Question

What is the wavelength in nanometers of infrared light for which \(\lambda=2.50 \times 10^{-5} \mathrm{~m}\) ? How many times longer is this wavelength than red light that has a wavelength of \(750 \mathrm{~nm}\) ?

Short answer

The wavelength of infrared light in nanometers is \(\lambda_{\rm infrared} \, (nm) = 2.50 \times 10^{4} \, {\rm nm}\). The ratio of the wavelengths of infrared and red light is \(\text{Ratio} = \frac{2.50 \times 10^{4}}{750} = 33.3\). Thus, the infrared light's wavelength is 33.3 times longer than the red light's wavelength.

Step-by-step solution

Convert the wavelength of infrared light to nanometers

First, remember that 1 meter is equal to \(10^9\) nanometers (\(1 \rm m = 10^9 nm\)). To convert the wavelength of infrared light to nanometers, multiply its given value in meters by \(10^9\): \(\lambda_{\rm infrared} \, (nm) = 2.50 \times 10^{-5} \, {\rm m} \times \frac{10^9 \, {\rm nm}}{1 \, {\rm m}}\)

Calculate the wavelength of infrared light in nanometers

Next, find the value of the wavelength of infrared light in nanometers by multiplying: \(\lambda_{\rm infrared} \, (nm) = 2.50 \times 10^{-5} \times 10^9\)

Find the ratio of the wavelengths

Now, to find out how many times longer the wavelength of infrared light is compared to the wavelength of red light, divide the wavelength of infrared light (in nanometers) by the wavelength of red light: \(\text{Ratio} = \frac{\lambda_{\rm infrared} \, (nm)}{\lambda_{\rm red} \, (nm)}\)

Substitute the values and calculate the answer

Finally, substitute the wavelengths of both infrared and red light into the formula and calculate the ratio: \(\text{Ratio} = \frac{2.50 \times 10^{-5} \times 10^9}{750}\) After calculating these values, we can find the wavelength of infrared light in nanometers and how many times longer it is than the wavelength of red light.

Key concepts

Infrared Light Wavelength

In the study of light and its properties, understanding the wavelength of various types of light is crucial. When we discuss infrared light, we're referring to a type of electromagnetic radiation that is not visible to the human eye but can be felt as heat. Its wavelength is typically longer than that of visible light, placing it just beyond the red end of the visible spectrum.

The wavelength of infrared light can vary from about 700 nanometers (nm) to 1 millimeter (mm). Because of its longer wavelength, infrared light has different properties and applications than visible light. For instance, remote controls often use infrared light, and it is crucial in thermal imaging as well as in telecommunications where it's used for fiber-optic communication systems. Understanding the wavelength of infrared light helps us exploit these unique applications.

Nanometers to Meters Conversion

The language of light often involves measuring wavelengths in units such as nanometers (nm) and meters (m). A nanometer is a unit of length in the metric system, equal to one billionth of a meter ().

Converting nanometers to meters is straightforward due to the metric system's base-10 structure. To convert nanometers to meters, you divide the number of nanometers by 1 billion (or multiply by

This conversion is pivotal in physics and engineering, especially when dealing with light wavelengths, as they are typically expressed in nanometers for convenience and practicality. For example, the visible spectrum ranges roughly from 380 nm to 750 nm, which in meters would be to . Therefore, when tackling problems or interpreting scientific data, students and professionals must be comfortable with converting between these units to grasp the scale and implications of their work.

Comparing Wavelengths of Light

When studying different types of light, one key aspect is the comparison of their wavelengths. The wavelength is the distance between consecutive peaks (or any two identical points) in a wave and is closely related to the energy and frequency of the light. A fundamental understanding of comparing these wavelengths helps us appreciate the electromagnetic spectrum, which includes not only visible light but also other forms of radiation such as ultraviolet, X-rays, microwaves, and radio waves.

For instance, when comparing infrared light with red light, we notice that infrared has a longer wavelength. This means it carries less energy and has a lower frequency than red light. The practical result of longer wavelengths like those of infrared is that the light has the ability to penetrate materials more deeply and can be utilized in applications such as night-vision devices or heating. Using the concept of ratios, we can quantify just how much longer one wavelength is compared to another, an invaluable tool for scientists and engineers in fields such as optics and telecommunications. It's these comparisons that allow us to understand and utilize the broad spectrum of light in various technologies.