Problem 2 - College Physics

Question

\(\cdot\) The electromagnetic spectrum. Electromagnetic waves, which include light, consist of vibrations of electric and magnetic fields, and they all travel at the speed of light. (a) FM radio. Find the wavelength of an FM radio station signal broadcasting at a frequency of 104.5 \(\mathrm{MHz}\) . (b) \(\mathrm{X}\) rays. X rays have a wavelength of about 0.10 \(\mathrm{nm}\) . What is their frequency? (c) The Big Bang. Microwaves with a wavelength of 1.1 \(\mathrm{mm}\) , left over from soon after the Big Bang, have been detected. What is their frequency? (d) Sunburn. Sunburn (and skin cancer) are caused by ultraviolet light waves having a frequency of around \(10^{16} \mathrm{Hz}\) . What is their wavelength? (e) SETI. It has been suggested that extraterrestrial civilizations (if they exist) might try to communicate by using electromagnetic waves having the same frequency as that given off by the spin flip of the electron in hydrogen, which is 1.43 GHz. To what wave-length should we tune our telescopes in order to search for such signals? (f) Microwave ovens. Microwave ovens cook food with electromagnetic waves of frequency around 2.45 \(\mathrm{GHz}\) . What wavelength do these waves have?

Step-by-Step Solution

Answer

(a) 2.87 m, (b) 3 × 10¹⁸ Hz, (c) 2.73 × 10¹¹ Hz, (d) 30 nm, (e) 0.210 m, (f) 0.122 m.

Step 1: Understand the Relationship

The speed of light (\(c\)) is related to the frequency (\(f\)) and wavelength (\(\lambda\)) of electromagnetic waves by the equation \( c = f \cdot \lambda \). Here, \( c \approx 3 \times 10^8 \; \text{m/s} \) is constant. We can use this equation to find either wavelength or frequency when one of them is known.

Step 2: Calculate the Wavelength of FM Radio - Part (a)

Given the frequency for the FM radio is 104.5 MHz. Convert this into Hz: \(104.5 \; \text{MHz} = 104.5 \times 10^6 \; \text{Hz}\). Use the formula \( \lambda = \frac{c}{f} \). Substitute \(c = 3 \times 10^8 \; \text{m/s}\) and \(f = 104.5 \times 10^6 \; \text{Hz}\) to find \(\lambda\): \[ \lambda = \frac{3 \times 10^8}{104.5 \times 10^6} \]. This gives \(\lambda \approx 2.87 \; \text{m}\).

Step 3: Calculate the Frequency of X-rays - Part (b)

Given the wavelength for X-rays is 0.10 nm. Convert this into meters: \(0.10 \; \text{nm} = 0.10 \times 10^{-9} \; \text{m}\). Use the formula \( f = \frac{c}{\lambda} \). Substitute \(c = 3 \times 10^8 \; \text{m/s}\) and \(\lambda = 0.10 \times 10^{-9} \; \text{m}\) to find \(f\): \[ f = \frac{3 \times 10^8}{0.10 \times 10^{-9}} \]. This gives \(f \approx 3 \times 10^{18} \; \text{Hz}\).

Step 4: Calculate the Frequency of Microwaves from the Big Bang - Part (c)

Given the wavelength is 1.1 mm, convert this into meters: \(1.1 \; \text{mm} = 1.1 \times 10^{-3} \; \text{m}\). Use the formula \( f = \frac{c}{\lambda} \). Substitute \(c = 3 \times 10^8 \; \text{m/s}\) and \(\lambda = 1.1 \times 10^{-3} \; \text{m}\) to find \(f\): \[ f = \frac{3 \times 10^8}{1.1 \times 10^{-3}} \]. This gives \(f \approx 2.73 \times 10^{11} \; \text{Hz}\).

Step 5: Calculate the Wavelength of Ultraviolet Light - Part (d)

Given frequency for ultraviolet light is \(10^{16} \; \text{Hz}\). Use the formula \( \lambda = \frac{c}{f} \). Substitute \(c = 3 \times 10^8 \; \text{m/s}\) and \(f = 10^{16} \; \text{Hz}\) to find \(\lambda\): \[ \lambda = \frac{3 \times 10^8}{10^{16}} \]. This gives \(\lambda \approx 3 \times 10^{-8} \; \text{m}\) or 30 nm.

