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; \((\mathrm{c})\) 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.

The relationship between frequency and wavelength for electromagnetic radiation is an inverse one, expressed by the equation

\( v = f \lambda \)

where \( v\) is the speed of light in a vacuum (approximately \(3 \cdot 10^8 \mathrm{m/s}\)), \(f\) is the frequency, and \(\lambda\) is the wavelength. When frequency increases, the wavelength decreases, and vice versa. This fundamental principle allows us to understand and predict the behaviors of different types of electromagnetic waves, including radio waves, visible light, and gamma rays.
In educational materials and homework help, this relationship is often visualized using graphs or animations to illustrate how waves with higher frequencies have shorter wavelengths, while those with lower frequencies have longer wavelengths. It's this core concept that helps students grasp why different types of electromagnetic radiation are sorted into an electromagnetic spectrum based on frequency or wavelength.

Gamma rays have the shortest wavelengths and highest frequencies in the electromagnetic spectrum, with frequencies above \(10^{19} \mathrm{Hz}\). They carry a significant amount of energy and are produced by various high-energy astronomical phenomena and nuclear reactions, including radioactive decay used in medical imaging.

Due to their penetrating power, gamma rays have the ability to pass through many materials, which is why they find application in sterilizing medical equipment, cancer treatment through radiotherapy, and in imaging techniques like PET scans. However, because of their high energy, they can also be harmful to biological tissues, which is why safety and protection measures are critical when working with gamma rays.

FM (Frequency Modulation) and AM (Amplitude Modulation) radio waves are used for broadcasting and communication purposes. They are located at the low-frequency end of the electromagnetic spectrum and, consequently, have longer wavelengths.

FM radio waves typically range from \(10^7\) to \(10^9 \mathrm{Hz}\), translating to wavelengths between 30 meters to 3 meters. They are less susceptible to interference and noise, which leads to better sound quality, explaining why they're widely used for music broadcasts.
On the other hand, AM radio waves range from \(10^5\) to \(10^7 \mathrm{Hz}\) with much longer wavelengths – up to hundreds of meters. They can travel longer distances and are great for talk shows, news broadcasts, and other spoken-word formats. However, they’re more prone to interference from electrical equipment and atmospheric conditions.

Light wavelength is a measure associated with visible light, which is the portion of the electromagnetic spectrum that the human eye can detect. Different wavelengths are perceived as different colors, with violet light having the shortest wavelength (approximately 380 nm) and red light having the longest (approximately 700 nm).

In our exercise, the yellow light from sodium vapor streetlights has a wavelength of about \(5.9 \cdot 10^{-7} \mathrm{m}\), and red light from an LED has a slightly longer wavelength of \(6.98 \cdot 10^{-7} \mathrm{m}\). These specific colors fall within a narrow band of the spectrum, illustrating the fine-tuning possible within light sources for specific purposes, such as street lighting or digital displays.

Speed of light, denoted by \( c \), is a fundamental constant of nature representing the speed at which all electromagnetic radiation travels in a vacuum. The established value is approximately \(3 \cdot 10^8 \mathrm{m/s}\). This constant is not just important for physicists but also plays a crucial role in technologies that depend on light and electromagnetic waves, including GPS, telecommunications, and various scientific measurements.

When discussing the speed of light in educational content, we emphasize that it's not just a speed limit for physical objects, but it also determines how quickly information can be transmitted across space. It's vital in calculations of cosmological distances and is used to define the meter, the basic unit of length in the International System of Units (SI).