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; \((\mathrm{d})\) 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 order of increasing wavelength for the given electromagnetic radiations is (a) Gamma rays (\(3 \times 10^{-10}\) m), (d) Yellow light from sodium vapor streetlights (\(589 \times 10^{-9}\) m), (e) Red light from a light-emitting diode (\(700 \times 10^{-9}\) m), (b) FM radio radiation (≈ 3.22 m), and (c) AM radio signal (≈ 441 m).

Step-by-step solution

Find the wavelength of gamma rays

Gamma rays have very high frequency, usually in the range of several exahertz (1 exahertz = \(10^18 Hz\)). For simplicity, let's take the lower end of this range and use \(f = 1 \times 10^{18}\) Hz for gamma rays. Using the wavelength formula: \(λ_{\text{gamma}} = \frac{3 \times 10^8}{1 \times 10^{18}}\) \(λ_{\text{gamma}} = 3 \times 10^{-10}\) m

Find the wavelength of FM radio radiation

Given, the frequency of FM radio station is 93.1 MHz. To convert it to Hz, multiply by \(10^6\), so \(f = 93.1 \times 10^6\) Hz. Now, using the wavelength formula: \(λ_{\text{FM}} = \frac{3 \times 10^8}{93.1 \times 10^6}\) \(λ_{\text{FM}} ≈ 3.22\) m

Find the wavelength of AM radio signal

Given, the frequency of AM radio station is 680 kHz. To convert it to Hz, multiply by \(10^3\), so \(f = 680 \times 10^3\) Hz. Using the wavelength formula: \(λ_{\text{AM}} = \frac{3 \times 10^8}{680 \times 10^3}\) \(λ_{\text{AM}} ≈ 441\) m

Find the wavelength of yellow light from sodium vapor streetlights

The yellow light emitted by sodium vapor lamps has a wavelength of approximately 589 nm. To convert it to meters, multiply by \(10^{-9}\), so: \(λ_{\text{yellow}} = 589 \times 10^{-9}\) m

Find the wavelength of red light from a light-emitting diode

The red light emitted by LEDs has a wavelength of approximately 700 nm. To convert it to meters, multiply by \(10^{-9}\), so: \(λ_{\text{red}} = 700 \times 10^{-9}\) m

Arrange the wavelengths in increasing order

Now that we have the wavelengths for each type of radiation, we can list them in increasing order: 1. Gamma rays: \(3 \times 10^{-10}\) m 2. Yellow light from sodium vapor streetlights: \(589 \times 10^{-9}\) m 3. Red light from a light-emitting diode: \(700 \times 10^{-9}\) m 4. FM radio radiation: ≈ 3.22 m 5. AM radio signal: ≈ 441 m Hence, the order of increasing wavelength is (a) Gamma rays, (d) Yellow light, (e) Red light, (b) FM radio radiation, and (c) AM radio signal.

Key concepts

Gamma Rays

Gamma rays are a type of electromagnetic radiation with the shortest wavelengths and the highest frequencies in the electromagnetic spectrum. They are produced by the most energetic and extreme events in the universe, such as nuclear explosions, radioactive decay, and cosmic phenomena. In medical imaging, gamma rays are used in diagnostic techniques like PET scans because they can penetrate the body and allow doctors to observe internal processes.

Due to their very short wavelength—typically less than a tenth of a nanometer (\(3 \times 10^{-10} \text{m}\) in our example)—they are highly penetrative and can be dangerous to living tissue, but can also be used in therapy to target cancer cells. Understanding gamma rays is crucial not only in astronomy and medicine but also in fields like nuclear physics and environmental monitoring.

FM Radio Radiation

Frequency Modulation (FM) radio radiation is commonly used for broadcasting music and speech with high fidelity over long distances. FM stations, such as the one at 93.1 MHz mentioned in the exercise, utilize frequencies in the very high frequency (VHF) range of the spectrum. This type of radiation uses variations in frequency to encode information, unlike AM radio which uses amplitude variations.

The FM radio wavelengths are usually measured in meters and, typically, are longer than those of visible light but shorter than those of AM radio signals. For instance, the wavelength of the mentioned FM radio station at 93.1 MHz is approximately 3.22 meters. FM radio is favored for its resilience to noise and interference, which is why music sounds clearer on FM radio compared to AM.

AM Radio Signal

Amplitude Modulation (AM) radio signals are one of the oldest methods of transmitting audio content through electromagnetic waves. An AM station broadcasting at 680 kHz—kilohertz—(as in our exercise) produces radio waves with lower frequencies than FM radio waves. The AM signal's longer wavelengths, which in this case are approximately 441 meters, allow them to diffract around obstacles and reach longer distances, particularly at night.

Though subject to more interference than FM signals, AM radio is still widely used, especially for talk radio, news broadcasts, and areas where the terrain or cost considerations make FM radio less feasible. Understanding AM radio is essential when studying how different frequencies behave and are utilized in communication technologies.

Yellow Light from Sodium Vapor

Yellow light from sodium vapor lamps is another intriguing instance of electromagnetic radiation, commonly seen in streetlights. These lamps contain a small amount of sodium, which, when heated, produces a distinctive yellow glow with a wavelength of about 589 nanometers (\(589 \times 10^{-9} \text{m}\) as given in our exercise).

The characteristic yellow light is due to the specific energy levels of sodium atoms, which when excited, emit photons in the yellow part of the visible light spectrum. This type of light is quite efficient and is used not only for street lighting but also for industrial and security lighting due to its high luminous efficacy and the ability to penetrate fog.

Red Light from LED

Red light emitted from Light-Emitting Diodes (LEDs), like those in calculator displays or indicator lights, is the result of electroluminescence from materials like gallium arsenide. Red LEDs operate with a longer wavelength in the visible light spectrum, approximately 700 nanometers (\(700 \times 10^{-9} \text{m}\) in our example), which gives them their distinctive red appearance.

LEDs are highly efficient, durable, and require very little energy to operate, making them an environmentally friendly lighting option. Red LEDs, in particular, are also used in remote controls, optical communication devices, and as indicator lights because the human eye is highly sensitive to red light, making it easily noticeable.