1. Yellow light of wavelength 5.5 x 10^-7 m is incident normally at a diffraction grating with 300 lines per mm. Calculate the angle between the first-order and second-order maxima.
2. Blue light of wavelength 4.5 x 10^-7 m is incident normally at a diffraction grating with 100 lines per mm. Calculate the maximum number of orders that can be observed.
To calculate the angle between the first-order and second-order maxima for the given information:
1. We can use the formula for the angle of diffraction for a grating:
d * sin(θ) = m * λ
where:
- d is the grating spacing (distance between adjacent lines), given as 1 / 300 lines per mm = 3.33 x 10^-3 mm = 3.33 x 10^-6 m
- θ is the angle between the incident light and the diffracted light
- m is the order of the maximum (in this case, first and second orders)
- λ is the wavelength of the light, given as 5.5 x 10^-7 m
For the first-order maximum (m = 1):
(3.33 x 10^-6) * sin(θ₁) = 1 * 5.5 x 10^-7
For the second-order maximum (m = 2):
(3.33 x 10^-6) * sin(θ₂) = 2 * 5.5 x 10^-7
2. Rearranging both equations to solve for sin(θ₁) and sin(θ₂), we get:
sin(θ₁) = (5.5 x 10^-7) / (3.33 x 10^-6) = 0.165
sin(θ₂) = (2 * 5.5 x 10^-7) / (3.33 x 10^-6) = 0.330
3. Taking the arcsin (inverse sine) of both values, we can find the angles:
θ₁ = arcsin(0.165) ≈ 9.54 degrees
θ₂ = arcsin(0.330) ≈ 19.63 degrees
Therefore, the angle between the first-order and second-order maxima is approximately 19.63 - 9.54 = 10.09 degrees.
Now, let's move to the second question and calculate the maximum number of orders that can be observed for blue light incident on a diffraction grating with 100 lines per mm.
To calculate the maximum number of orders:
1. We can use the formula for the maximum number of orders (m_max):
m_max = n * d / λ
where:
- d is the grating spacing (distance between adjacent lines), given as 1 / 100 lines per mm = 1 x 10^-2 mm = 1 x 10^-5 m
- λ is the wavelength of the light, given as 4.5 x 10^-7 m
- n is the index of refraction (typically assumed to be 1 for air)
2. Substituting the given values into the formula:
m_max = (1 x 10^-5) / (4.5 x 10^-7) = 22.22
Rounding down to the nearest whole number, the maximum number of orders that can be observed for blue light incident on this diffraction grating is 22 orders.
Please note that in practice, higher-order maxima can become weak and less well-separated, so only a few orders may be clearly visible.
To solve both of these questions, we need to use the formula for the angle of diffraction:
sinθ = mλ / d
where:
θ is the angle of diffraction,
m is the order of the maximum,
λ is the wavelength of light, and
d is the spacing between adjacent lines on the diffraction grating.
1. Let's solve the first question:
Given:
Wavelength λ = 5.5 x 10^-7 m
Number of lines per mm = 300
First, convert the number of lines per mm to the spacing between adjacent lines (d) in meters:
d = 1mm / (number of lines per mm) = 1mm / 300 = 3.33 x 10^-6 m
Using the formula, let's calculate the angle between the first-order and second-order maxima:
For the first-order maximum (m = 1):
sinθ1 = (1)(5.5 x 10^-7 m) / (3.33 x 10^-6 m)
θ1 = sin^(-1) [(1)(5.5 x 10^-7 m) / (3.33 x 10^-6 m)]
For the second-order maximum (m = 2):
sinθ2 = (2)(5.5 x 10^-7 m) / (3.33 x 10^-6 m)
θ2 = sin^(-1) [(2)(5.5 x 10^-7 m) / (3.33 x 10^-6 m)]
The angle between the first-order and second-order maxima is:
Angle = θ2 - θ1
2. Now let's solve the second question:
Given:
Wavelength λ = 4.5 x 10^-7 m
Number of lines per mm = 100
Convert the number of lines per mm to the spacing between adjacent lines (d) in meters:
d = 1mm / (number of lines per mm) = 1mm / 100 = 1 x 10^-5 m
Using the formula, let's calculate the maximum number of orders (m) that can be observed:
For the maximum order (m), the sine of the angle of diffraction (sinθ) should be less than or equal to 1.
So we can rearrange the formula to solve for the maximum order (m):
m = d / λ
m = (1 x 10^-5 m) / (4.5 x 10^-7 m)
The maximum number of orders that can be observed is the closest integer less than or equal to the calculated value of m.
Round down the value of m to the nearest whole number to get the answer.
I hope this helps! Let me know if you have any further questions.