91Ó°ÊÓ

Standing waves are fascinating phenomena where specific points, called nodes, remain stationary while other parts of the wave vibrate with maximum amplitude. In the context of an electromagnetic wave, standing waves occur when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. These waves create a pattern of alternating nodes and antinodes.
Nodes are points of complete destructive interference where no movement occurs, while antinodes exhibit the maximum movement. In a standing electromagnetic wave, both the electric field (\(\overrightarrow{\boldsymbol{E}}\)) and the magnetic field (\(\overrightarrow{\boldsymbol{B}}\)) have nodes and antinodes. Studying these patterns help us understand properties like the wavelength and speed of the wave.
In the given exercise, knowing that nodal planes are where the fields have zero intensity helps us identify distances between nodes to find more information about the wave.

The wavelength of a wave is the physical distance from one point on a wave to the next identical point. In standing waves, this means measuring from node to node or antinode to antinode. The relationship between nodes in standing waves simplifies calculation: the distance between consecutive nodes is half the wavelength.
Given the distance between nodal planes, 3.55 mm, we recognize this as half the wavelength (\(\frac{\lambda}{2}\)). Therefore, the full wavelength is calculated by doubling this distance, as shown: \[\lambda = 2 \times 3.55 = 7.10 \text{ mm}\] Or in meters, \(7.10 \times 10^{-3} \text{ m}\). This valuable piece of information reveals the full length of one cycle of the wave as it propagates through the medium.

The speed of propagation is a core characteristic of a wave, indicating how fast the wave travels through its medium. When talking about electromagnetic waves, this speed in a vacuum is approximately the speed of light, but it varies in different materials.
For calculating the speed of propagation, use the formula \[v = f \lambda\] where \(v\) is the speed, \(f\) is the frequency, and \(\lambda\) is the wavelength. In our exercise, the frequency \(f = 2.20 \times 10^{10}\) Hz and the wavelength \(\lambda = 7.10 \times 10^{-3}\) m. When we substitute:\[v = 2.20 \times 10^{10} \times 7.10 \times 10^{-3} = 1.562 \times 10^8 \text{ m/s}\] This calculation shows how the frequency and wavelength directly impact the speed of the wave within the material.

Nodal planes are fundamental in the study of standing waves as they mark positions of no displacement. In electromagnetic waves, nodal planes occur for both the electric fields (\(\overrightarrow{\boldsymbol{E}}\)) and magnetic fields (\(\overrightarrow{\boldsymbol{B}}\)). These planes are essential for diagnosing wave patterns and understanding the distribution of energy.
In a standing electromagnetic wave, the nodal planes of the magnetic field \(\overrightarrow{\boldsymbol{B}}\), with a given spacing of 3.55 mm, serve as a reference for additional calculations. The electric field (\(\overrightarrow{\boldsymbol{E}}\)) nodal planes occur at a distance which is half of that of the magnetic field, as they are in quadrature, meaning out of phase by 90 degrees.
Therefore, identifying the distance between \(\overrightarrow{\boldsymbol{E}}\) nodes is straightforward:\[\frac{3.55}{2} = 1.775 \text{ mm} \text{ or } 1.775 \times 10^{-3} \text{ m} \] Understanding these distances aids in visualizing the structure and behavior of standing electromagnetic waves.