/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 46 Acetone A thin film of acetone \... [FREE SOLUTION] | 91影视

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Chapter 36: Problem 46

Acetone A thin film of acetone \((n=1.25)\) coats a thick glass plate \((n=1.50)\). White light is incident normal to the film. In the reflections, fully destructive interference occurs at \(600 \mathrm{~nm}\) and fully constructive interference at \(700 \mathrm{~nm}\). Calculate the thickness of the acetone film.

Short Answer

Expert verified
The thickness of the acetone film is 840 nm.

Step by step solution

01

Understand the problem

Determine the thickness of an acetone film on a glass plate by using the principles of thin-film interference, given the refractive indices and the wavelengths at which destructive and constructive interference occur.
02

Identify given parameters

Refractive index of acetone: \(n_{1} = 1.25\), refractive index of glass: \(n_{2} = 1.50\), wavelength for destructive interference: \( \text{位}_{destructive} = 600 \text{ nm} \), and wavelength for constructive interference: \( \text{位}_{constructive} = 700 \text{ nm} \).
03

Consider the phase change

Since light reflects off a medium with a higher refractive index (glass) and acetone (lower refractive index), there will be a phase change of half a wavelength (i.e., \( \text{位}/2 \)) associated with the reflection.
04

Equation for destructive interference

The condition for destructive interference in a thin film involves a half-wavelength phase difference. Formula: \(2 n_{1} t = (m + 0.5) \text{位}_{destructive} \).
05

Equation for constructive interference

The condition for constructive interference in a thin film involves a full wavelength phase difference. Formula: \(2 n_{1} t = m \text{位}_{constructive} \).
06

Construct equations with given wavelengths

For destructive interference: \(2 \times 1.25 \times t = (m + 0.5) \times 600 \text{ nm}\). For constructive interference: \(2 \times 1.25 \times t = m \times 700 \text{ nm}\).
07

Solve the equations

We have two equations: Equation 1) \(2.5t = (m+0.5) \times 600\) Equation 2) \(2.5t = m \times 700\). Set them equal to each other: \((m+0.5) \times 600 = m \times 700\) Solve for m: \(600m + 300 = 700m\) \(300 = 100m\) \(m = 3\).
08

Substitute back to find thickness

Substitute \(m = 3\) into either equation, e.g., \(2.5t = 3 \times 700\) \(2.5t = 2100\) \(t = \frac{2100}{2.5} = 840 \text{ nm}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Refractive Index
The refractive index is a measure of how much light slows down when it travels through a material. It is important in thin-film interference as it affects the phase change and interference pattern.

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Most popular questions from this chapter

Immersed in Water A double-slit arrangement produces interference fringes for sodium light \((\lambda=589 \mathrm{~nm})\) that are \(0.20^{\circ}\) apart. What is the angular fringe separation if the entire arrangement is immersed in water \((\bar{n}=1.33)\) ?

Soap Film Light of wavelength \(624 \mathrm{~nm}\) is incident perpendicularly on a soap film (with \(n=1.33\) ) suspended in air. What are the least two thicknesses of the film for which the reflections from the film undergo fully constructive interference?

Two Waves In Fig. \(36-3\), assume that two waves of light in air, of wave length \(400 \mathrm{~nm}\), are initially in phase. One travels through a glass layer of index of refraction \(n_{1}=\) \(1.60\) and thickness \(L\). The other travels through an equally thick plastic layer of index of refraction \(n_{2}=1.50 .\) (a) What is the least value \(L\) should have if the waves are to end up with a phase difference of \(5.65 \mathrm{rad} ?\) (b) If the waves arrive at some common point after emerging, what type of interference do they undergo?

Two Glass Plates Two glass plates are held together at one end to form a wedge of air that acts as a thin film. A broad beam of light of wavelength \(480 \mathrm{~nm}\) is directed through the plates, perpendicular to the first plate. An observer intercepting light reflected from the plates sees on the plates an interference pattern that is due to the wedge of air. How much thicker is the wedge at the sixteenth bright fringe than it is at the sixth bright fringe, counting from where the plates touch?

Newtons's Rings One In a Newton's rings experiment (see Problem 49), the radius of curvature \(R\) of the lens is \(5.0 \mathrm{~m}\) and the lens diameter is \(20 \mathrm{~mm}\). (a) How many bright rings are produced? Assume that \(\lambda=589 \mathrm{~nm}\). (b) How many bright rings would be produced if the arrangement were immersed in water \((n=1.33)\) ?
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