Two radio towers are a distance d apart as shown in the...

Two radio towers are a distance $d$ apart as shown in the overhead view. Each antenna by itself would radiate equally in all directions in a horizontal plane. The radio waves have the same wavelength $\lambda$ and start out in phase. A detector is moved in a circle all the way around the towers $\left(-180^{\circ}<\theta \leq+180^{\circ}\right)$ at a distance much greater than \lambda. The power $P$ measured by the detector is found to vary with the angle $\theta$. (a) Is the power detected at $\theta=0$ a maximum or a minimum? Explain. (b) For what values of $d$ (in terms of \lambda) would the power be minimum at $\theta=90^{\circ} ?$

Two radio towers are a distance $d$ apart as shown in the overhead view. Each antenna by itself would radiate equally in all directions in a horizontal plane. The radio waves have the same wavelength $\lambda$ and start out in phase. A detector is moved in a circle all the way around the towers $\left(-180^{\circ}<\theta \leq+180^{\circ}\right)$ at a distance much greater than
\lambda. The power $P$ measured by the detector is found to vary with the angle $\theta$.
(a) Is the power detected at $\theta=0$ a maximum or a minimum? Explain.
(b) For what values of $d$ (in terms of
\lambda) would the power be minimum at $\theta=90^{\circ} ?$

Key Concepts

Path Difference

Path difference is the extra distance traveled by one wave relative to another before they meet at a detector. This extra distance introduces a phase shift proportional to the path difference divided by the wavelength. In the context of interference, whether the waves add constructively or destructively depends on whether the path difference is an integer multiple of the wavelength or an odd multiple of one?half of the wavelength, respectively.

Phase Difference

The phase difference between waves, arising from the path difference or initial phase offsets, is central to predicting interference outcomes. Given that the waves start in phase, any phase difference at the detector comes solely from differences in travel distance. A phase difference of 0 (or multiples of 2?) leads to constructive interference, whereas a phase difference of ? (or an odd multiple thereof) produces destructive interference.

Far?Field (Fraunhofer) Approximation

In problems where the detector is located much farther away than the wavelength (and usually also than the source separation), the far?field approximation can be applied. This approximation simplifies the geometry by assuming that the distance from the sources to the detector is nearly the same for both sources except for a small difference that depends linearly on the source separation and the angle of observation. This leads to the commonly used expression for the phase difference in terms of the cosine of the angle.

Wave Interference

This concept refers to the phenomenon that occurs when two or more coherent (that is, same frequency and fixed phase relation) waves overlap. Their amplitudes add according to their phase difference, giving rise to regions of constructive interference (when the waves reinforce each other) or destructive interference (when they cancel each other). In problems like these, understanding how the superposition of two wavefronts leads to spatial variations in the observed intensity is crucial.

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