Problem 32 - University Physics with Modern Physics

Question

An electromagnetic standing wave in air has frequency 75.0 MHz. (a) What is the distance between nodal planes of the \(\vec{E}\) field? (b) What is the distance between a nodal plane of \(\vec{E}\) and the closest nodal plane of $\vec{B}?

Step-by-Step Solution

Answer

(a) 2 m; (b) 1 m.

Step 1: Understand the Problem

We need to find the distance between nodal planes for an electromagnetic wave, given the frequency, and calculate the distance between odal planes of the electric field (\( \vec{E} \)) and the magnetic field (\( \vec{B} \)).

Step 2: Determine the Wavelength

First, we calculate the wavelength of the electromagnetic wave using the formula\[\lambda = \frac{c}{f},\]where \( \lambda \) is the wavelength, \( c = 3 \times 10^8 \text{ m/s} \) is the speed of light, and \( f = 75 \times 10^6 \text{ Hz} \) is the frequency. Substituting the given values:\[\lambda = \frac{3 \times 10^8}{75 \times 10^6} = 4 \text{ m}.\]

Step 3: Find the Distance Between Nodal Planes of \(\vec{E}\)

The distance between nodal planes of the electric field component of a standing wave is half the wavelength:\[\Delta x_E = \frac{\lambda}{2} = \frac{4 \text{ m}}{2} = 2 \text{ m}.\]

Step 4: Find the Distance Between \(\vec{E}\) and \(\vec{B}\) Nodal Planes

In an electromagnetic wave, the nodal planes of \( \vec{E} \) and \( \vec{B} \) fields are separated by a quarter wavelength:\[\Delta x_{EB} = \frac{\lambda}{4} = \frac{4 \text{ m}}{4} = 1 \text{ m}.\]

Step 5: Conclusion

Thus, the distance between nodal planes of the electric field is 2 m, and the distance between a nodal plane of the electric field and the closest nodal plane of the magnetic field is 1 m.

Key Concepts

Standing Waves

Standing waves are a fascinating phenomenon that occur in wave physics. They are formed when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. This interference leads to the creation of fixed points in the wave pattern known as nodes, where there is no net movement. In contrast to nodes, antinodes are points where the wave reaches its maximum amplitude.
For electromagnetic standing waves, both electric (\( \vec{E} \)) and magnetic (\( \vec{B} \)) fields behave similarly, forming their own respective nodes and antinodes.

• Nodes: Points of zero amplitude where the wave does not move.
• Antinodes: Points where the wave reaches maximum amplitude.

The beauty of standing waves is that they help us visualize how waves can be controlled or manipulated, such as in musical instruments and optical research equipment.

Electric Field

The electric field (\( \vec{E} \)) is a vector field representing the force that a charged particle would experience in the presence of another charge. In the context of electromagnetic waves, it oscillates perpendicular to the direction of the wave propagation.
For a standing electromagnetic wave in air, the wavelength (\( \lambda \)) is crucial in determining the behavior of the electric field. The distance between nodal planes of this electric field is half of the wavelength, as demonstrated in the solution.

• Wavelength (\( \lambda \)): The distance over which the wave's shape repeats. For the given problem, \( \lambda = 4 \text{ m} \).
• Distance Between Nodes: This is \( \frac{\lambda}{2} \), which equals 2 meters for the electric field node separation.

The electric field in a standing wave forms nodes and antinodes, just like any other wave patterns. These aid in understanding how energy transfers through electromagnetic fields.

Magnetic Field

A magnetic field (\( \vec{B} \)) is another vector field which describes the magnetic force on moving electric charges. In electromagnetic waves, the magnetic field oscillates perpendicular to both the electric field and the direction of wave propagation.
When examining electromagnetic waves, the \( \vec{B} \) field also forms nodes and antinodes, akin to the \( \vec{E} \) field, but interestingly their nodes are positioned differently within the wave cycle.

• Nodal Plane Separation: In a standing wave, nodes of the \( \vec{B} \) and \( \vec{E} \) fields are separated by a quarter wavelength. For the problem in question, this distance is \( \frac{\lambda}{4} \), equating to 1 meter.
• Orthogonality: The magnetic field is always orthogonal to its electric counterpart, maintaining a perpendicular orientation, which is a hallmark of electromagnetic waves.

Understanding the placement of nodes in both electric and magnetic fields is essential in fields like radio broadcasting and antenna design, as it enables precision in tuning in to the desired frequency and minimizing interference.