Step 6: Calculate the Wavelength for SETI Signal - Part (e)

Given the frequency is 1.43 GHz. Convert this into Hz: \(1.43 \; \text{GHz} = 1.43 \times 10^9 \; \text{Hz}\). Use the formula \( \lambda = \frac{c}{f} \). Substitute \(c = 3 \times 10^8 \; \text{m/s}\) and \(f = 1.43 \times 10^9 \; \text{Hz}\) to find \(\lambda\): \[ \lambda = \frac{3 \times 10^8}{1.43 \times 10^9} \]. This gives \(\lambda \approx 0.210 \; \text{m}\).

Step 7: Calculate the Wavelength in a Microwave Oven - Part (f)

Given frequency is 2.45 GHz. Convert this into Hz: \(2.45 \; \text{GHz} = 2.45 \times 10^9 \; \text{Hz}\). Use the formula \( \lambda = \frac{c}{f} \). Substitute \(c = 3 \times 10^8 \; \text{m/s}\) and \(f = 2.45 \times 10^9 \; \text{Hz}\) to find \(\lambda\): \[ \lambda = \frac{3 \times 10^8}{2.45 \times 10^9} \]. This gives \(\lambda \approx 0.122 \; \text{m}\).

Key Concepts

Wavelength Calculation

The concept of wavelength is central when dealing with electromagnetic waves. Wavelength, denoted by \( \lambda \), is the distance between successive crests of a wave. It determines the nature of the electromagnetic wave, distinguishing between radio waves, infrared, visible light, and other segments of the electromagnetic spectrum.

To find the wavelength of any electromagnetic wave, we use the fundamental relationship:

• \( \lambda = \frac{c}{f} \)

where \( c \) is the speed of light approximately equal to \( 3 \times 10^8 \) m/s, and \( f \) is the frequency of the wave.

For instance, calculating the wavelength of an FM radio signal, like one broadcasting at a frequency of 104.5 MHz, involves:

• Convert MHz to Hz: \( 104.5 \times 10^6 \) Hz.
• Substitute values into the formula: \( \lambda = \frac{3 \times 10^8}{104.5 \times 10^6} \).
• Resulting in a wavelength of approximately 2.87 m.

Understanding how to calculate wavelength helps in various applications from understanding radio broadcasts to interpreting cosmic microwave backgrounds.

Frequency Calculation

Frequency is another crucial concept in understanding electromagnetic waves. Frequency, represented by \( f \), is the number of wave cycles that pass a point in one second. It's measured in Hertz (Hz) and is inversely related to wavelength, meaning higher frequency corresponds to shorter wavelength and vice versa.

To calculate frequency when you know the wavelength, use:

• \( f = \frac{c}{\lambda} \)

Consider the example of X-rays with a known wavelength of 0.10 nm:

• Convert nm to meters: \( 0.10 \times 10^{-9} \) m.
• Substitute into the formula: \( f = \frac{3 \times 10^8}{0.10 \times 10^{-9}} \).
• Resulting in a high frequency of about \( 3 \times 10^{18} \) Hz.

Calculating frequency is essential in medical imaging with X-rays or understanding microwave frequencies from the Big Bang.

Speed of Light Formula

The speed of light in a vacuum is a universal constant, denoted by \( c \), and it's crucial to understanding the electromagnetic spectrum. It is always approximately \( 3 \times 10^8 \) meters per second. Light speed links wavelength and frequency in the electromagnetic wave equation:

• \( c = f \cdot \lambda \)

This relationship allows us to switch between concepts of frequency and wavelength depending on which specifics are given in a problem or scenario.

For practical usage, consider microwave ovens, which operate at around 2.45 GHz:

• Convert GHz to Hz: \( 2.45 \times 10^9 \) Hz.
• Use the formula for wavelength: \( \lambda = \frac{3 \times 10^8}{2.45 \times 10^9} \).
• Giving us a wavelength of approximately 0.122 meters.

The speed of light formula acts as the backbone of many calculations in the realm of physics, especially in technologies relying on electromagnetic waves.