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Chapter 11: Problem 16

Light visible to humans consists of electromagnetic waves with wavelengths (in air) in the range \(400-700 \mathrm{nm}\) $\left(4.0 \times 10^{-7} \mathrm{m} \text { to } 7.0 \times 10^{-7} \mathrm{m}\right) .$ The speed of light in air is \(3.0 \times 10^{8} \mathrm{m} / \mathrm{s} .\) What are the frequencies of electromagnetic waves that are visible?

Short Answer

Expert verified
Answer: The range of frequencies for visible light is from \(4.3 \times 10^{14} \thinspace Hz\) to \(7.5 \times 10^{14} \thinspace Hz\).

Step by step solution

01

Recall the speed-wavelength-frequency relationship

For electromagnetic waves, we have the relationship: \(c = \lambda \nu\) where: - \(c\) is the speed of light in air, which is approximately \(3.0 \times 10^{8} \thinspace m/s\), - \(\lambda\) is the wavelength, and - \(\nu\) is the frequency. We will use this equation to solve for the frequencies of visible light.
02

Find the frequency at the smallest wavelength (400 nm)

We will first find the frequency corresponding to the smallest wavelength, \(400 \thinspace nm\). We have to convert this wavelength to meters, which is \(4.0 \times 10^{-7} \thinspace m\). Then, we can use the equation to find the frequency: \(\nu_{min} = \frac{c}{\lambda_{min}} = \frac{3.0 \times 10^{8} \thinspace m/s}{4.0 \times 10^{-7} \thinspace m} = 7.5 \times 10^{14} \thinspace Hz\)
03

Find the frequency at the largest wavelength (700 nm)

Next, we will find the frequency corresponding to the largest wavelength, \(700 \thinspace nm\). Again, we need to convert this wavelength to meters, which is \(7.0 \times 10^{-7} \thinspace m\). Then, we can use the equation to find the frequency: \(\nu_{max} = \frac{c}{\lambda_{max}} = \frac{3.0 \times 10^{8} \thinspace m/s}{7.0 \times 10^{-7} \thinspace m} = 4.3 \times 10^{14} \thinspace Hz\)
04

State the range of frequencies for visible light

We have found the frequencies corresponding to the smallest and largest wavelengths of visible light. Therefore, the range of frequencies for visible light is from \(4.3 \times 10^{14} \thinspace Hz\) to \(7.5 \times 10^{14} \thinspace Hz\).

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