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Chapter 4: Problem 50

At what angle does a diffraction grating produce a second-onder maximum for light having a first-order maximum at \(20.0^{\circ}\) ?

Short Answer

Expert verified
The angle at which a diffraction grating produces a second-order maximum for light having a first-order maximum at \(20.0^{\circ}\) is approximately \(43.5^{\circ}\).

Step by step solution

01

Write down the diffraction grating equation.

The equation used to describe the diffraction grating maxima is: \[n\lambda = d \sin\theta\] where \(n\) is the order of the maximum, \(\lambda\) is the wavelength of the light, \(d\) is the distance between the slits in the grating, and \(\theta\) is the angle of the maxima measured from the normal to the grating.
02

Determine the first-order maximum and set up the equation.

Since we know the angle for the first-order maximum (n=1) is 20.0 degrees, we can set up the equation as follows: \[1\cdot\lambda = d\sin(20.0^{\circ})\]
03

Introduce the second-order maximum and set up the equation.

Now we want to find the angle for the second-order maximum (n=2). We can set up another equation but keep in mind we don't know the wavelength \(\lambda\) and the distance between slits \(d\). Our equation becomes: \[2\cdot\lambda =d\sin(\theta_{2})\]
04

Eliminate unknown variables.

To eliminate the unknown variables \(\lambda\) and \(d\), we can first solve for \(\lambda\) from the first equation by dividing both sides by \(d\): \[\lambda = \frac{d\sin(20.0^{\circ})}{1}\] Now, we can substitute this expression for \(\lambda\) into the second equation: \[2\cdot\frac{d\sin(20.0^{\circ})}{1} =d\sin(\theta_{2})\]
05

Solve for the angle.

Now, we can simplify the equation by dividing both sides by \(d\) and find the angle of the second-order maximum: \[2\sin(20.0^{\circ})=\sin(\theta_{2})\] To find the angle, we need to take the inverse sine (arcsin) of both sides of the equation: \[\theta_{2}=\arcsin(2\sin(20.0^{\circ}))\] Now, we can calculate the angle: \[\theta_{2}=\arcsin(2\sin(20.0^{\circ}))\approx 43.5^{\circ}\] So, the angle at which a diffraction grating produces a second-order maximum for light having a first-order maximum at 20.0 degrees is approximately 43.5 degrees.

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Most popular questions from this chapter

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.

How many complete orders of the visible spectrum \((400 \mathrm{nm}<\lambda<700 \mathrm{nm})\) can be produced with a diffraction grating that contains 5000 lines per centimeter?

What is the minimum diameter mirror on a telescope that would allow you to see details as small as \(5.00 \mathrm{km}\) on the moon some \(384,000 \mathrm{km}\) away? Assume an average wavelength of \(550 \mathrm{nm}\) for the light received.

Structures on a bird feather act like a reflection grating having 8000 lines per centimeter. What is the angle of the first-order maximum for 600 -nm light?

A single slit of width \(2100 \mathrm{nm}\) is illuminated normally by a wave of wavelength 632.8 nm. Find the phase difference between waves from the top and one third from the bottom of the slit to a point on a screen at a horizontal distance of \(2.0 \mathrm{m}\) and vertical distance of \(10.0 \mathrm{cm}\) from the center.
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