Question

A 1 ns pulse of electromagnetic waves would be \(30 \mathrm{~cm}\) Iong. (a) Consider such a pulse of \(633 \mathrm{~nm}\) wavelength laser light. Find its central wave number and the range of wave numbers it comprises. (b) Repeat part (a), bue for a 1 ns pulse of \(100 \mathrm{MHz}\) radio waves.

Short answer

For the laser light with a wavelength of 633 nm, the central wave number is approximately \(9.93 x 10^{6} m^{-1}\) and the range of wave numbers is approximately \(20.94 m^{-1}\). For the radio waves with a frequency of 100 MHz, their central wave number is about \(0.00209 m^{-1}\) and the wave number range is the same as the laser light, \(20.94 m^{-1}\).

Step-by-step solution

Compute for the Central Wave Number for the Laser Light

To find the central wave number of the laser light with wavelength of 633 nm, convert the wavelength into meters, then substitute it into the wave number formula \(k = 2π/λ\). This will result in \( k = 2π/(633 x 10^{-9})\)

Calculate the Range of Wave Numbers for the Laser Light

The range of wave numbers for the laser light can be found using the formula Δk=2π/Δx, where Δx = 30 cm or 0.3 m. Thus, Δk= 2π/(0.3).

Compute for the Central Wave Number for the Radio Waves

Finding the central wave number for the radio waves requires converting frequency into wavelength then use the wave number formula. Using the speed of light \(c = 3 x 10^{8} m/s\), convert the frequency of 100 MHz (1 MHz = \(10^6 Hz\)) to wavelength by using the formula \(λ = c/f = 3 x 10^{8}/(100 x 10^6)\). After determining the wavelength, substitute it into the formula to get the wave number \( k = 2π/λ\).

Calculate the Range of Wave Numbers for the Radio Waves

The range of wave numbers can be found using the formula Δk=2π/Δx, where Δx = 30 cm or 0.3 m, the same as in step 2. Thus, Δk= 2π/(0.3) as well.

Key concepts

Wave Number Calculation

Understanding the wave number helps grasp the nature of different electromagnetic waves. The wave number is inversely related to the wavelength, with its unit being reciprocal meters \( m^{-1} \). Generally, it's defined as \( k = \frac{2\pi}{\lambda} \), where \( \lambda \) is the wavelength and \( \pi \) is the mathematical constant pi, approximately 3.14159. This relationship highlights that shorter wavelengths have higher wave numbers and vice versa.

For example, in a 1 ns pulse of electromagnetic waves, calculating the wave number provides us with valuable information about the spatial frequency of the wave—essentially telling us how many wave cycles can exist over a certain distance. It's a fundamental concept in understanding the behavior of waves, from the light of lasers to the transmission of radio signals.

Laser Light Wavelength

Laser light is often defined by its wavelength because it falls within the visible to infrared range of the electromagnetic spectrum. The wavelength of laser light affects its color and energy. For instance, a 633 nm wavelength corresponds to the red portion of the spectrum.

To connect this to our exercise, imagining a pulse of laser light as a physical entity 30 cm long helps us visualize the compactness of that light. The central wave number, which we calculate by inverting the wavelength, represents the most common number or the 'average' spatial frequency of the wave within that pulse.

Real-World Applications

• Laser Precision: The precise wavelength of laser light is crucial for applications like cutting, medical procedures, and optical communications.
• Wavelength and Energy: Shorter wavelengths, like those of blue or ultraviolet laser light, carry more energy, which is important in applications like laser imaging or spectroscopy.

Radio Waves Frequency

Radio waves are a type of electromagnetic radiation with frequencies as low as 3 kHz to as high as 300 GHz. A frequency of 100 MHz, which falls in the radio spectrum, is commonly used for FM radio broadcasting. Unlike the laser light wavelength, radio waves have much longer wavelengths and thus lower frequencies.

In our problem, converting from frequency to wavelength allows us to apply the same wave number concept to these much larger and longer-pulse radio waves. The central wave number gives us a measure of the number of wave cycles per unit distance within the radio pulse, which is crucial for understanding signal propagation and antenna design.

Importance in Communication

• Frequency Bandwidth: Different frequencies are used for various communication services, with certain frequencies penetrating buildings and traveling longer distances better than others.
• Antenna Design: The wavelength of the radio waves determines the size and type of the antenna needed to transmit or receive those waves effectively.