By Fabrizio Frezza

This ebook is a concise creation to electromagnetics and electromagnetic fields that covers the points of so much importance for engineering purposes via a rigorous, analytical therapy. After an creation to equations and easy theorems, issues of primary theoretical and applicative significance, together with airplane waves, transmission traces, waveguides and Green's services, are mentioned in a intentionally basic approach. Care has been taken to make sure that the textual content is quickly available and self-consistent, with conservation of the intermediate steps within the analytical derivations. The e-book bargains the reader a transparent, succinct direction in easy electromagnetic thought. it is going to even be an invaluable search for software for college students and designers.

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We will denote the two field solutions with a prime and with a double prime respectively. The impressed currents (field sources) must obviously be the same in both cases. It can then be written: ⎧ ⎧ ∂B ∂B ⎪ ⎪ ⎨ ∇ × E = −Jmi − ∂t ⎨ ∇ × E = −Jmi − ∂t , . ⎪ ⎪ ∂D ∂D ⎩ ⎩ ∇ × H = Ji + Jc + ∂t ∇ × H = Ji + Jc + ∂t Defining now some “difference” fields and currents: E d = E − E , Bd = B − B , Hd = H − H , Jcd = Jc − Jc , Dd = D − D . Subtracting member to member the Maxwell’s equations for the two cases, homogeneous Maxwell’s equations are obtained for the difference field: 36 1 Fundamental Theorems and Equations of Electromagnetism ⎧ ⎪ ⎨ ∇ × Ed = − ∂ Bd ∂t ⎪ ⎩∇×H = J + d cd ∂ Dd ∂t .

The frequency instead does not depend on the medium. 1 Helmholtz Equation We derive now the non-homogeneous Helmholtz equation starting from the Maxwell’s equations in the transformed domain. Let us consider again a homogeneous, isotropic, but also generally dispersive medium ∇ × E = −Jmi − jωμ H . ∇ × H = Ji + σ E + jωε E = Ji + (σ + jωε)E For compactness of notation it is usually put: jωεc = σ + jωε ⇒ with: εc = ε + ∇ × H = Ji + jωεc E, σ σ =ε− j jω ω equivalent dielectric constant (complex in general).

T ∂t On the other hand (again in the assumption of homogeneity of the medium): ∇·E = Then: ∇ 2 E − μσ ρ . ε ∂2 E ∂E ∇ρ − με 2 = . ∂t ∂t ε 40 1 Fundamental Theorems and Equations of Electromagnetism Assuming the absence of free charges the following homogeneous equation is obtained: ∂2 E ∂E − με 2 = 0. ∇ 2 E − μσ ∂t ∂t The well-known d’Alembert equation follows in the particular case of nondissipative medium (σ = 0): ∇ 2 E − με ∂2 E 1 ∂2 E 2 = ∇ E − = 0, ∂t 2 v 2 ∂t 2 having introduced the light speed in the medium: 1 v=√ .