Problem 43 - University Physics with Modern Physics

Question

In a certain experiment, a radio transmitter emits sinusoidal electromagnetic waves of frequency 110.0 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

The nodal planes are \(1.36\,\text{m}\) apart, and the cavity is \(10.9\,\text{m}\) long.

Step 1: Understand the Problem

We need to calculate two things: the distance between the nodal planes for magnetic waves in a standing wave pattern, and the length of the cavity given the standing wave pattern is in its eighth harmonic.

Step 2: Calculate the Wavelength

The frequency of the waves is given as 110.0 MHz. We start by calculating the wavelength using the formula \( \lambda = \frac{c}{f} \), where \( c = 3 \times 10^8 \) m/s is the speed of light, and \( f = 110.0 \times 10^6 \) Hz.

Step 3: Calculate the Distance Between Nodal Planes

For standing waves, the distance between nodal planes (nodes) is half the wavelength. So, calculate \( \frac{\lambda}{2} \) using the wavelength obtained in Step 2.

Step 4: Identify Harmonic Relationship

For the eighth harmonic in a standing wave, the length of the cavity \( L \) is \( \frac{8\lambda}{2} \), as each harmonic adds a node.

Step 5: Calculate the Cavity Length

Using the relation found in Step 4, substitute the wavelength from Step 2 to find the length of the cavity as \( L = 8 \times \frac{\lambda}{2} \).

Key Concepts

Standing Wave

Standing waves are fascinating patterns that form when waves travel in opposite directions within a confined space, such as a cavity with reflective ends. These waves appear to be stationary or "standing," rather than moving through the medium. This phenomenon occurs due to the interference of incoming and reflected waves.

Standing waves have specific points called nodes and antinodes. Nodes are points where the amplitude is always zero, while antinodes are points with maximum amplitude. In an electromagnetic standing wave, nodes can refer to magnetic or electric fields, depending on the context of the problem. Knowing where these nodes lie helps in determining properties such as the length of the medium hosting the standing wave pattern.

Standing waves are a key concept in understanding how wave patterns interact within bounded surfaces, making them essential in fields like acoustics and electromagnetism.

Frequency

Frequency is the measure of how many wave cycles pass a given point in space per unit time. It is denoted by the symbol \( f \) and is measured in hertz (Hz). In the context of electromagnetic waves, the frequency determines the energy and the type of wave, such as radio waves and microwaves.

Higher frequencies mean more cycles in a given time frame. This concept is fundamental in understanding how waves like the sinusoidal electromagnetic waves in our problem, which have a frequency of 110.0 MHz, behave.

The relationship between frequency \( f \) and wavelength \( \lambda \) is given by the equation \( \lambda = \frac{c}{f} \), where \( c \) is the speed of light, a constant \( 3 \times 10^8 \) m/s. This equation is vital for determining one property if the other is known, linking frequency to other wave characteristics.

Harmonics

Harmonics are integral multiples of the fundamental frequency or the simplest mode of vibration in a standing wave. Each harmonic corresponds to a unique standing wave pattern, characterized by a series of nodes and antinodes. The first harmonic is the fundamental frequency itself, with the simplest standing wave pattern, while subsequent harmonics have more nodes and higher frequencies.

The eighth harmonic, as in our exercise, means that the frequency of the standing wave is eight times that of the fundamental frequency. - Each harmonic corresponds to a specific standing wave pattern. - More nodes and antinodes appear as the harmonic order increases. - Higher harmonics reflect higher frequencies and complex waveforms.

Understanding harmonics is crucial for analyzing musical instruments, the transmission of sound waves, and specific vibrations in confined spaces, helping to determine characteristics like the cavity length for different harmonics.

Wavelength Calculation

Calculating wavelength is fundamental when analyzing wave motion. It involves using the relationship between the speed of the wave, its frequency, and wavelength itself. Wavelength (\( \lambda \)) is calculated using the formula \( \lambda = \frac{c}{f} \).

In our example, with a frequency of 110.0 MHz: 1. Convert MHz to Hz: \( f = 110.0 \times 10^6 \) Hz. 2. Use the speed of light \( c = 3 \times 10^8 \) m/s in the formula. 3. Calculate \( \lambda \) by substituting in the values: \( \lambda = \frac{3 \times 10^8}{110.0 \times 10^6} \) meters.

This computation provides the wavelength of electromagnetic waves in our scenario. Breaking down these calculations helps ascertain essential wave properties, such as node placement or harmonics within standing waves. Understanding these techniques support broader applications in fields like telecommunications and physics.