Electrical Dipole Radiation: Hertzian Dipoles, Near and Far Fields, Poynting Vector - Prof, Assignments of Guiding Electromagnetic Systems

Download Electrical Dipole Radiation: Hertzian Dipoles, Near and Far Fields, Poynting Vector - Prof and more Assignments Guiding Electromagnetic Systems in PDF only on Docsity! An electrical dipole consists of two equal charges with opposite sign (or phase). When the phase oscillates, it becomes an oscillating electrical dipole which will have non-zero Poynting vector in the far field. The total outward power flow in the far field is defined as radiation.  Hertzian dipole is a special class of electrical dipole in which the dipole moment remained constant while the its dimension becomes very so small that it is smaller than any length of interests in the problem. Hertzian dipole simplifies the mathematics while maintains the essential features of an electrical dipole.  There are more than one approaches to this problem. D. Griffiths use non-phasor approach; its takes more effort in tracking the details but I think he did a great job in that. Below I will use phasor approach, following D. Staelin s2.3.  Electrical dipole phasor: Current phasor: "Classical Electromagnetic Theory" by Jack Vanderlinde, section 10.2.a. Kong::1.4 & 5.3. Note that Kong handles this topic very different from mine. You can look through what topic he has used but they are not easy to follow. I will still assign appropriate HWs from Kong but you should be able to handle using the approaches taught in class. b. (optional) Jackson::9.1, Staelin 2.3, Griffith 11.1.2c. Reading: Electrical Dipole Radiation Saturday, December 01, 2007 4:01 PM Dipole Radiation Page 4 For an Hertzian dipole, d << r and d << l. The current can be approximated constant over such small dimension. The vector potential thus becomes:  The H-field is Dipole Radiation Page 5 Near field is the field at distance such that kr << 1, or r << l. It plays an important role in optics as it enables super-resolution imaging and lithography.  E- & H-fields as derived above can be simplified at near field as shown below. Near-field E-field: d << r << l Near-field H-field: Near Field of Oscillating Electrical Dipole Saturday, December 01, 2007 4:01 PM Dipole Radiation Page 8 Near field of oscillating electrical dipole is quasi-electrostatics; i.e. for oscillating electrical dipole, near-field E-field is much stronger than the H-field and E-field resembles closely to that of a static electrical dipole.  JAK:p1.4.2; JAK:p1.4.4; Dipole Radiation Page 9 As opposed to near field, far field is the field at distance such that kr >> 1, or r >> l. It represents the fields we normally perceive and therefore is responsible for the visualization of our world.  E- & H-fields as derived above can be shown to be orthogonal to each other and are spherical waves as shown below.  Far-field E-field: d << l << r JAK:p1.4.1; JAK:p5.3.2; Far Field of Oscillating Electrical Dipole Saturday, December 01, 2007 4:01 PM Dipole Radiation Page 10 The first important implication of the above result is that an IDEAL electrical dipole radiation exhibits high degree of spectral dispersion; i.e. it radiates more the blue light (shorter wavelength) than the red light (longer wavelength). This fact is true as long as the dimension of the dipole is much smaller than the wavelength, an is called Rayleigh scattering. • (in-class exercise) Scattering of light by air molecules can be considered that, quantum mechanically, the incident light is absorbed by the molecules to induce its dipole oscillation, which in turn radiates as an ideal dipole. At noon, the sky is white atop and blue elsewherea. At sunset the sky is red/orange at sun, then gradually turns blue/black as it departs from the sun (DG::ex11.1 partial). b. This phenomenon explains the following two phenomena: Pasted from <http://stochastix.wordpress.com/2007 /02/28/sunset-physics/> Please refer to the above link for more information on "sunset physics". The second important implication of the above result is that an IDEAL electrical dipole radiation exhibits high degree of angular dispersion; i.e. it radiates more light in the • Rayleigh Scattering Wednesday, November 19, 2008 11:58 AM Dipole Radiation Page 13 radiation exhibits high degree of angular dispersion; i.e. it radiates more light in the equatorial plane and no power at all along the north and south poles. (in-class exercise; DG: Ex11.1) Scattering of light by air molecules can be considered that, quantum mechanically, the incident light is absorbed by the molecules to induce its dipole oscillation, which in turn radiates as an ideal dipole. This phenomenon explains the why the blue sky at noon is highly polarized: It is shown that Poynting vector is highly anisotropic: it is zero along the direction of oscillation and is the strongest in the orthogonal directions. At a fixed distance from the source (e.g. if you are on the surface of a sphere), the intensity variation has a donut-like shape as in the following (this is also called "polar radiation pattern"): Total radiated power is proportional to the square of the dipole strength (and of course the fourth power of the frequency). • Dipole Radiation Page 14 (DG::p11.3) radiation resistance of an electrical dipole. (OPTI6104 skips) JAK: p1.4.3; JAK:p1.4.5; JAK:p1.4.6; Dipole Radiation Page 15