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Chapter 34: Problem 26

In a double-slit experiment, He-Ne laser light of wavelength \(633 \mathrm{nm}\) produced an interference pattern on a screen placed at some distance from the slits. When one of the slits was covered with a thin glass slide of thickness \(12.0 \mu \mathrm{m},\) the central fringe shifted to the point occupied earlier by the 10 th dark fringe (see figure). What is the refractive index of the glass slide? (a) Without the glass slide (b) With glass slide

Short Answer

Expert verified
Answer: The refractive index of the glass slide is approximately 1.53.

Step by step solution

01

Identifying known values and the formula for path difference

We are given: - Wavelength of light (λ) = 633 nm = 633 x 10^-9 m - Thickness of the glass slide (t) = 12.0 μm = 12.0 x 10^-6 m - The central fringe shifts to the previous 10th dark fringe when the glass slide is placed. We need to find the path difference (Δ) caused by the glass slide. The formula for path difference in a double-slit experiment is given by: Δ = mλ where m = the order of the dark fringe (in this case, m=10).
02

Calculate the path difference

Using the given values and the path difference formula, we can calculate the path difference: Δ = 10λ = 10 * (633 * 10^-9 m) = 6.33 * 10^-6 m
03

Determine the extra distance traveled in the glass slide

When the glass slide is placed over one slit, the light from that slit has to travel an extra distance (d) through the slide compared to the other slit. This extra distance is given by: d = (n - 1) t where n = refractive index of the glass slide.
04

Use the path difference to find the refractive index

We have the path difference (Δ) and the extra distance (d) formulas: Δ = 6.33 * 10^-6 m d = (n - 1) t Since the path difference is equal to the extra distance traveled in the glass slide, we can equate the two formulas: 6.33 * 10^-6 m = (n - 1) * (12.0 * 10^-6 m) Now, we can solve for the refractive index (n): n = (6.33 * 10^-6 m / (12.0 * 10^-6 m)) + 1
05

Calculate the refractive index

Plugging the values into the equation, we get: n = (6.33 / 12) + 1 = 1.5275 Thus, the refractive index of the glass slide is approximately 1.53.

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