Abstract
After some elements about differential operators and their analytical properties, the general basic characteristics of electromagnetic fields are described, both in time and in frequency domain. Maxwell’s equations, boundary conditions, constitutive relations are treated. The material media properties are investigated and an introductory treatment of dispersion is given. The fundamental Poynting and uniqueness theorems are derived. The wave equation is obtained. Finally, electromagnetic potentials are introduced.
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Notes
- 1.
A dyad is defined in Sect. 1.7.
- 2.
For example, the current flowing in a transmitting antenna.
- 3.
For example, the current on the receiving antenna.
- 4.
Even a resonator, which is the typical homogeneous system (the so-called free oscillations, i.e. without forcing) actually has losses and requires excitation.
- 5.
Note that \(\nabla \) is an operator called real, as it consists of derivations with respect to real variables and real unit vectors, and so if it operates on a real-valued function, then the result also is a real function. It is also said that \(\nabla \) commutes with the operation of conjugation:
$$\begin{aligned} \left( \nabla \!\times \!\underline{H}\right) ^*=\nabla \!\times \!\underline{H}^*=\underline{J}^*_{\,i}+\underline{J}^*_{\,c}-j\omega \,\underline{D}^*. \end{aligned}$$.
- 6.
Recall that for real vectors the quantity:
$$\begin{aligned} A^2=\underline{A}\,\mathbf{\cdot }\,\underline{A}=A_x^2+A_y^2+A_z^2, \end{aligned}$$can be defined as the square of a vector and so it is defined as positive, and its square root \(|\underline{A}|=+\sqrt{\underline{A}\,\mathbf{\cdot }\,\underline{A}}\,\) can be assumed as the modulus of the vector. In the case of complex vectors, \(A^2=\underline{A}\,\mathbf{\cdot }\,\underline{A}\) is not even real in general. To obtain a positive real quantity the modulus is defined as:
$$\begin{aligned} |\underline{A}|=+\sqrt{\underline{A}\,\mathbf{\cdot }\,\underline{A}^*}=+\sqrt{A_xA_x^*+A_yA_y^*+A_zA_z^*}=+\sqrt{|A_x|^2+|A_y|^2+|A_z|^2}. \end{aligned}$$Often in the literature we talk about “amplitude” instead of the “modulus” although, however, the term amplitude may also sometimes denote a multiplicative complex factor.
- 7.
Power that goes back and forth in capacitors and inductors.
- 8.
This case corresponds as already seen to the possible presence of dielectric losses.
- 9.
And so there are no dielectric losses.
- 10.
With \(\underline{\varepsilon }^{{\scriptstyle T}*}\) we mean the conjugate transpose of \(\underline{\underline{\varepsilon }}\).
- 11.
Then the light in material media is slower than in a vacuum.
- 12.
So, for example, using dielectrics with high \(\varepsilon _r\) we can miniaturize components, such as for example resonators.
- 13.
Note that in frequency domain the corresponding formula shows a division by \(j\omega \) instead of an integration with respect to \(t\).
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© 2015 Springer International Publishing Switzerland
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Frezza, F. (2015). Fundamental Theorems and Equations of Electromagnetism. In: A Primer on Electromagnetic Fields. Springer, Cham. https://doi.org/10.1007/978-3-319-16574-5_1
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DOI: https://doi.org/10.1007/978-3-319-16574-5_1
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