Question

List the following types of electromagnetic radiation in order of increasing wavelength: (a) the gamma rays produced by a radioactive nuclide used in medical imaging; (b) radiation from an FM radio station at 93.1 \(\mathrm{MHz}\) on the dial; (c) a radio signal from an AM radio station at 680 \(\mathrm{kHz}\) on the dial; ( d ) the yellow light from sodium vapor streetlights; (e) the red light of a light-emitting diode, such as in a calculator display.

Short answer

The electromagnetic radiations can be arranged in the order of increasing wavelengths as follows: (a) gamma rays, (d) yellow light (589-591 nm), (e) red light (620-750 nm), (b) FM radio station, and (c) AM radio station.

Step-by-step solution

Speed of light and formula connection

Recall that the speed of light is constant in a vacuum, approximately equal to \(c = 3.0 \times 10^8 \mathrm{m/s}\). We can use the speed of light along with the frequency to calculate the wavelength for FM and AM radio stations as follows: \(c = \lambda \nu\)

Calculate the wavelength for the FM radio station

The given frequency of the FM radio station is 93.1 MHz which is equal to \(93.1 \times 10^6 \mathrm{Hz}\). Now, to calculate the wavelength, we can plug the values into the formula: \(\lambda_{FM} = \dfrac{c}{\nu_{FM}} = \dfrac{3.0 \times 10^8 \mathrm{m/s}}{93.1 \times 10^6 \mathrm{Hz}}\)

Calculate the wavelength for the AM radio station

The given frequency of the AM radio station is 680 kHz which is equal to \(680 \times 10^3 \mathrm{Hz}\). Now, calculating the wavelength: \(\lambda_{AM} = \dfrac{c}{\nu_{AM}} = \dfrac{3.0 \times 10^8 \mathrm{m/s}}{680 \times 10^3 \mathrm{Hz}}\)

Recognize the position of other electromagnetic radiations in the spectrum

For gamma rays, yellow light, and red light, we know their position in the electromagnetic spectrum: 1. Gamma rays have the shortest wavelength and the highest frequency. 2. The yellow light from sodium vapor streetlights has a wavelength range around 589-591 nm. 3. The red light of a light-emitting diode (LED) has a wavelength range around 620-750 nm.

Arrange the electromagnetic radiations in increasing order of wavelengths

Based on the wavelengths, we can arrange the types of electromagnetic radiation as follows: 1. Gamma rays (the shortest wavelength) 2. Yellow light from sodium vapor streetlights (589-591 nm) 3. Red light from a light-emitting diode (620-750 nm) 4. Radiation from an FM radio station at 93.1 MHz 5. Radio signal from an AM radio station at 680 kHz (the longest wavelength) So, the final order is: (a) gamma rays, (d) yellow light, (e) red light, (b) FM radio station, and (c) AM radio station.

Key concepts

Electromagnetic Spectrum

The concept of the electromagnetic spectrum is essential in understanding how different forms of electromagnetic radiation relate to each other. Imagine a vast continuum of waves, each with different energies, frequencies, and wavelengths. On one end, we have gamma rays, with very short wavelengths and high frequencies, often produced by cosmic events or radioactive materials. Such rays are known for their penetrating ability and are used in medical imaging.

As we move along the spectrum, we encounter visible light, like the yellow light from sodium vapor streetlights and the red light of LEDs commonly found in electronic displays. These fall roughly in the middle of the spectrum and are the only types of electromagnetic radiation visible to the human eye. Their wavelengths range from about 380 nm (violet) to 750 nm (red).

Moving further, we reach radio waves, which have much longer wavelengths and lower frequencies. FM radio waves and AM radio waves belong here. FM radio stations, like the one mentioned at 93.1 MHz, transmit in shorter wavelengths than AM stations, like the one at 680 kHz. The full spectrum is a vital tool for various technologies, including communication, imaging, and even cooking in the case of microwaves, which reside between infrared waves and radio waves.

Speed of Light Formula

The speed of light in a vacuum is one of the fundamental constants of nature, denoted by the symbol 'c'. Its universally accepted value is approximately \(3.0 \times 10^8 \mathrm{m/s}\). This incredible speed holds the key to understanding how light and all forms of electromagnetic radiation travel.

When we discuss the speed of light formula, we're looking at the relationship \(c = \lambda u\) where \(\lambda\) represents the wavelength of the light (or other electromagnetic radiation), and \(u\) stands for its frequency. This equation tells us how wavelength and frequency balance against the constant speed of light — when one increases, the other must decrease, and vice versa.

This formula is crucial for calculating wavelengths of electromagnetic waves when frequency is known and vice versa. It's an essential tool for scientists and engineers who work with telecommunications, astrophysics, and a spectrum of other fields.

Wavelength and Frequency Relationship

The relationship between wavelength and frequency is fundamental to understanding electromagnetic waves. Given by the formula \(c = \lambda u\), this inverse relationship shows that as the frequency \(u\) of a wave increases, its wavelength \(\lambda\) decreases, provided the speed of light \(c\) remains constant. This principle applies across the electromagnetic spectrum.

For instance, the frequency of an FM radio station might be in the MHz range, and we can calculate its wavelength in meters by rearranging the speed of light formula to \(\lambda = \frac{c}{u}\). Applying this to an FM station broadcasting at 93.1 MHz, a higher frequency than an AM station at 680 kHz, we'd find the FM wave has a shorter wavelength.

Understanding this relationship helps explain why different types of electromagnetic radiation behave the way they do — higher frequencies like gamma rays can pass through materials that lower frequencies cannot, and long-wavelength radio waves can bend around obstacles, making them ideal for long-distance communication. Such principles are key in designing antennas and transmitters, as well as in medical imaging technologies and even in our understanding of the universe.