Basic Equations and Notations
This chapter is dedicated to introduce the most important equations needed for the main part of the thesis. One of the basic tasks of electrodynamics is to find the electric and magnetic fields for given charge and current distributions. For this sake, Ma
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xwell’s Equations Maxwell’s equations are solved for given initial or boundary conditions. In differential form they are explicitly given as [9, p. 238] · E( r, t) = 1 ρ(r, t) , ∇ ε0 · B( r, t) = 0 , ∇
(3.1) (3.2)
r, t) , × E( r, t) = − ∂ B( ∇ ∂t r, t) . × B( r, t) = μ0j(r, t) + μ0 ε0 ∂ E( ∇ ∂t
(3.3) (3.4)
Equation (3.3) is also called Faraday’s law and Eq. (3.4) is Maxwell’s version of Ampere’s law. Eq. (3.1) is Gauss’s law and Eq. (3.2) is Gauss’s law for magnetism. r, t) indicates the electric field, B( r, t) the magnetic field, j(r, t) the electric current density E( and ρ(r, t) the electric charge density. All quantities are given in SI units and evaluated at point r and time t. ε0 is the vacuum permittivity and μ0 the vacuum permeability. They are related via ε0 μ0 = 1/c2 , where c is the speed of light in vacuum. Note the difference in the meaning of density: The current density is given as current per area, whereas the charge density has the units of charge per volume. The above set of differential equations has to be completed with the continuity equation [9, p. 238], · j(r, t) = 0 , ρ( ˙ r, t) + ∇
(3.5)
which relates the charge density of the system to the current which is flowing out of the system. In our systems, (3.5) is always fulfilled, but in the general case, the right side of the equation can have a non-zero value due to e.g. external fields and sources. The dot on top of ρ is the short notation of the time derivative ∂/∂t. We will mainly use the dot in this thesis.
S. Nanz, Toroidal Multipole Moments in Classical Electrodynamics, BestMasters, DOI 10.1007/978-3-658-12549-3_3, © Springer Fachmedien Wiesbaden 2016
3 Basic Equations and Notations
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3.2 Wave Equation and Helmholtz Equation Maxwell’s equations describe the electromagnetic field, but it is not directly obvious that they also include an equation for describing an electromagnetic wave. This task is accomplished when Faraday’s law (3.3) and Maxwell’s version of Ampere’s law (3.4) are combined, which yields two wave equations for the propagation of the electric and the magnetic field, respectively [9, p. 246]:
∂ 1 ∂2 1 E(r, t) = μ0 j(r, t) + ∇ρ( r, t) , c2 ∂t2 ∂t ε0 2 r, t) − 1 ∂ B( r, t) = −μ0 ∇ × j(r, t) . ΔB( c2 ∂t2 r, t) − ΔE(
(3.6) (3.7)
Since every time-dependent function can be represented as a decomposition into Fourier components [9, p. 407], we will use, where useful, harmonic time dependence e −iωt , by just considering one term of the representation j(r, t) = √1 2π
˜ e −iωtj(r, ω) dω .
(3.8)
Of course, the same decomposition is made for the charge density and all quantities derived from this, e.g. electric and magnetic fields. The sign of the argument of the exponential function is arbitrary, we use the minus sign in this thesis. Regarding the prefactor we choose the convention √
of symmetric prefactors 1/
2π
for direct and inverse transformation. To keep notation clear,
we will omit the tilde from now on at quantities given in the frequency domain;
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