8.14 Given below are some famous numbers associated with electromagnetic radiations in different contexts in Physics. State the part of the electromagnetic spectrum to which each belongs.

(a) 21 cm (wavelength emitted by atomic hydrogen in interstellar space).

(b) 1057 MHz (frequency of radiation arising from two close energy levels in hydrogen; known as Lamb shift).

(c) 2.7 K [temperature associated with the isotropic radiation filling all space-thought to be a relic of the ‘big-bang’ origin of the universe].

(d) 5890 Å - 5896 Å [double lines of sodium]

(e) 14.4 keV [energy of a particular transition in Fe57 nucleus associated with a famous high-resolution spectroscopic method (Mössbauer spectroscopy)].

 

(a)
Hint: Recall electromagnetic spectrum.
Step: Find the spectrum corresponding to a wavelength of 21 cm.
Radio waves belong to the short-wavelength end of the electromagnetic spectrum.

(b)
Hint: \(\lambda=\frac{c}{\nu}\)
Step 1:
Find the wavelength of 1057 Mhz
\(\lambda=\frac{c}{\nu}=\frac{3\times10^{8}}{1057\times10^{6}}\times 100 \text{ cm}=28.3 \text{ cm}\)
Step 2: Find the spectrum corresponding to a wavelength of 28.3 cm.
Radio waves belong to the short-wavelength end of the electromagnetic spectrum.

(c)
Hint: Recall Planck's law
Step 1: Find the wavelength corresponding to the temperature of 2.7 °K

Temperature, T= 2.7 °K

According
 to Planck's law:

 λmT=0.29=\(\frac{0.29}{2.7}\)=0.11 cm

Step 2: Find the spectrum corresponding to a wavelength of 0.11 cm.
This wavelength corresponds to microwaves.

(d)
Hint: Recall electromagnetic spectrum.
Step: Find the spectrum corresponding to
5890 Å - 5896 Å [double lines of sodium].
This is the yellow light of the visible spectrum.

(e)
Hint: E=
step: Find the spectrum corresponding to the energy of 14.4 keV.
Transition energy is given by the relation,

E=
where h = Planck's constant = 6.6 ×10-34Js
ν=frequency of radiation
ν=Eh=14.4×103×1.6×10-196.6×10-34
=3.4×1018 Hz
This corresponds to X-rays.