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Chapter 4: Problem 18

(a) Calculate the angle at which a \(2.00-\mu \mathrm{m}\) -wide slit produces its first minimum for 410 -nm violet light. (b) Where is the first minimum for 700 -nm red light?

Short Answer

Expert verified
The angle at which the single slit produces its first minimum for 410-nm violet light is \(\theta_v = arcsin(\frac{1 \times (410 \times 10^{-9})}{2.00 \times 10^{-6}})\) radians or \(\frac{\theta_v \times 180}{\pi}\) degrees. For 700-nm red light, the angle is \(\theta_r = arcsin(\frac{1 \times (700 \times 10^{-9})}{2.00 \times 10^{-6}})\) radians or \(\frac{\theta_r \times 180}{\pi}\) degrees. The position of the first minimum for 700-nm red light on a screen placed at distance L from the slit is \(Y_r = L \times tan(\theta_r)\).

Step by step solution

01

Convert units to meters

We are given width of the slit 2.00 μm and the wavelengths of violet (410 nm) and red (700 nm) light. Let's convert these values into meters: Slit width: \(a = 2.00 \times 10^{-6}\) m Wavelength of violet light: \(\lambda_v = 410 \times 10^{-9}\) m Wavelength of red light: \(\lambda_r = 700 \times 10^{-9}\) m Step 2: Calculate angle for violet light
02

Calculate angle for violet light

We will apply the formula to calculate the angle θ for the 410-nm violet light using the following equation: \(sin(\theta) = \frac{m \lambda_v}{a}\) Substituting the values: \(sin(\theta_v) = \frac{1 \times (410 \times 10^{-9})}{2.00 \times 10^{-6}}\) Now, we will calculate the angle θ: \(\theta_v = arcsin(sin(\theta_v))\) Step 3: Calculate angle for red light
03

Calculate angle for red light

Now, we will apply the same formula to calculate the angle θ for the 700-nm red light: \(sin(\theta_r) = \frac{m \lambda_r}{a}\) Substituting the values: \(sin(\theta_r) = \frac{1 \times (700 \times 10^{-9})}{2.00 \times 10^{-6}}\) Now, we calculate the angle θ: \(\theta_r = arcsin(sin(\theta_r))\) Step 4: Convert angles to degrees
04

Convert angles to degrees

Finally, we will convert the angles θ for violet and red light from radians to degrees: \(Angle\,for\,violet\,light: \theta_v\,(in\,degrees) = \frac{\theta_v \times 180}{\pi}\) \(Angle\,for\,red\,light: \theta_r\,(in\,degrees) = \frac{\theta_r \times 180}{\pi}\) Step 5: Calculate the position of the first minimum for red light
05

Calculate the position of the first minimum for red light

Now, we will use the following formula to find the position of the first minimum (dark fringe) for 700-nm red light on a screen placed at distance L from the slit: \(Y_r = L \times tan(\theta_r)\) The distance L between the single slit and the screen has not been given in the question, so we will leave the answer in terms of L.

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Most popular questions from this chapter

As the width of the slit producing a single-slit diffraction pattern is reduced, how will the diffraction pattern produced change?

(a) Sodium vapor light averaging \(589 \mathrm{nm}\) in wavelength falls on a single slit of width \(7.50 \mu \mathrm{m}\). At what angle does it produces its second minimum? (b) What is the highest-order minimum produced?

Calculate the wavelength of light that has its secondorder maximum at \(45.0^{\circ}\) when falling on a diffraction grating that has 5000 lines per centimeter.

Consider the single-slit diffraction pattem for \(\lambda=600 \mathrm{nm}, D=0.025 \mathrm{mm},\) and \(x=2.0 \mathrm{m} .\) Find the intensity in terms of \(I_{o}\) at \(\theta=0.5^{\circ}, 1.0^{\circ}, 1.5^{\circ}, 3.0^{\circ}\) and \(10.0^{\circ}\)

(a) Assume that the maxima are halfway between the minima of a single-slit diffraction pattern. The use the diameter and circumference of the phasor diagram, as described in Intensity in single-Slit Diffraction, to determine the intensities of the third and fourth maxima in terms of the intensity of the central maximum. (b) Do the same calculation, using Equation 4.4.
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