Problem 30 - University Physics with Modern Physics

Question

A standing electromagnetic wave in a certain material has frequency 2.20 \(\times\) 10\(^{10}\) Hz. The nodal planes of \(\vec{B}\) are 4.65 mm apart. Find (a) the wavelength of the wave in this material; (b) the distance between adjacent nodal planes of the \(\vec{E}\) field; (c) the speed of propagation of the wave.

Step-by-Step Solution

Answer

(a) 9.30 mm; (b) 4.65 mm; (c) 2.046 x 10^8 m/s.

Step 1: Determine the Wavelength of the Wave

In a standing wave, adjacent nodes are separated by half a wavelength. Given that the nodal planes of \( \vec{B} \) are 4.65 mm apart, this distance represents half the wavelength. So to find the full wavelength, multiply the given distance by 2.\[ \lambda = 2 \times 4.65 \text{ mm} = 9.30 \text{ mm} \]

Step 2: Calculate the Distance Between Nodal Planes of the E Field

For electromagnetic waves, the nodal and antinodal planes of the electric field \( \vec{E} \) are staggered by a quarter wavelength compared to those of \( \vec{B} \). Thus, the distance between adjacent nodal planes of the \( \vec{E} \) field is equivalent to this distance. Using the wavelength from Step 1:\[ \text{Distance between nodal planes of } \vec{E} = \frac{9.30 \text{ mm}}{2} = 4.65 \text{ mm} \]

Step 3: Compute the Speed of Propagation

The speed of a wave is given by the equation \( v = f \lambda \), where \( f \) is the frequency and \( \lambda \) is the wavelength. Substitute the frequency \( 2.20 \times 10^{10} \text{ Hz} \) and the wavelength \( 9.30 \text{ mm} \) converted to meters (0.00930 m).\[ v = (2.20 \times 10^{10} \text{ Hz}) \times 0.00930 \text{ m} \]\[ v = 2.046 \times 10^{8} \text{ m/s} \]

Key Concepts

Wavelength Calculation

In a standing wave pattern, calculating the wavelength is essential to understanding its properties. A wavelength of a wave is the distance over which the wave's shape repeats. For a standing electromagnetic wave, this involves recognizing the relationship between nodal points. Nodal points, or nodes, are locations along the wave where the amplitude of displacement is consistently zero. For a standing wave, each adjacent node is separated by exactly half a wavelength.
When given the distance between adjacent nodes, as in the scenario above with a distance of 4.65 mm, this distance represents half of the full wavelength. To find the complete wavelength (\( \lambda \)), you can simply double the distance between these nodes. This leads us to the expression:

• Calculate the full wavelength using: \( \lambda = 2 \times \text{distance between nodes} \)
• Given \( 4.65 \text{ mm} \) as the distance between nodes, the full wavelength is: \( \lambda = 9.30 \text{ mm} \)

Node and Antinode

Understanding the behavior of nodes and antinodes in electromagnetic waves is crucial. Nodes are stationary points where destructive interference causes the amplitude to be consistently zero, while antinodes are points of maximum amplitude where constructive interference occurs.
In electromagnetic waves, nodes and antinodes can refer to specific qualities of the electric field (\( \vec{E} \)) and magnetic field (\( \vec{B} \)). These nodes do not align directly with each other. Instead, the nodal planes of one field are staggered with respect to the other. For instance, the nodal planes of \( \vec{E} \) are staggered and are a quarter of a wavelength away from those of \( \vec{B} \). Thus, the distance between adjacent nodal planes of the \( \vec{E} \) field can be calculated using:

• \( \text{Distance between adjacent \( \vec{E} \) nodal planes} = \frac{\text{full wavelength}}{2} \)
• With a wavelength of \( 9.30 \text{ mm} \), the distance becomes: \( 4.65 \text{ mm} \)

Understanding this is key to analyzing the patterns and behaviors of standing waves in various materials.

Wave Speed Calculation

The speed of a wave is a fundamental property that reflects how fast the wave travels through a medium. It can be calculated using the formula:\[ v = f \lambda \]where:

• \( v \) is the wave speed,
• \( f \) is the frequency of the wave,
• \( \lambda \) is the wavelength.

To find the speed of the electromagnetic wave with a frequency of \( 2.20 \times 10^{10} \text{ Hz} \) and a wavelength of \( 9.30 \text{ mm} \), firstly convert the wavelength into meters for consistency (\( 9.30 \text{ mm} = 0.00930 \text{ m} \)).
Then, apply the values to the wave speed formula:\[ v = (2.20 \times 10^{10} \text{ Hz}) \times 0.00930 \text{ m} \]By solving this, we find that the speed of the wave is \( 2.046 \times 10^{8} \text{ m/s} \). Understanding this equation and calculation helps in exploring the behavior of waves in different contexts.