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Chapter 35: Problem 28

A researcher measures the thickness of a layer of benzene \((n=1.50)\) floating on water by shining monochromatic light onto the film and varying the wavelength of the light. She finds that light of wavelength \(575 \mathrm{nm}\) is reflected most strongly from the film. What does she calculate for the minimum thickness of the film?

Short Answer

Expert verified
The minimum thickness of the benzene layer that results in maximum constructive interference for light with a wavelength of 575 nm is approximately 192 nm.

Step by step solution

01

Understand What We're Looking For

We want to find the minimum thickness of the benzene layer that would result in the maximum constructive interference for light with a given wavelength. This happens when the path difference between the two beams is an integer multiple of the wavelength.
02

Applying the Constructive Interference Formula

The formula for constructive interference in thin films is \(2nt = m\lambda\), where \(n\) is the refractive index of the medium, \(t\) is the thickness of the film, \(m\) is the order of interference (integer), and \(\lambda\) is the wavelength of light. Since we're looking for the minimum thickness, we want to find the thickness for the first order of interference, which is when \(m = 1\). Plugging in the given values, we get \(2 \cdot 1.50 \cdot t = 1 \cdot 575 \cdot 10^{-9}\) meters.
03

Solving for the Thickness \(t\)

Rearranging the equation above, we get \(t = \frac{575 \cdot 10^{-9}}{2 \cdot 1.50}\) meters. Doing the math, we find that \(t \approx 191.67 \cdot 10^{-9}\) meters, or approximately \(192\) nanometers.

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Key Concepts

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

Constructive Interference
When you see colorful patterns on the surface of a thin film, like oil on water, you're observing constructive interference. This phenomenon occurs when light waves reflected from the top and bottom surfaces of the film interact in such a way that they strengthen each other.

For constructive interference to happen, the waves need to be in phase. This means that their crests and troughs line up perfectly. The mathematical condition for constructive interference in a thin film is given by the equation:
  • \(2nt = m\lambda\)
Here, \(n\) is the refractive index of the film, \(t\) is the thickness of the film, \(m\) is an integer representing the order of interference, and \(\lambda\) is the wavelength of light. In this case, the light reflects most strongly because the waves combine constructively, amplifying the intensity of the reflected light.

Constructive interference explains why we see vivid colors in thin films, as different thicknesses enhance or diminish certain wavelengths of light.
Refractive Index
The refractive index, often symbolized by \(n\), characterizes how much a material slows down light as it passes through. It is a critical factor in determining how light behaves in materials like the benzene layer described in the exercise.

Light travels slower in benzene than in air because the refractive index of benzene is higher. In the equation for constructive interference, \(2nt = m\lambda\), the factor \(n\) modifies the effective optical path length of the light within the benzene film.
  • A higher refractive index means light has a longer optical path.
  • A refractive index of 1.50 for benzene indicates it bends light more than air (with \(n\approx 1\)).
Understanding the refractive index helps explain how light waves change speed and direction in different substances, affecting interference patterns and the thickness needed for certain wavelengths of light to interfere constructively.
Monochromatic Light
Monochromatic light consists of waves of a single wavelength. This purity simplifies calculations in interference problems because we only consider one wavelength.

In the problem you're exploring, light with a wavelength of \(575 \text{ nm}\) is used. This makes the calculations simpler because we don't need to consider the effects of multiple wavelengths, which would create complex color patterns due to different wavelengths interfering at different thicknesses.
  • Monochromatic light is often a specific color, like red or green.
  • It is used in experiments to precisely measure thicknesses and other optical properties.
By using monochromatic light, we can precisely calculate the thickness of films necessary for constructive interference. This is especially useful in scientific research and technological applications like sensors and coatings.

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Most popular questions from this chapter

Jan first uses a Michelson interferometer with the \(606 \mathrm{nm}\) light from a krypton-86 lamp. He displaces the movable mirror away from him, counting 818 fringes moving across a line in his field of view. Then Linda replaces the krypton lamp with filtered \(502 \mathrm{nm}\) light from a helium lamp and displaces the movable mirror toward her. She also counts 818 fringes, but they move across the line in her field of view opposite to the direction they moved for Jan. Assume that both Jan and Linda counted to 818 correctly. (a) What distance did each person move the mirror? (b) What is the resultant displacement of the mirror?

Consider two antennas separated by \(9.00 \mathrm{~m}\) that radiate in phase at \(120 \mathrm{MHz},\) as described in Exercise \(35.1 .\) A receiver placed \(150 \mathrm{~m}\) from both antennas measures an intensity \(I_{0} .\) The receiver is moved so that it is \(1.8 \mathrm{~m}\) closer to one antenna than to the other. (a) What is the phase difference \(\phi\) between the two radio waves produced by this path difference? (b) In terms of \(I_{0},\) what is the intensity measured by the receiver at its new position?

Figure \(\mathbf{P} 35.56\) shows an interferometer known as Fresnel's biprism. The magnitude of the prism angle \(A\) is extremely small. (a) If \(S_{0}\) is a very narrow source slit, show that the separation of the two virtual coherent sources \(S_{1}\) and \(S_{2}\) is given by \(d=2 a A(n-1)\), where \(n\) is the index of refraction of the material of the prism. (b) Calculate the spacing of the fringes of green light with wavelength \(500 \mathrm{nm}\) on a screen \(2.00 \mathrm{~m}\) from the biprism. Take \(a=0.200 \mathrm{~m}\) \(A=3.50 \mathrm{mrad},\) and \(n=1.50\)

The professor returns the apparatus to the original setting. She then adjusts the speakers again. All of the students who had heard nothing originally now hear a loud tone, while you and the others who had originally heard the loud tone hear nothing. What did the professor do? (a) She turned off the oscillator. (b) She turned down the volume of the speakers. (c) She changed the phase relationship of the speakers. (d) She disconnected one speaker.

After a laser beam passes through two thin parallel slits, the first completely dark fringes occur at \(\pm 19.0^{\circ}\) with the original direction of the beam, as viewed on a screen far from the slits. (a) What is the ratio of the distance between the slits to the wavelength of the light illuminating the slits? (b) What is the smallest angle, relative to the original direction of the laser beam, at which the intensity of the light is \(\frac{1}{10}\) the maximum intensity on the screen?
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