By Fabrizio Frezza
This ebook is a concise advent to electromagnetics and electromagnetic fields that covers the features of such a lot importance for engineering functions via a rigorous, analytical remedy. After an creation to equations and simple theorems, subject matters of basic theoretical and applicative significance, together with aircraft waves, transmission traces, waveguides and Green's services, are mentioned in a intentionally common method. Care has been taken to make sure that the textual content is quickly obtainable and self-consistent, with conservation of the intermediate steps within the analytical derivations. The publication deals the reader a transparent, succinct path in simple electromagnetic conception. it's going to even be an invaluable search for instrument for college students and designers.
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Extra info for A Primer on Electromagnetic Fields
The equation to be solved in this case is: ∇ 2 F + k 2 F = −Jmi . Let us mention now the corresponding procedure in the time domain. Let us suppose, for simplicity, that the medium is isotropic non-dispersive and non-dissipative. In the case Jmi = 0, it follows: ⎧ ∂H ⎪ ⎨ ∇ × E = −μ ∂t ∂E . ⎪ ⎩ ∇ × H = Ji + ε ∂t 46 1 Fundamental Theorems and Equations of Electromagnetism Applying again the divergence to the first equation for homogeneous media and then using the Schwarz theorem, we obtain: ∂ (∇ · H ) = 0 ∂t ∇ · H = constant ⇒ ⇒ ∇× E + μ ⇒ ∂A ∂t ∇·H =0 =0 ⇒ ⇒ E = −μ H = ∇×A ∂A − ∇V.
We therefore expressed the electromagnetic field in terms of a pair of potentials. This is not the only possible choice: for example, the so-called Hertz potentials can also be used. Note now that if one changes the vector potential according to the transformation A = A + ∇ (which leaves unchanged the magnetic field, as it should be) then one has to change the scalar potential in order to maintain the electric field unchanged too. In fact it follows: E = − jωμ (A − ∇ ) − ∇V = − jωμ A + jωμ ∇ − ∇V = = − jωμ A − ∇(V − jωμ ) = − jωμ A − ∇V 44 1 Fundamental Theorems and Equations of Electromagnetism having put V = V − jωμ .
We start this time from the transformed Maxwell’s equations: ∇ × E = −Jmi − jω B . ∇ × H = Ji + Jc + jω D We scalar multiply the first equation by H ∗ , while we conjugate the second5 and then multiply it by E: H ∗ · ∇ × E = −H ∗ · Jmi − jω H ∗ · B, E · ∇ × H ∗ = E · J i∗ + E · J c∗ − jω E · D ∗ . Subtracting member to member, we have: H ∗ · ∇×E − E · ∇× H ∗ = −Jmi · H ∗ − J i∗ · E − J c∗ · E − jω B · H ∗ + jω E · D ∗ . 5 Note that ∇ 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.
A Primer on Electromagnetic Fields by Fabrizio Frezza