By Orfanidis S.J.

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**Example text**

We assume that the wavevector k in S lies in the xz-plane and forms an angle θ with the z-axis as shown. 69 Eq. 14). Because ky = 0 and the transverse components of k do not change, we will have ky = ky = 0, that is, the wavevector k will still lie in the xz-plane of the S frame. 5) For Doppler radar this doubles to Δf /fa = 2(va − vb )/c because the wave suffers two Doppler shifts, one for the transmitted and one for the reﬂected wave. This is further discussed in Sec. 8. For electromagnetic waves,† the correct Doppler formula depends only on the relative velocity between observer and source and is given by the relativistic generalization of Eq.

09 μm Thus, the skin depth is extremely small for good conductors at RF. Because δ is so small, the ﬁelds will attenuate rapidly within the conductor, depending on distance like e−γz = e−αz e−jβz = e−z/δ e−jβz . The factor e−z/δ effectively conﬁnes the ﬁelds to within a distance δ from the surface of the conductor. This allows us to deﬁne equivalent “surface” quantities, such as surface current and surface impedance. With reference to Fig. 6) ˆ = −ˆ where n z is the outward normal to the conductor.

60 2. 9. 1) A good conductor corresponds to the limit τ 1, or, σ ω . Using the approximations of Eqs. 3) πf μσ where we replaced ω = 2πf . 8 × 107 Siemens/m. 5) where we deﬁned the surface impedance Zs = γ/σ . In the good-conductor limit, Zs is equal to ηc . Indeed, it follows from Eqs. 09 μm Thus, the skin depth is extremely small for good conductors at RF. Because δ is so small, the ﬁelds will attenuate rapidly within the conductor, depending on distance like e−γz = e−αz e−jβz = e−z/δ e−jβz .

### Electromagnetic waves and antennas by Orfanidis S.J.

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