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 \(\overrightarrow{\boldsymbol{E}} \) field? (b) What is the distance between a nodal plane of \(\vec{E}\) and the closest nodal plane of \(\overrightarrow{\boldsymbol{B}} ?\)

Step-by-Step Solution

Answer

(a) 2.0 meters; (b) 1.0 meter.

Step 1: Calculate the Wavelength

The frequency of the wave is given as 75.0 MHz. To find the wavelength, we use the formula \( c = \lambda \cdot f \), where \( c \) is the speed of light (\( 3 \times 10^8 \) m/s), \( \lambda \) is the wavelength, and \( f \) is the frequency. Rearranging the formula gives \( \lambda = \frac{c}{f} \). Substituting the values, we have:\[\lambda = \frac{3 \times 10^8 \text{ m/s}}{75.0 \times 10^6 \text{ Hz}} = 4.0 \text{ meters}\]

Step 2: Find the Distance Between Nodal Planes of \(\vec{E}\)

In a standing wave, the distance between nodal planes is half of the wavelength. Since we calculated the wavelength to be 4.0 meters, the distance between nodal planes is:\[\text{Distance between nodal planes of } \vec{E} = \frac{\lambda}{2} = \frac{4.0 \text{ meters}}{2} = 2.0 \text{ meters}\]

Step 3: Find the Distance Between \(\vec{E}\) Nodal Plane and the Closest \(\vec{B}\) Nodal Plane

In a standing electromagnetic wave, the electric field (\(\vec{E}\)) and magnetic field (\(\vec{B}\)) are out of phase by a quarter of a wavelength. This means the nodal plane of \(\vec{E}\) and a nearby nodal plane of \(\vec{B}\) are separated by one quarter of the wavelength:\[\text{Distance between } \vec{E} \text{ and } \vec{B} \text{ nodal planes} = \frac{\lambda}{4} = \frac{4.0 \text{ meters}}{4} = 1.0 \text{ meter}\]

Key Concepts

Understanding Standing Waves

Standing waves are fascinating phenomena that occur when waves of the same frequency and amplitude travel in opposite directions and interfere with each other. This creates points of no motion, called nodes, and points of maximum amplitude, known as antinodes. In electromagnetic standing waves, such as the one given in the exercise, both electric (\( \vec{E} \)) and magnetic (\( \vec{B} \)) fields form standing waves. These waves are fixed in space and only oscillate in time.

For electromagnetic waves:

• The nodes of the electric field refer to locations where the field strength is zero.
• The antinodes represent areas of maximum field strength.

A key characteristic of standing waves is that the distance between successive nodes (or antinodes) is half the wavelength (\(\frac{\lambda}{2}\)). This is important for understanding how nodes are spaced, which was part of the exercise tackled earlier.

Frequency and Wavelength Relation

Frequency and wavelength are core components of all waves, including electromagnetic waves. The relationship ties directly to the speed of the wave. For electromagnetic waves, such as light or radio waves, this relationship is expressed through the equation \( c = \lambda \cdot f \), where \( c \) is the speed of light, \( \lambda \) is the wavelength, and \( f \) is the frequency.

Key considerations include:

• Frequency is the number of cycles per second (measured in Hertz, Hz).
• Wavelength is the physical length of one wave cycle (measured in meters).
• The speed of light \( c \) is approximately \( 3 \times 10^8 \) m/s.

The exercise involved using this formula to find the wavelength once the frequency was known. Rearranging gives \( \lambda = \frac{c}{f} \), which allows one to calculate the distance over which the wave completes one cycle. This relation is foundational in understanding how electromagnetic waves behave and propagate.

Interplay Between Electric and Magnetic Fields

Electromagnetic waves uniquely consist of oscillating electric and magnetic fields, which are perpendicular to each other and to the direction of wave travel. One intriguing feature is that in a standing electromagnetic wave:

• The \( \vec{E} \) field and \( \vec{B} \) field combine constructively and destructively, leading to a pattern of nodes and antinodes.
• These fields are equidistant from each other by one-quarter of a wavelength (\(\frac{\lambda}{4}\)).

This spatial phase difference is why the distance between a node of the electric field and a node of the magnetic field is a quarter of the wavelength, as calculated in the exercise. This property not only reflects the periodic nature of electromagnetic waves but also highlights how electric and magnetic fields are interdependent within wave dynamics. Understanding this relationship is critical for applications including wireless communication and optical devices.