Problem 49 - Physics

Question

In a traveling electromagnetic wave, the electric field is represented mathematically as $$ E=E_{0} \sin \left[\left(1.5 \times 10^{10} \mathrm{~s}^{-1}\right) t-\left(5.0 \times 10^{1} \mathrm{~m}^{-1}\right) x\right] $$ where \(E_{0}\) is the maximum field strength. (a) What is the frequency of the wave? (b) This wave and the wave that results from its reflection can form a standing wave, in a way similar to that in which standing waves can arise on a string (see Section 17.5). What is the separation between adjacent nodes in the standing wave?

Step-by-Step Solution

Answer

(a) The frequency is approximately \(2.39 \times 10^9 \, \text{Hz}\). (b) The node separation is approximately \(0.063 \, \text{m}\).

Step 1: Identify the Angular Frequency

The equation for the electric field of a traveling wave is given by \(E = E_{0} \sin \left[\omega t - k x\right]\). Here, \(\omega\) represents the angular frequency and it is given as \(1.5 \times 10^{10} \mathrm{~s}^{-1}\).

Step 2: Determine the Frequency

The frequency \(f\) of the wave is related to the angular frequency \(\omega\) by the equation \(f = \frac{\omega}{2\pi}\). Plugging in the value of \(\omega\), we get:\[f = \frac{1.5 \times 10^{10}}{2\pi}\approx 2.39 \times 10^9 \, \mathrm{Hz}\]

Step 3: Understand the Formation of a Standing Wave

A standing wave is formed when two identical waves traveling in opposite directions interfere, such as a wave and its reflection. The nodes are points of zero amplitude on the standing wave.

Step 4: Identify the Wavenumber

The wavenumber \(k\) in the given wave equation is \(5.0 \times 10^1 \mathrm{~m}^{-1}\). This parameter is crucial for finding node separation.

Step 5: Calculate Node Separation

The separation between adjacent nodes in a standing wave is half the wavelength \(\lambda\). Using the relationship \(k = \frac{2\pi}{\lambda}\), solve for \(\lambda\):\[k = \frac{2\pi}{\lambda} \implies \lambda = \frac{2\pi}{5.0 \times 10^1} \approx 0.1256 \, \mathrm{m}\]Thus, the separation between nodes is \(\frac{\lambda}{2} = \frac{0.1256}{2} \approx 0.063 \, \mathrm{m}\).

Key Concepts

Standing Waves

Standing waves occur when two identical waves move in opposite directions, and their interference creates a pattern of fixed nodes and antinodes. In the context of electromagnetic waves as described in the original exercise, standing waves can form through reflection.

• Nodes: These are points of zero amplitude where the waves always cancel out.
• Antinodes: These are points of maximum amplitude where the wave crests meet.

In simpler terms, standing waves appear to 'stand still' despite being formed by the dynamic interaction of waves moving through space. Understanding standing waves helps us see how wave phenomena can create stable patterns in various physical contexts, including optics and acoustics.

Angular Frequency

Angular frequency (\(\omega\)) is a key concept when discussing waves and oscillations. It tells us how many radians a wave progresses in a second. This is distinct from the regular frequency, which counts cycles per second. The formula to find angular frequency is: \[\omega = 2\pi f\]- Where in the exercise given, \(\omega\) is provided directly as \(1.5 \times 10^{10} \mathrm{~s}^{-1}\).- This angular frequency hints at how fast the given electric field section is "rotating" in a graphical representation of the wave.Angular frequency is crucial for linking the time-dependency of wave phenomena to the cyclical nature of calculations using radians.

Wavenumber

The wavenumber (\(k\)) is a wave characteristic that describes the number of wave cycles present in a unit distance. In the exercise, \(k\) is given as \(5.0 \times 10^1 \mathrm{~m}^{-1}\), which indicates that it describes the spatial frequency of the wave:- Mathematically, the wavenumber is the spatial equivalent to angular frequency, showing how many "waves" fit into a given length.- The relationship between wavelength (\(\lambda\)) and wavenumber is expressed as:\[k = \frac{2\pi}{\lambda}\]Understanding the wavenumber helps in visualizing how tightly packed the wave crests and troughs are, which ties into how the wave's energy is distributed spatially. It’s pivotal for applications involving wave interference and diffraction patterns.

Node Separation

Node separation in standing waves corresponds to the distance between two nodes, where destructive interference causes no wave activity. In the context of the electromagnetic wave problem, this is particularly important in determining the structure of the wave. To find node separation:

• You first calculate the wavelength \(\lambda\) using the formula \(\lambda = \frac{2\pi}{k}\).
• Apply this to get \(\lambda = \frac{2\pi}{5.0 \times 10^1} \approx 0.1256 \mathrm{~m}\).
• Then, node separation is simply half the wavelength: \(\frac{\lambda}{2} \approx 0.063 \mathrm{~m}\).

This calculation demonstrates how node separation is dictated by the wavelength and informs the precise positioning of zero-amplitude points in a standing wave, which is critical for designing optical cavities and analyzing wave interference phenomena.