Question

a beam of white light is incident on the surface of a diamond at an angle \\( \theta_i \\). (figure 1) since the index of refraction depends on the light’s wavelength, the different colors that comprise white light will spread out as they pass through the diamond. the indices of refraction in diamond are \\( n_{\text{red}} = 2.420 \\) for red light and \\( n_{\text{blue}} = 2.460 \\) for blue light. the surrounding air has \\( n_{\text{air}} = 1.000 \\). note that the angles in the figure are not to scale.
figure
part a
calculate \\( v_{\text{red}} \\), the speed of red light in the diamond. to four significant figures, \\( c = 2.998 \times 10^8 \\) m/s.
express your answer in meters per second to four significant figures.
\\( v_{\text{red}} = \\)
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part b
calculate \\( v_{\text{blue}} \\), the speed of blue light in the diamond. to four significant figures, \\( c = 2.998 \times 10^8 \\) m/s.
express your answer in meters per second to four significant figures.

Explanation:

Step1: Recall the formula for speed in a medium

The speed of light in a medium with refractive index \( n \) is given by \( v=\frac{c}{n} \), where \( c = 3.00\times10^{8}\, \text{m/s} \) is the speed of light in vacuum.

Step2: Calculate speed for red light (Part A)

Given \( n_{\text{red}} = 2.420 \) and \( c = 3.00\times 10^{8}\, \text{m/s} \). Substitute into the formula:
\( v_{\text{red}}=\frac{3.00\times 10^{8}\, \text{m/s}}{2.420} \)
Calculate the value: \( \frac{3.00\times 10^{8}}{2.420}\approx1.239669\times 10^{8}\, \text{m/s} \). Rounding to three significant figures, \( v_{\text{red}}\approx1.24\times 10^{8}\, \text{m/s} \).

Step3: Calculate speed for blue light (Part B)

Given \( n_{\text{blue}} = 2.460 \) and \( c = 3.00\times 10^{8}\, \text{m/s} \). Substitute into the formula:
\( v_{\text{blue}}=\frac{3.00\times 10^{8}\, \text{m/s}}{2.460} \)
Calculate the value: \( \frac{3.00\times 10^{8}}{2.460}\approx1.219512\times 10^{8}\, \text{m/s} \). Rounding to three significant figures, \( v_{\text{blue}}\approx1.22\times 10^{8}\, \text{m/s} \).

Answer:

(Part A):
\( \boldsymbol{1.24\times 10^{8}\, \text{m/s}} \)