Power Density Radiated by a Short Horizontal Current Element: 17-Phys-A3 Electromagnetics, December 2016, Q8

Question 8: Power Density Radiated by a Short Horizontal Current Element (equal value)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Also: the source's geometry names only a horizontal distance and an elevation angle, with no compass bearing given. Absent a figure, the polar angle $\theta$ (measured from the dipole's own horizontal axis) is taken equal to the stated elevation angle — i.e. the second point is read as lying in the vertical plane containing the current element itself, the natural reading of "away from the element" for a HORIZONTAL dipole (contrast Question 5, which uses a VERTICAL dipole, for which elevation instead maps to $\theta=90^\circ-\text{elevation}$ measured from the vertical axis).

Given. Short (Hertzian) horizontal current element, $f=10$ MHz; at $r_1=1$ km directly above (broadside, $\theta_1=90^\circ$ from the element's own axis) the field is $E_1=100\ \mu\text{V/m}$; second point at horizontal distance $2$ km along the element's axis, elevation $30^\circ$ (so $\theta_2=30^\circ$ from the axis, per the check note above).

Point A is broadside ($\theta=90^\circ$, maximum response); point 2 is read off at $\theta=30^\circ$ from the element's own horizontal axis.

Find. (i) Power density at point 2 (10 MHz); (ii) the power density at the same point if $f$ is reduced to 5 MHz.

Approach. A short dipole's far field is $E(r,\theta)=C\sin\theta/r$ for a fixed frequency and current; use the reference point to find $C$, then evaluate at point 2's $(r,\theta)$ and convert to power density with $S=E^2/\eta_0$. For part (ii), use that $E\propto f$ for a Hertzian dipole of fixed physical length and current (since $E\propto\beta=\omega/c$), so $S\propto f^2$.

Calibrate the pattern constant from point A. $E(r,\theta)=C\sin\theta/r$ with $E_1=100\ \mu\text{V/m}$ at $r_1=1000$ m, $\theta_1=90^\circ$: $$C=\frac{E_1 r_1}{\sin\theta_1}=(100\times10^{-6})(1000)=0.1\ \text{V}.$$
Slant range to point 2. With horizontal distance $2$ km and elevation $30^\circ$, $$r_2=\frac{2000}{\cos30^\circ}=\boxed{2{,}309\ \text{m}}.$$
Field and power density at point 2 (10 MHz). $$E_2=\frac{C\sin\theta_2}{r_2}=\frac{(0.1)(\sin30^\circ)}{2309}=21.65\ \mu\text{V/m},$$ $$S_2=\frac{E_2^2}{\eta_0}=\frac{(21.65\times10^{-6})^2}{376.8}=\boxed{1.244\times10^{-12}\ \text{W/m}^2}.$$
Part (ii): halve the frequency. For a Hertzian dipole of fixed length and current, $E\propto\beta\propto f$, so $S\propto f^2$. Reducing $10\to5$ MHz scales $S_2$ by $(5/10)^2=1/4$: $$S_{2,5\text{MHz}}=\frac{1}{4}S_2=\boxed{3.11\times10^{-13}\ \text{W/m}^2}.$$

Short horizontal dipole — radiated power density
Quantity	Result
Pattern constant $C$	0.1 V
Slant range $r_2$	2,309 m
$S_2$ at 10 MHz	$1.244\times10^{-12}$ W/m²
$S_2$ at 5 MHz	$3.11\times10^{-13}$ W/m²

Topic: Far-field radiation pattern and frequency scaling of a short (Hertzian) dipole.

Key relations: $|E_\theta|=\dfrac{\eta_0\beta I_0\,d\ell\sin\theta}{4\pi r}\propto\dfrac{f\sin\theta}{r}$; $S=E^2/\eta_0\propto f^2$; radiation resistance of a short dipole $R_{rad}\propto(d\ell/\lambda)^2\propto f^2$, consistent with $S\propto f^2$ at fixed current.

Why this works: a short dipole's far field comes from the time-varying vector potential, which brings down one factor of $\omega$ (equivalently $\beta$) relative to the near-field terms; halving the frequency (same physical antenna, same current) halves the far-field amplitude and quarters the power density.

Common pitfall: assuming $E\propto f^2$ (confusing the field with the power, or with the loop-antenna case, where the SAME current instead gives $E\propto f^2$ because the loop's moment is set by $I\omega$, not $I$ alone) — for a straight current element the field itself scales as the first power of frequency.

Source: Balanis, Antenna Theory, Ch. 4 (infinitesimal dipole far field, radiation resistance); Sadiku, Elements of Electromagnetics, Ch. 13.

Question 8 of 8: Power Density Radiated by a Short Horizontal Current Element

Question 8: Power Density Radiated by a Short Horizontal Current Element (equal value)