Diffraction Grating: Solving The Spectrum Width
Hey everyone, let's dive into a fascinating physics problem involving diffraction gratings! This is a super common topic, so understanding it is crucial. We've got a diffraction grating with 100 lines per millimeter, placed 2 meters away from a screen. White light shines on it, ranging from a maximum wavelength of 760 nm to a minimum of 380 nm. Our mission? To figure out the width of the first-order spectrum. Sounds fun, right?
Understanding the Basics of Diffraction Gratings
First off, let's get friendly with the diffraction grating. Imagine it as a series of super tiny, evenly spaced slits. When light waves hit these slits, they bend and interfere with each other. This interference creates a pattern of bright and dark bands on a screen. These bright bands are where the light waves constructively interfere, and their position depends on the wavelength of the light, the spacing of the slits, and the angle at which the light is observed. This is how we see the spectrum! This is because each wavelength of light bends at a slightly different angle, which is why we see the colors of the rainbow. Think of a prism, but a grating does it even better and is easier to measure.
The key equation that describes the behavior of a diffraction grating is:
d * sin(θ) = m * λ
Where:
- d is the distance between the slits (also known as the grating spacing).
- θ is the angle of diffraction.
- m is the order of the spectrum (0, 1, 2, etc.). The first order is where we're focusing on in our problem.
- λ is the wavelength of the light.
Now, let's break down each element. The grating spacing, d, is the distance between adjacent slits on the grating. For example, if a grating has 100 lines per millimeter, then d = 1mm / 100 lines = 1/100 mm = 1 x 10^-5 m. The order of the spectrum, m, indicates which band you are looking at. The zero-order (m=0) is the central bright band, where all the wavelengths of light overlap. The first order (m=1) is the first band on either side of the central band, second order (m=2) is the second band, etc. Each order corresponds to a different angle, and the angle depends on the wavelength of the light. The wavelength, λ, is the measure of the light's color, like the 760 nm and 380 nm mentioned in the question.
The most important thing about a diffraction grating is its ability to separate colors based on their wavelengths. This is why the white light entering the diffraction grating separates into its constituent colors, producing a spectrum.
Calculating Grating Spacing
Before we can solve this problem, we need to find the grating spacing, d. The question states that the grating has 100 lines per millimeter. This means that in one millimeter, there are 100 slits. So, the distance between each slit is:
d = 1 mm / 100 lines = 0.01 mm
To keep our units consistent, let's convert this to meters:
d = 0.01 mm * (1 m / 1000 mm) = 1 x 10^-5 m
Got it? This is a crucial step to solve diffraction grating problems. Remember, always start by determining the grating spacing based on the given lines per unit length!
Finding the Angles of Diffraction
Now, let's use the diffraction grating equation: d * sin(θ) = m * λ. We want to find the angle θ for both the maximum and minimum wavelengths. We'll be using m = 1 (first-order spectrum).
For the maximum wavelength (λ = 760 nm = 760 x 10^-9 m):
1 x 10^-5 m * sin(θ_max) = 1 * 760 x 10^-9 m
sin(θ_max) = (760 x 10^-9 m) / (1 x 10^-5 m) = 0.076
θ_max = arcsin(0.076) ≈ 4.36 degrees
For the minimum wavelength (λ = 380 nm = 380 x 10^-9 m):
1 x 10^-5 m * sin(θ_min) = 1 * 380 x 10^-9 m
sin(θ_min) = (380 x 10^-9 m) / (1 x 10^-5 m) = 0.038
θ_min = arcsin(0.038) ≈ 2.18 degrees
So, the maximum wavelength is diffracted at an angle of roughly 4.36 degrees, and the minimum wavelength is diffracted at approximately 2.18 degrees. We're getting closer to solving the problem, aren't we?
Determining the Width of the Spectrum
Alright, we now know the angles at which the extreme wavelengths of the spectrum are diffracted. The width of the spectrum on the screen is directly related to the angular separation between these wavelengths and the distance to the screen. To find the width, we'll need to use trigonometry and the fact that the screen is 2 meters away from the grating. Since our angles are small, we can make the small-angle approximation: tan(θ) ≈ θ (in radians).
First, we need to convert our angles from degrees to radians:
θ_max (in radians) = 4.36 degrees * (π / 180) ≈ 0.076 rad
θ_min (in radians) = 2.18 degrees * (π / 180) ≈ 0.038 rad
Now, let's determine the positions of the maximum and minimum wavelengths on the screen using the tangent function. We know that tan(θ) = (opposite side) / (adjacent side), and the adjacent side is the distance to the screen (2 m). The opposite side is the distance from the central maximum to the position of the wavelength on the screen.
Position of the maximum wavelength on the screen: x_max = 2 m * tan(θ_max) ≈ 2 m * 0.076 ≈ 0.152 m
Position of the minimum wavelength on the screen: x_min = 2 m * tan(θ_min) ≈ 2 m * 0.038 ≈ 0.076 m
Finally, to find the width of the spectrum, we subtract the position of the minimum wavelength from the position of the maximum wavelength:
Width = x_max - x_min = 0.152 m - 0.076 m = 0.076 m
Therefore, the width of the first-order spectrum on the screen is approximately 0.076 meters, or 7.6 centimeters. Isn't that cool?
Conclusion: We Did It!
Congratulations, guys! We've successfully solved this diffraction grating problem. We've gone through the process step-by-step, from understanding the basics to calculating the width of the spectrum. We learned the importance of grating spacing and how it affects the separation of light into its colors. We also used the diffraction grating equation and basic trigonometry to find the angles and positions on the screen. Mastering these concepts provides a strong foundation for understanding wave phenomena and optics. Keep practicing, and you'll become a diffraction grating expert in no time! Keep exploring the wonderful world of physics, and never stop being curious!