To determine the number of lines per centimeter of a grating based on the given diffraction pattern, we can use the grating equation, which relates the angle of diffraction to the wavelengths of light and the spacing of the grating lines. The equation is expressed as:
Understanding the Grating Equation
The grating equation is given by:
d sin(θ) = mλ
Where:
- d = distance between adjacent grating lines (grating spacing)
- θ = angle of diffraction
- m = order of the diffraction pattern
- λ = wavelength of the light
Given Information
From your question, we have:
- Wavelength 1 (λ₁) = 736 nm
- Wavelength 2 (λ₂) = 589 nm
- Angle of diffraction (θ) = 47 degrees
- Order of diffraction for λ₁ = m
- Order of diffraction for λ₂ = m + 1
Calculating the Grating Spacing
We can first convert the wavelengths from nanometers to meters for consistency:
- λ₁ = 736 nm = 736 x 10-9 m
- λ₂ = 589 nm = 589 x 10-9 m
Now, using the grating equation for both wavelengths:
For λ₁:
d sin(47°) = m(736 x 10-9)
For λ₂:
d sin(47°) = (m + 1)(589 x 10-9)
Setting Up the Equations
Since both equations equal d sin(47°), we can set them equal to each other:
m(736 x 10-9) = (m + 1)(589 x 10-9)
Solving for m
Expanding the equation gives:
736m = 589m + 589
Rearranging terms leads to:
736m - 589m = 589
147m = 589
m = 589 / 147 ≈ 4.0
Finding the Grating Spacing (d)
Now that we have m, we can substitute it back into one of the original equations to find d:
d sin(47°) = 4(736 x 10-9)
Calculating sin(47°) gives approximately 0.7314:
d(0.7314) = 2944 x 10-9
d = 2944 x 10-9 / 0.7314 ≈ 4.03 x 10-6 m
Calculating Lines per Centimeter
The number of lines per centimeter (N) is the reciprocal of the grating spacing (d) in centimeters:
N = 1/d
First, convert d to centimeters:
d = 4.03 x 10-6 m = 4.03 x 10-4 cm
Now, calculate N:
N = 1 / (4.03 x 10-4) ≈ 2475 lines/cm
Final Result
The number of lines per centimeter on the grating is approximately 2475 lines/cm. This calculation illustrates how different wavelengths and their orders can be used to determine the properties of a diffraction grating effectively.
