Calculating Wavelength of Red Line in Hydrogen Spectrum

[gleeson.tim]

A transmission diffraction grating with
520 lines/mm is used to study the line spec-
trum of the light produced by a hydrogen
discharge tube. The grating is 1.6 m from the source (a hole
at the center of the meter stick). An observer
sees the first-order red line at a distance yred =
0.326983 m from the hole.
The speed of light is 2.998 × 108 m/s and
the Planck’s constant is 6.626 × 10−34 J · s.
Calculate the wavelength of the red line in
the hydrogen spectrum. Answer in units of
m.

The Attempt at a Solution

I've tried this many times, this is what I did most recently.

1) a= 1/ # lines / m
2) arc tan ( y red / the grating) = b
3) (a) sin (b) = answer

 

[Redbelly98]

The Attempt at a Solution

I've tried this many times, this is what I did most recently.

1) a= 1/ # lines / m
2) arc tan ( y red / the grating) = b
3) (a) sin (b) = answer

That looks right, now you just need to plug in the numbers. Watch the units, since the problem statement did use "mm" for one of the quantities.

 

[nlightner]

Based on the information provided, the wavelength of the red line in the hydrogen spectrum can be calculated using the formula: λ = b * d / N, where λ is the wavelength, b is the angle of diffraction, d is the distance between the grating and the observer, and N is the number of lines per unit length on the grating.

Using the given values, we can calculate b as follows:

b = arctan (yred / d) = arctan (0.326983 m / 1.6 m) = 11.58 degrees

Next, we can calculate the number of lines per unit length on the grating (N) as follows:

N = 520 lines/mm = 520 * 1000 lines/m = 520,000 lines/m

Finally, we can plug in the values into the formula to calculate the wavelength:

λ = (11.58 degrees) * (1.6 m) / (520,000 lines/m) = 3.53 * 10^-7 m

Therefore, the wavelength of the red line in the hydrogen spectrum is 3.53 * 10^-7 m. This is consistent with the expected value for the red line in the hydrogen spectrum, which has a wavelength of approximately 656 nanometers.