Question

(a) What is the frequency of radiation that has a wavelength of \(10 \mu \mathrm{m}\), about the size of a bacterium? (b) What is the wavelength of radiation that has a frequency of \(5.50 \times 10^{14} \mathrm{~s}^{-1} ?(\mathrm{c})\) Would the radiations in part (a) or part (b) be visible to the human eye? (d) What distance does electromagnetic radiation travel in \(50.0 \mu \mathrm{s} ?\)

Short answer

The frequency of radiation with a wavelength of \(10 \mu m\) is \(3 \times 10^{13} s^{-1}\). The wavelength of radiation with a frequency of \(5.50 \times 10^{14} s^{-1}\) is approximately \(5.45 \times 10^{-7} m\). The radiation from part (a) would not be visible to the human eye, while the radiation from part (b) would be visible. Electromagnetic radiation travels 15,000 meters (or 15 km) in 50.0 microseconds.

Step-by-step solution

1. Find the Frequency of Radiation with Wavelength \(10 \mu m\)

To determine the frequency, we use the formula \(c = \lambda \nu\). First, we need to convert the given wavelength from micrometers to meters: \(10 \mu m = 10 \times 10^{-6} m\). Next, we can solve for the frequency using the speed of light value, \(c = 3 \times 10^8 m/s\): \[\nu = \frac{c}{\lambda} = \frac{3 \times 10^8 m/s}{10 \times 10^{-6} m} = 3 \times 10^{13} s^{-1}\] The frequency of radiation with a wavelength of \(10 \mu m\) is \(3 \times 10^{13} s^{-1}\).

2. Find the Wavelength of Radiation with Frequency \(5.50 \times 10^{14} s^{-1}\)

To determine the wavelength, we use the same formula \(c = \lambda \nu\), but this time we have the frequency and need to solve for \(\lambda\): \[\lambda = \frac{c}{\nu} = \frac{3 \times 10^8 m/s}{5.5 \times 10^{14} s^{-1}} \approx 5.45 \times 10^{-7} m\] The wavelength of radiation with frequency \(5.50 \times 10^{14} s^{-1}\) is approximately \(5.45 \times 10^{-7} m\).

3. Determine Visibility to the Human Eye

Visible light has wavelengths ranging from approximately 380 nm to 750 nm. We need to check if the wavelengths from parts (a) and (b) fall within this range: 1. Wavelength from part (1): \(10 \mu m = 10 \times 10^3 nm = 10000 nm\), which is not within the visible light range. 2. Wavelength from part (2): \(5.45 \times 10^{-7} m = 545 nm\), which is within the visible light range. Thus, the radiation from part (a) would not be visible to the human eye, while the radiation from part (b) would be visible.

4. Calculate the Distance Electromagnetic Radiation Travels in \(50.0 \mu s\)

To determine the distance traveled, we need to use the formula \(d = ct\), where \(c\) is the speed of light and \(t\) is the time. First, we need to convert the given time from microseconds to seconds: \(50.0 \mu s = 50.0 \times 10^{-6} s\). Then, we can plug in the values to find the distance: \[d = (3 \times 10^8 m/s)(50.0 \times 10^{-6} s) = 15 \times 10^3 m\] Electromagnetic radiation travels 15,000 meters (or 15 km) in 50.0 microseconds.

Key concepts

Wavelength and Frequency Relationship

Understanding the relationship between wavelength and frequency is crucial when studying electromagnetic radiation. Both of these physical quantities are inversely related, meaning that as one increases, the other decreases. This relationship is governed by the formula: \[u = \frac{c}{\lambda}\] where \(u\) represents the frequency, \(c\) is the speed of light, and \(\lambda\) denotes the wavelength.

When we talk about the size of a bacterium, which could be approximately \(10 \mu m\), the frequency for radiation of this wavelength can be calculated using the above formula. For instance, substituting the values in the formula, one finds that radiation with a wavelength of \(10 \mu m\) would have a frequency of \(3 \times 10^{13} s^{-1}\), using the speed of light value of \(3 \times 10^8 m/s\).

This conversion from wavelength to frequency and vice versa is fundamental to understanding the nature of electromagnetic waves, which is essential for fields ranging from astronomy to telecommunications.

Visible Light Spectrum

The visible light spectrum is a small but significant part of the entire electromagnetic spectrum that the human eye can detect. It ranges approximately from 380 nm to 750 nm. This range encompasses all the colors visible to us, from violet at the lower end to red at the higher end.

The significance of the visible light spectrum becomes apparent when considering if a certain electromagnetic radiation is visible to us or not. For example, a wavelength calculated as \(5.45 \times 10^{-7} m\), or 545 nm, falls within this range. Thus, this radiation would appear to us as a certain color, in this case, green. In contrast, a wavelength such as \(10 \mu m\), or 10000 nm, is well beyond the visible spectrum and therefore invisible to the human eye.

Knowing where a wavelength fits within the visible spectrum helps us understand the variety of colors we see in everyday life and is instrumental for technology such as cameras and screens that need to mimic this spectrum.

Speed of Light

The speed of light is a fundamental constant in physics, denoted as \(c\), and its value is approximately \(3 \times 10^8 m/s\) in a vacuum. This constancy is a principle of Einstein's theory of relativity, indicating that light travels at a constant speed regardless of the motion of the source or observer.

Using the speed of light, we can calculate how far light travels in a given time. If, for example, you were to measure the distance electromagnetic radiation could cover in \(50.0 \mu s\), you would find it's 15 kilometers - a distance light can travel almost instantaneously to the human perception. This swift travel is what allows us to have technologies such as instant global communication via satellites.

• Instantaneous, but not infinite - light speed sets the ultimate limit for how quickly information can travel.
• It also underlies the calculation of astronomical distances and has implications in technologies like GPS and laser precision tools.

Electromagnetic Spectrum

The electromagnetic spectrum includes all types of electromagnetic radiation, which vary in wavelength and frequency. At one end of the spectrum, we find gamma rays with very short wavelengths and high frequencies. As we move through X-rays, ultraviolet light, the visible spectrum, and infrared, we notice that wavelengths get longer and frequencies decrease until we reach radio waves at the other end, featuring the longest wavelengths and lowest frequencies.

Each type of electromagnetic radiation serves a different purpose and interacts uniquely with matter. For example, X-rays can pass through soft tissues but are absorbed by denser materials such as bone, which is why they are used in medicine for imaging. Radio waves, with their longer wavelengths, are ideal for carrying information over large distances and are used in broadcast and communication technologies.

The exploration of the electromagnetic spectrum is ongoing, with new applications and discoveries in fields like medical imaging and telecommunications, demonstrating the versatility and significance of electromagnetic radiation in both science and daily life.