Problem 50 - University Physics with Modern Physics

Question

In a certain experiment, a radio transmitter emits sinusoidal electromagnetic waves of frequency 110.0 \(\mathrm{MHz}\) in opposite directions inside a narrow cavity with reflectors at both ends, causing a standing wave pattern to occur. (a) How far apart are the nodal planes of the magnetic field? (b) If the standing wave pattern is determined to be in its eighth harmonic, how long is the cavity?

Step-by-Step Solution

Answer

(a) Nodal planes are 1.364 meters apart. (b) The cavity is 10.912 meters long.

Step 1: Understand the problem

We're given electromagnetic waves with a frequency of 110.0 MHz in a cavity that form standing waves. We need to find the distance between nodal planes of the magnetic field and the cavity length for the eighth harmonic.

Step 2: Calculate the wavelength of the waves

The frequency of the waves is given as 110.0 MHz which is 110.0 x 10^6 Hz. The speed of electromagnetic waves is given by the speed of light, approximately 3.0 x 10^8 m/s. The wavelength \( \lambda \) can be found using the formula: \( \lambda = \frac{c}{f} \) where \( c \) is the speed of light and \( f \) is the frequency. Thus, \( \lambda = \frac{3.0 \times 10^8}{110.0 \times 10^6} = 2.727 \) meters.

Step 3: Determine the distance between nodal planes

A nodal plane in a standing wave occurs at every half-wavelength. Since the wavelength is 2.727 m, the distance between nodal planes is \( \frac{\lambda}{2} = \frac{2.727}{2} = 1.364 \) meters.

Step 4: Identify the relationship of harmonics

The standing wave pattern in its eighth harmonic means that there are eight half-wavelengths along the length of the cavity. The length of the cavity \( L \) can be identified by the formula \( L = n \frac{\lambda}{2} \), where \( n \) is the harmonic number.

Step 5: Calculate the length of the cavity

Substitute the eighth harmonic and previously calculated half-wavelength: \( L = 8 \times \frac{2.727}{2} = 8 \times 1.364 = 10.912 \) meters. Therefore, the length of the cavity is 10.912 meters.

Key Concepts

Standing Waves

Standing waves are a fascinating phenomenon that occurs when waves of the same frequency interfere, usually through reflection, in such a way that the pattern of wave crests and troughs stand still, hence the name 'standing'. Imagine plucking a guitar string: the resulting vibration creates standing waves. In an electromagnetic context, this could happen when waves meet a reflective surface, as seen in cavities with reflectors at both ends.
Standing waves exhibit points of zero amplitude called nodes, where destructive interference suppresses motion completely. These are surrounded by antinodes, points of maximum amplitude due to constructive interference.

• Nodes: Points where the medium doesn't move.
• Antinodes: Points where the medium moves most.

Understanding this concept is key to solving problems involving standing waves, like calculating distances between nodes or finding specific wave characteristics such as harmonic frequencies.

Wavelength Calculation

In wave physics, the wavelength is the distance after which the wave's shape repeats. Calculating the wavelength in an electromagnetic wave can be straightforward when knowing the wave's speed and frequency.
The formula for this is:\[\lambda = \frac{c}{f}\]where \( \lambda \) is the wavelength, \( c \) is the speed of light, typically valued at \( 3.0 \times 10^8 \) m/s, and \( f \) is the frequency of the wave. This relationship shows that a higher frequency yields a shorter wavelength, and vice versa.
In practical terms, mastering this calculation allows you to determine how wave properties change when conditions vary, such as frequency adjustments in the radio transmitter described in the original exercise which emits waves at 110.0 MHz, resulting in a calculated wavelength of approximately 2.727 meters.

Harmonic Frequencies

Harmonic frequencies arise when standing waves form within a medium. Each harmonic corresponds to a specific wave pattern and is defined by the number of nodes and antinodes along the wave's path. For a simple wave trapped within a cavity or any bounded medium, only certain frequencies can form standing waves due to the boundary conditions.
As in the exercise, we often deal with harmonics like the eighth harmonic. With electromagnetic waves in a cavity, harmonics are calculated by considering the integer multiple of half-wavelengths filling the length of the cavity. Thus, the harmonic number tells you how many half-wavelengths fit within the boundary:

• First harmonic: one half-wavelength fits in the medium.
• Second harmonic: two half-wavelengths fit, and so on.
• Eighth harmonic: eight half-wavelengths fit.

Understanding these relationships allows you to solve complex wave problems like finding the length of a cavity that supports a specific harmonic, such as the length of 10.912 meters for the eighth harmonic, as determined using the wavelength calculated earlier. This involves understanding how electromagnetic waves set up in constrained environments as well as interpreting harmonic series to gain insights into wave behavior.