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Chapter 3: Problem 21

What is the wavelength of light falling on double slits separated by \(2.00 \mu \mathrm{m}\) if the third-order maximum is at an angle of \(60.0^{\circ} ?\)

Short Answer

Expert verified
The wavelength of the light falling on the double slits is approximately \(5.77 \times 10^{-7} \thinspace m\) or \(577 \thinspace nm\).

Step by step solution

01

1: Identify the double-slit interference formula

The double-slit interference formula is given by: \[d \sin{\theta} = m \lambda\] where \(d\) is the distance between the slits, \(\theta\) is the angle for the interference order, \(m\) is the order number, and \(\lambda\) is the wavelength.
02

2: Convert the given values into standard units

We are given the distance between the slits \(d = 2.00 \mu m\), which must be converted to meters: \[d = 2.00 \times 10^{-6} m\] We are also given the third-order maximum angle \(\theta = 60.0^{\circ}\), which must be converted to radians: \[\theta = \frac{(60.0)(\pi)}{180} \thinspace radians\]
03

3: Plug the given values and order number into the formula

We use the formula with our given values and \(m = 3\) since it's the third-order maximum: \[2.00 \times 10^{-6} \thinspace m \times \sin{\frac{(60.0)(\pi)}{180}} = 3\lambda\]
04

4: Solve for the wavelength \(\lambda\)

Now, we can solve for the wavelength, \(\lambda\): \[\lambda = \frac{2.00 \times 10^{-6} \thinspace m \times \sin{\frac{(60.0)(\pi)}{180}}}{3}\] \[\lambda = \frac{2.00 \times 10^{-6} \thinspace m \times \frac{\sqrt{3}}{2}}{3}\]
05

5: Calculate the result

Finally, we calculate the wavelength: \[\lambda = \frac{2.00 \times 10^{-6} \thinspace m \times \frac{\sqrt{3}}{2}}{3} \approx 5.77 \times 10^{-7} m\] So, the wavelength of the light falling on the double slits is approximately \(5.77 \times 10^{-7} \thinspace m\) or \(577 \thinspace nm\).

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Most popular questions from this chapter

(a) As a soap bubble thins it becomes dark, because the path length difference becomes small compared with the wavelength of light and there is a phase shift at the top surface. If it becomes dark when the path length difference is less than one-fourth the wavelength, what is the thickest the bubble can be and appear dark at all visible wavelengths? Assume the same index of refraction as water. (b) Discuss the fragility of the film considering the thickness found.

A hydrogen gas discharge lamp emits visible light at four wavelengths, \(\lambda=410,434,486,\) and \(656 \mathrm{nm}\) (a) If light from this lamp falls on a \(N\) slits separated by \(0.025 \mathrm{mm},\) how far from the central maximum are the third maxima when viewed on a screen \(2.0 \mathrm{m}\) from the slits? (b) By what distance are the second and third maxima separated for \(l=486 \mathrm{nm}\) ?

How is the difference in paths taken by two originally in-phase light waves related to whether they interfere constructively or destructively? How can this be affected by reflection? By refraction?

Into one arm of a Michelson interferometer, a plastic sheet of thickness \(75 \mu \mathrm{m}\) is inserted, which causes a shift in the interference pattern by 86 fringes. The light source has wavelength of \(610 \mathrm{nm}\) in air. What is the index of refraction of this plastic?

What is the highest-order maximum for 400-nm light falling on double slits separated by \(25.0 \mu \mathrm{m}\) ?
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