A transmission diffraction grating with 600 lines/mm is used to study the line
spectrum of the light produced by a hydrogen discharge tube with the setup shown above.
The grating is 1.0 m from the source (a hole at the center of the meter stick). An observer sees the first-order red line at a distance yr = 428 mm from the hole.
Calculate the wavelength of the red line in the hydrogen spectrum.
I know the answer is 657. I want to know, if it was a reflection grating, well the answer change ?
To calculate the wavelength of the red line in the hydrogen spectrum, we can use the formula for the diffraction grating:
λ = d * sin(θ)
Where λ represents the wavelength of light, d is the grating spacing (in this case, the reciprocal of the number of lines per unit length), and θ is the angle of diffraction.
In the case of a transmission diffraction grating, the angle of diffraction θ can be approximated by:
sin(θ) ≈ yr / D
Where yr is the distance at which the observer sees the first-order red line (428 mm in this case) and D is the distance from the grating to the source (1.0 m or 1000 mm in this case).
Using the given values, we can calculate the angle of diffraction:
sin(θ) ≈ 428 mm / 1000 mm
sin(θ) ≈ 0.428
Now, let's calculate the wavelength using the formula for the diffraction grating. Since the grating spacing is given as 600 lines/mm, we can express it as:
d = 1 / (600 lines/mm)
d = 1 / (600 x 10^-3 mm)
d ≈ 1.67 x 10^-6 mm
Plugging in the values, we get:
λ = (1.67 x 10^-6 mm) * sin(θ)
λ ≈ 1.67 x 10^-6 mm * 0.428
λ ≈ 7.16 x 10^-7 mm
λ ≈ 716 nm
Therefore, the wavelength of the red line in the hydrogen spectrum is approximately 716 nm.
Now, let's consider if it was a reflection grating instead of a transmission grating. In that case, the formula for the angle of diffraction would be different. The angle of diffraction for a reflection grating is given by:
sin(θ) ≈ λ / (2 * d)
Where λ is the wavelength of light and d is the grating spacing.
Using the given value of the wavelength (716 nm) and the calculated value of the grating spacing (1.67 x 10^-6 mm), we can calculate the angle of diffraction.
sin(θ) ≈ 716 nm / (2 * 1.67 x 10^-6 mm)
sin(θ) ≈ 2.14 x 10^5
However, the sine of an angle cannot be greater than 1. Therefore, the angle of diffraction calculated using this formula exceeds the limit.
Hence, for a reflection grating, the answer would not change, and the calculated value would still be approximately 716 nm.