Question

An electric current through hydrogen gas produces several distinct wavelengths of visible light.

a) What are the wavelengths of the hydrogen spectrum if they form first-order maximum at angles of 24.2, 25.7, 29.1 and 41.0 degrees when projected on a diffraction grating having 10,000 lines per cm?

b) What do the minimum and maximum angles become if a 1000 line-per-cm diffraction grating is used? Sketch result.

c) Discuss relationship between further reductions in lines-per-cm and the new angles for the minimum and maximum wavelengths.

d) Why is this important when designing spectroscopic experiments?

Answer 1

(a) Here, we use the equation of grating

m = d*sin

where m is the order number.

First of all, we find value of d (slit width)

d = 1 / 10000 = 1e-4 cm = 1e-6 m

so, we have

for = 24.2

= 1e-6 * sin 24.2 / 1

= 409.9 nm

for = 25.7

= 1e-6 * sin 25.7 / 1

= 433.7 nm

for = 29.1

= 486.33 nm

for = 41

= 656.05  nm

-------------------------------------------------------------------------------------------------------

(b) If only 1000 lines / cm are used then

d = 1 / 1000 = 1e-3 cm = 1e-5 m

so,

minimum angle (for first order)

= arcsin (1 * 409.9e-9 / 1e-5)

= 2.34 degree

maximum angle

= arcsin (1 * 656.05e-9 / 1e-5)

= 3.76 degree

-----------------------------------------------------------------------------------

(c) As we further reduce the lines per cm, the angle of minimum and maximum wavelength decreases and factor by which the lines per centimeter is reduced is the order of the maximum which will have the same angle.

-----------------------------------------------------------------------------------------------------

(d) spectroscopy is used to measure different wavelengths of light passing through various apertures. Thus it is important to have proper relation between the slit width and the angle at which the light ( maximum and minimums) will fall on the screen.