By Fabrizio Frezza
This booklet is a concise advent to electromagnetics and electromagnetic fields that covers the points of such a lot value for engineering purposes via a rigorous, analytical therapy. After an creation to equations and simple theorems, themes of primary theoretical and applicative value, together with airplane waves, transmission traces, waveguides and Green's capabilities, are mentioned in a intentionally common manner. Care has been taken to make sure that the textual content is instantly obtainable and self-consistent, with conservation of the intermediate steps within the analytical derivations. The ebook deals the reader a transparent, succinct direction in easy electromagnetic idea. it is going to even be an invaluable search for instrument for college kids and designers.
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Extra resources for A Primer on Electromagnetic Fields
It can be shown that, as for any gradient, its direction is the direction in which the phase variation is maximum; moreover β is orthogonal to equiphase surfaces; finally, obviously, β ≡ 0 for a standing wave. As it should be known, the superposition of two progressive waves having same module and of opposite phases generates a standing wave. To show this, let us write: A1 (x, y, z) = M(x, y, z) e− j (x,y,z) , A2 (x, y, z) = M(x, y, z) e j (x,y,z) , it follows: A1 (x, y, z) + A2 (x, y, z) = M(x, y, z) e j = 2 M(x, y, z) cos (x,y,z) + e− j (x,y,z) (x, y, z) , which is a real function.
2 Let us now put in the transformed domain: w E (r ) = 1 E(r ) · D ∗ (r ) 4 ⇒ p E (r ) = −2 jω w E (r ). Exploiting again the phasor relations we obtain: w E (r, t) = Re 1 1 E(r ) · D(r ) e2 jωt + Re E(r ) · D ∗ (r ) . 4 4 By averaging with respect to time, we have: t w E (r, t) = Re[w E (r )]. Let us now consider the expression of w E (r ). It was, for an isotropic medium: D = εE, with ε real (and positive) independent of ω. Therefore w E (r ) = 1 1 ε E · E ∗ = ε |E|2 4 4 is a real quantity, so we can omit the real part in the previous relation.
As an elastic force. Finally it needs to be considered in general also a damping force due to collisions (and thus a dissipative term, which indicates a transfer of energy from one form to another). This damping force can be considered proportional to the velocity d d d dt of the charge, and expressed as −β v = −β dt = β dt eo (viscous friction). At this point, by applying the second law of dynamics F = m a (where a = d2 dt2 = −d 2 dt2 eo ) projected in the direction of eo we obtain: −m d2 dt 2 = −q E + k + β d .
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