conductivity of the medium, “×,” the cross product, and

∇ = i ∂ ∂x

+ j ∂ ∂y

+ k ∂ ∂z . (1.5)

represents a linear vector diﬀerential operator.