Q1Electromagnetics Waves

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Q.1 Explain the method of antenna radiation for a Hertz dipole.

Answer 1

Setting up the source: a Hertz (infinitesimal) dipole is an idealized current element of length dl ≪ λ, located at the origin along the z-axis, carrying a uniform time-harmonic current I(t) = I0cos(ωt), so the phasor current density is effectively a point source J = I0 dl δ(r) âz.

Vector potential: using the retarded-potential formula derived generally in Part C, Q.3 of the companion 2024 (Main/Back) paper on this subject, with the source confined to a point at the origin, the vector potential at an observation point a distance r away (spherical coordinates) is simply the point-source Green's function scaled by the current moment:

A = \overset{a}{^}_{z} \frac{μ _{0} I _{0} d l}{4 π r} e^{- j k r}

Converting to spherical components and finding H: resolving âz into spherical unit vectors (âz = cosθ âr − sinθ âθ) and applying H = (1/μ0)∇×A, the only nonzero H component (by symmetry, independent of φ) is Hφ:

H_{ϕ} = \frac{j k I _{0} d l sin θ}{4 π r} (1 + \frac{1}{j k r}) e^{- j k r}

Finding E: applying E = (1/jωε0)∇×H (source-free region away from the point source) yields two nonzero components, Er and Eθ:

E_{r} = \frac{η _{0} I _{0} d l cos θ}{2 π r ^{2}} (1 + \frac{1}{j k r}) e^{- j k r}, E_{θ} = \frac{j η _{0} k I _{0} d l sin θ}{4 π r} (1 + \frac{1}{j k r} - \frac{1}{( k r ) ^{2}}) e^{- j k r}

Far-field limit (kr ≫ 1, r ≫ λ): retaining only the leading 1/r terms (the '1' term in each bracket) and discarding the faster-decaying 1/r² and 1/r³ near-field terms:

E_{θ} \approx \frac{j η _{0} k I _{0} d l sin θ}{4 π r} e^{- j k r}, H_{ϕ} \approx \frac{j k I _{0} d l sin θ}{4 π r} e^{- j k r}, E_{r} \to 0

confirming that in the far zone only the transverse components Eθ and Hφ survive, related by Eθ/Hφ = η0 exactly as for a local plane wave, and both proportional to the characteristic sinθ radiation pattern — maximum broadside to the dipole (θ=90°) and zero along its axis (θ=0°,180°).

Radiated power and radiation resistance: integrating the time-average Poynting vector Sr = |Eθ|²/(2η0) over a large sphere surrounding the dipole gives the total radiated power, and equating this to ½I0²Rrad yields the standard Hertz-dipole radiation resistance result:

R_{r a d} = 80 π^{2} (\frac{d l}{λ})^{2} Ω

This method — computing A from the source, differentiating to find H then E, and separating near- and far-field terms by their power of 1/r — is the standard procedure for finding the radiated fields of any antenna once its current distribution is known, with the Hertz dipole serving as the fundamental building block from which more complex antennas (finite dipoles, loops, arrays) are constructed by integrating this elemental result over the actual current distribution.

Q1Electromagnetics Waves

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