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Chapter 34: Problem 62

Coherent monochromatic light passes through parallel slits and then onto a screen that is at a distance \(L=2.40 \mathrm{~m}\) from the slits. The narrow slits are a distance \(d=2.00 \cdot 10^{-5} \mathrm{~m}\) apart. If the minimum spacing between bright spots is \(y=6.00 \mathrm{~cm},\) find the wavelength of the light.

Short Answer

Expert verified
Answer: The wavelength of the light is 10 nm.

Step by step solution

01

Finding the angle θ using the minimum spacing between bright spots, y

We'll use the small angle approximation formula: $$ \theta \approx tan(\theta) = \frac{y}{L} $$ where \(L\) is the distance from the slits to the screen, and \(y\) is the minimum spacing between bright spots. Plugging the given values: $$ \theta = \frac{6.00 \times 10^{-2} \mathrm{~m}}{2.40 \mathrm{~m}} $$ Calculate the value of \(\theta\): $$ \theta = 0.025 $$
02

Finding the order m for the first minimum using the angle θ

For the first minimum (dark spot) in the interference pattern, the path difference between two rays is half a wavelength. Therefore, the corresponding angle will be: $$ \sin(\theta') = \frac{\lambda}{2d} $$ Using the small angle approximation and knowing that \(\theta' = \theta\), $$ \theta' \approx \frac{\lambda}{2d} $$ We can now find the order of the bright spot \(m\) by multiplying both sides of the equation by \(2\): $$ m = 2\theta' = \frac{\lambda}{d} $$ By using the value of \(\theta\) calculated in Step 1, $$ m = 2 \times 0.025 = 0.05 $$
03

Finding the wavelength λ using the constructive interference formula

Now, we can use the constructive interference formula to find the wavelength of the light: $$ \sin(\theta) = \frac{m\lambda}{d} $$ Plugging the calculated values for \(m\), \(d\), and \(\theta\): $$ 0.025 = \frac{0.05\lambda}{2.00 \times 10^{-5} \mathrm{~m}} $$ Solve for \(\lambda\): $$ \lambda = \frac{0.025 \times 2.00 \times 10^{-5} \mathrm{~m}}{0.05} $$ Calculate the value of \(\lambda\): $$ \lambda = 1.00 \times 10^{-5} \mathrm{~m} $$ So, the wavelength of the light is \(1.00 \times 10^{-5} \mathrm{~m}\) or \(10\,\mathrm{nm}\).

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Most popular questions from this chapter

Think of the pupil of your eye as a circular aperture \(5.00 \mathrm{~mm}\) in diameter. Assume you are viewing light of wavelength \(550 \mathrm{nm}\), to which your eyes are maximally sensitive. a) What is the minimum angular separation at which you can distinguish two stars? b) What is the maximum distance at which you can distinguish the two headlights of a car mounted \(1.50 \mathrm{~m}\) apart?

It is common knowledge that the visible light spectrum extends approximately from \(400 \mathrm{nm}\) to \(700 \mathrm{nm}\). Roughly, \(400 \mathrm{nm}\) to \(500 \mathrm{nm}\) corresponds to blue light, \(500 \mathrm{nm}\) to \(550 \mathrm{nm}\) corresponds to green, \(550 \mathrm{nm}\) to \(600 \mathrm{nm}\) to yelloworange, and above \(600 \mathrm{nm}\) to red. In an experiment, red light with a wavelength of \(632.8 \mathrm{nm}\) from a HeNe laser is refracted into a fish tank filled with water with index of refraction 1.33. What is the wavelength of the same laser beam in water, and what color will the laser beam have in water?

What is the wavelength of the X-rays if the first-order Bragg diffraction is observed at \(23.0^{\circ}\) related to the crystal surface, with inter atomic distance of \(0.256 \mathrm{nm} ?\)

What minimum path difference is needed to cause a phase shift by \(\pi / 4\) in light of wavelength \(700 . \mathrm{nm} \)

When using a telescope with an objective of diameter \(12.0 \mathrm{~cm},\) how close can two features on the Moon be and still be resolved? Take the wavelength of light to be \(550 \mathrm{nm}\), near the center of the visible spectrum.
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