Q2Antenna And Wave Propagation

(a) Radiation and Reception via Maxwell's Equations

Radiation of an electromagnetic wave from an antenna begins when a time-varying current is established on the antenna's conducting structure (driven by the transmitter's oscillating source voltage) — by Ampere's law (including Maxwell's displacement current term, as discussed in the corresponding earlier answer), this time-varying current produces an associated time-varying magnetic field in the surrounding space; by Faraday's law, this time-varying magnetic field in turn induces an associated time-varying electric field; and by the displacement-current-augmented Ampere's law once again, this new time-varying electric field itself generates a further time-varying magnetic field, and so the process continues, propagating outward from the antenna at the speed of light as a mutually self-sustaining, coupled electric-and-magnetic-field disturbance that eventually detaches entirely from the antenna's near-field region and propagates as a genuine radiating electromagnetic wave into the far field, carrying energy away from the source indefinitely.

Reception is essentially the reverse process: an incident electromagnetic wave's time-varying electric field, upon reaching a receiving antenna's conducting structure, exerts a force on the free electrons within the conductor (via the Lorentz force law), driving a time-varying induced current along the antenna structure — this induced current is then delivered to the receiver's input circuitry (via the transmission line connecting the antenna to the receiver) as the desired received signal. Both radiation and reception are governed by the same underlying set of Maxwell's equations, and indeed, by the fundamental reciprocity theorem of antenna theory, any given antenna's transmitting and receiving properties (radiation pattern, gain, polarization characteristics) are identical, meaning an antenna's transmit-mode analysis (using Maxwell's equations to determine the radiated field from a specified current distribution) directly yields its receive-mode behavior as well.

Why an Isotropic Radiator Cannot Be Practically Designed

An isotropic radiator is a hypothetical, idealized antenna that radiates electromagnetic energy with exactly equal intensity in every direction in three-dimensional space (a perfectly spherically-symmetric radiation pattern), serving purely as a convenient mathematical reference standard against which the directivity and gain of any real antenna are defined and compared (as discussed in the corresponding earlier answer on directivity). However, Maxwell's equations, applied to any physically realizable radiating current distribution, can be shown to never produce a perfectly spherically-symmetric radiated field — this is a direct consequence of the vector (rather than scalar) nature of the electromagnetic field: any real antenna current element necessarily has a specific spatial direction (a current flows along some particular direction in space), and the radiated electric field from any current element is fundamentally tied to, and varies with, the angle between the observation direction and the current element's own direction (following the well-known sin(theta) radiation pattern of an elementary dipole, for instance) — since a real antenna's current must flow along some physical conductor with a definite direction at every point, its radiated field can never be perfectly uniform in all directions including along the current's own axis (where, in fact, radiation from any linear current element is always exactly zero). Only a mathematically idealized point source lacking any inherent directional current structure whatsoever could achieve perfectly isotropic radiation, and no such physically realizable radiating structure exists — every real antenna, no matter how small or symmetric its physical construction, exhibits at least some directional variation in its radiated field, meaning the isotropic radiator remains permanently a purely theoretical reference concept, never a practically achievable design.

Near-Field and Far-Field Radiation Zones

The space surrounding a radiating antenna is conventionally divided into three regions, distinguished by how the field structure varies with distance from the antenna. The reactive near-field region, extending out to approximately 0.62sqrt(D^3/lambda) from the antenna (D being the antenna's largest physical dimension), is dominated by reactive (stored, non-propagating) energy, analogous to the reactive field surrounding a lumped inductor or capacitor, with field components that do not vary simply as a traveling wave and that decay very rapidly with distance. The radiating near-field (Fresnel) region, extending from this reactive boundary out to the conventional far-field distance 2D^2/lambda, is a transitional zone in which the angular field distribution already begins to resemble the eventual far-field pattern but still varies appreciably with radial distance, requiring the full, un-approximated radiation integral for accurate analysis. Beyond the far-field (Fraunhofer) distance, 2*D^2/lambda, lies the far-field region, where the radiated fields have settled into their essentially fixed angular (radiation pattern) distribution, decaying simply as 1/r with distance while maintaining a locally plane-wave-like structure (mutually perpendicular E and H fields, related by the free-space intrinsic impedance, both transverse to the direction of propagation) — virtually all practical antenna radiation-pattern measurements, gain and directivity specifications, and link-budget calculations (including the Friis transmission equation) are defined with reference to this far-field region, since it is only here that the antenna's radiation pattern becomes independent of the observation distance.

Practical Antennas Approximating Isotropic Behavior

Although a mathematically perfect isotropic radiator can never be built, several practical antenna types are deliberately engineered to approximate omnidirectional (though not truly isotropic) behavior over a restricted angular range, and these are frequently and loosely referred to in engineering practice as being close to isotropic for the purpose of comparison. A vertical quarter-wave monopole or half-wave dipole, for instance, radiates uniformly in all azimuthal (horizontal) directions around its axis, exhibiting an omnidirectional pattern in the horizontal plane even though its vertical-plane pattern is distinctly non-uniform (following the familiar figure-eight-like dipole pattern, with nulls along the antenna's own axis) — such antennas are commonly described as omnidirectional rather than isotropic, precisely to distinguish their genuine (horizontal-plane-only) uniformity from the impossible, fully three-dimensional uniformity of a true isotropic source. Small loop antennas and turnstile/crossed-dipole antenna combinations are likewise used where near-uniform coverage in one plane, combined with acceptable (if imperfect) coverage in the orthogonal plane, is required, such as in broadcast, mobile communication base-station, and satellite telemetry applications — the isotropic radiator's enduring practical value is thus not as an achievable design target, but as the universal, direction-independent 0 dBi reference level against which the gain of all of these real, imperfectly-directional antennas is conventionally quoted.

(b) Numerical: Radiated Power and Directivity

Note on the given expression: the radiation intensity U(theta,phi)=10sin(theta)sin(phi) W/sr, exactly as literally stated, integrates to exactly zero total radiated power over the specified range (since sin(phi) is positive for 0<phi<pi and equally negative for pi<phi<2pi, so these contributions cancel exactly), which is not physically meaningful for a genuine antenna radiation intensity (radiation intensity, being proportional to the magnitude-squared of the far-field, must be non-negative everywhere). This is very likely an OCR/transcription error from the well-known standard textbook problem (a classic example from Balanis' Antenna Theory) in which the radiation intensity is U(theta,phi)=B0sin(theta)sin^2(phi), with sin(phi) squared rather than sin(phi) to the first power — the calculation below uses this corrected, physically consistent form with B0=10 W/sr, which is almost certainly the problem's intended statement.

Given (corrected): U(theta,phi)=B0sin(theta)sin^2(phi) W/sr, for 0<theta<pi, 0<phi<2pi, with B0=10 W/sr.

Radiated power is found by integrating the radiation intensity over the entire solid angle (4*pi steradians):

Evaluating each integral separately: the integral of sin^2(theta) from 0 to pi equals pi/2, and the integral of sin^2(phi) from 0 to 2*pi equals pi. Therefore:

Directivity: the maximum radiation intensity occurs at theta=90 degrees (where sin(theta)=1) and phi=90 degrees (where sin^2(phi)=1), giving Umax=B0=10 W/sr. The directivity is then:

Result: radiated power Prad approximately 49.35 W, and directivity D0 approximately 2.55 (4.06 dB), a well-known standard result for this specific radiation intensity pattern, illustrating the general two-step procedure — integrate the given radiation intensity over the full sphere to obtain total radiated power, then divide 4*pi times the maximum intensity by this radiated power to obtain directivity — used for any antenna whose radiation intensity function is analytically specified.