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Ch 35: Interference
Young & Freedman Calc - University Physics 15th Edition
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoYoung & Freedman Calc 15th EditionCh 35: InterferenceProblema 23
Capitolo 34, Problema 23

What is the thinnest film of a coating with n = 1.42 on glass (n = 1.52) for which destructive interference of the red component (650 nm) of an incident white light beam in air can take place by reflection?

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Determine the condition for destructive interference. For thin films, destructive interference occurs when the optical path difference between the reflected waves is equal to an odd multiple of half the wavelength in the film. This can be expressed as: \( 2n t = (m + \frac{1}{2}) \lambda_{film} \), where \( n \) is the refractive index of the film, \( t \) is the thickness of the film, \( m \) is an integer (0, 1, 2,...), and \( \lambda_{film} \) is the wavelength of light in the film.
Calculate the wavelength of light in the film. The wavelength in the film is related to the wavelength in air by \( \lambda_{film} = \frac{\lambda_{air}}{n} \), where \( \lambda_{air} \) is the wavelength of light in air (650 nm in this case) and \( n \) is the refractive index of the film (1.42).
Substitute \( \lambda_{film} \) into the interference condition. Replace \( \lambda_{film} \) in the equation \( 2n t = (m + \frac{1}{2}) \lambda_{film} \) with \( \frac{\lambda_{air}}{n} \). This gives \( 2n t = (m + \frac{1}{2}) \frac{\lambda_{air}}{n} \).
Solve for the thickness \( t \). Rearrange the equation to isolate \( t \): \( t = \frac{(m + \frac{1}{2}) \lambda_{air}}{2n^2} \).
Find the thinnest film by using the smallest value of \( m \), which is \( m = 0 \). Substitute \( m = 0 \), \( \lambda_{air} = 650 \text{ nm} \), and \( n = 1.42 \) into the equation for \( t \) to calculate the minimum thickness of the film.

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Thin Film Interference

Thin film interference occurs when light waves reflect off the boundaries of a thin layer, such as a coating on glass. The reflected waves can interfere constructively or destructively depending on their phase difference, which is influenced by the film's thickness and the refractive indices of the materials involved.
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Wave Interference & Superposition

Refractive Index

The refractive index (n) is a dimensionless number that describes how light propagates through a medium. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium. In this context, the refractive indices of the coating and glass determine how light behaves at their interfaces, affecting interference patterns.
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Index of Refraction

Destructive Interference

Destructive interference occurs when two light waves combine in such a way that their amplitudes cancel each other out, resulting in reduced or no intensity. For thin films, this typically happens when the path difference between reflected waves is equal to an odd multiple of half the wavelength, leading to specific conditions for thickness to achieve this effect.
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Wave Interference & Superposition
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