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Chapter 34: Problem 30

Sometimes thin films are used as filters to prevent certain colors from entering a lens. Consider an infrared filter, designed to prevent 800.0 -nm light from entering a lens. Find the minimum film thickness for a layer of \(\mathrm{MgF}_{2}\) \((n=1.38)\) to prevent this light from entering the lens.

Short Answer

Expert verified
Answer: The minimum thickness of the magnesium fluoride film is 435nm.

Step by step solution

01

Identify the Conditions for Destructive Interference

The film causes destructive interference for a particular wavelength when the total phase difference between the light waves reflecting from the upper and lower surfaces of the film is a half-integer multiple of the wavelength in air. In this problem, we want the destructive interference for 800nm light. The phase difference is given by \(2nt \cos{\theta_2} - m\lambda\), where \(n\) is the index of refraction of the film, \(t\) is the thickness of the film, \(\lambda\) is the wavelength in air, \(\theta_2\) is the angle of refraction inside the film, and \(m\) is an integer (usually 0 for normal incidence).
02

Determine the Angle of Refraction

Since the angle of incidence is not given in the problem, we can assume normal incidence. For normal incidence, the angle of incidence, \(\theta_1 = 0^\circ, \) and according to Snell's law, the angle of refraction \(\theta_2\) will also be \(0^\circ\). Therefore, the term \(\cos{\theta_2}\) will be 1.
03

Calculate the Required Thickness for Destructive Interference

We must find the minimum thickness \(t\) such that \(2nt - m\lambda = \dfrac{(2m-1)\lambda}{2}\), where \(m = 1, 2, 3, ...\). For the minimum thickness, we can choose \(m = 1\). Now, we can substitute the values of the variables: \(2 \times 1.38 \times t - 800.0 \times 10^{-9} = \dfrac{(2 - 1)800.0 \times 10^{-9}}{2}\) Solve for \(t\): \(2 \times 1.38 \times t = 800.0 \times 10^{-9} + \dfrac{800.0 \times 10^{-9} }{2}\) \(t = \dfrac{1200.0 \times 10^{-9}}{2 \times 1.38}\) \(t \approx 4.35 \times 10^{-7}\,\text{m}\)
04

Express the Answer in Nanometers

To express the answer in nanometers, multiply by \(10^{9}\): \(t \approx 4.35 \times 10^{-7}\,\text{m} \times 10^{9} = 435\,\text{nm}\) Therefore, the minimum thickness of the magnesium fluoride film that can prevent 800nm light from entering the lens is 435nm.

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Most popular questions from this chapter

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