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Chapter 25: Problem 16

A thin film of oil \((n=1.50)\) is spread over a puddle of water \((n=1.33) .\) In a region where the film looks red from directly above $(\lambda=630 \mathrm{nm}),$ what is the minimum possible thickness of the film? (tutorial: thin film).

Short Answer

Expert verified
Answer: The minimum thickness of the oil film is 210 nm.

Step by step solution

01

Understand thin film interference

Thin film interference occurs when light waves reflect off two surfaces separated by a thin film, causing constructive or destructive interference between the reflected waves. In this case, the film is the oil on top of the water. Constructive interference leads to a brighter color, while destructive interference results in a darker color.
02

Find the interference condition for a minimum thickness

We will use the following equation for the constructive interference of the thin film: $$2nt = m\lambda_{air}$$ where \(n\) is the refractive index of the oil, \(t\) is the film thickness, \(m\) is an integer representing the order of constructive interference, and \(\lambda_{air}\) is the wavelength of light in air. In our case, \(n=1.50\), \(\lambda_{air}=630\,\text{nm}\), and we want to find the smallest possible \(t\) for constructive interference, so we will consider the case when \(m=1\).
03

Solve for the minimum thickness of the film

Plug in the given values and solve for \(t\): $$2(1.50)t = 1(630\,\text{nm})$$ $$t = \frac{630\,\text{nm}}{2(1.50)}$$ $$t = 210\,\text{nm}$$ Therefore, the minimum possible thickness of the film for it to appear red when viewed from above is 210 nm.

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Most popular questions from this chapter

A Michelson interferometer is set up using white light. The arms are adjusted so that a bright white spot appears on the screen (constructive interference for all wavelengths). A slab of glass \((n=1.46)\) is inserted into one of the arms. To return to the white spot, the mirror in the other arm is moved $6.73 \mathrm{cm} .$ (a) Is the mirror moved in or out? Explain. (b) What is the thickness of the slab of glass?
A thin layer of an oil \((n=1.60)\) floats on top of water \((n=1.33) .\) One portion of this film appears green \((\lambda=510 \mathrm{nm})\) in reflected light. How thick is this portion of the film? Give the three smallest possibilities.
The radio telescope at Arecibo, Puerto Rico, has a reflecting spherical bowl of \(305 \mathrm{m}(1000 \mathrm{ft})\) diameter. Radio signals can be received and emitted at various frequencies with appropriate antennae at the focal point of the reflecting bowl. At a frequency of \(300 \mathrm{MHz}\), what is the angle between two stars that can barely be resolved? (Tutorial:radio telescope).
In a double-slit interference experiment, the wavelength is \(475 \mathrm{nm}\), the slit separation is \(0.120 \mathrm{mm},\) and the screen is $36.8 \mathrm{cm}$ away from the slits. What is the linear distance between adjacent maxima on the screen? [Hint: Assume the small-angle approximation is justified and then check the validity of your assumption once you know the value of the separation between adjacent maxima.] (tutorial: double slit 1 ).
A soap film has an index of refraction \(n=1.50 .\) The film is viewed in reflected light. (a) At a spot where the film thickness is $910.0 \mathrm{nm},$ which wavelengths are missing in the reflected light? (b) Which wavelengths are strongest in reflected light?
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