Question

A discharge lamp utilizes collisions between electrons and hydrogen atoms to produce a
set of line spectra. If 90 percent(the other 10 % is wasted as heat) of the electrical
energy results in transitions from the ground state to the 3rd(n=4) state for the
hydrogen. (The Electric Power is to be 200W)

a. What potential is needed between the electrodes of the lamp to accelerate the
electrons to the necessary energy to excite the hydrogen to n=4?
b. What wavelengths of light will be emitted from the bulb?
c. Assuming the linewidth is gaussian and approximately 5 nm (FWHM) wide, what is
the total radiant power for light in the range of 475 to 495 nm. (Approximations are
fine, just state them clearly).

Answer 1

Solution:

(a) As per the Bohr,s atomic model, energy needed to make the transition between ni=4 state to nf=1 state is given by

E = 13.6 [1/nf2 - 1/ni2] eV

Thus, E = 13.6 (1-1/16) eV

Or, E = 13.6(15/16) eV

Or, E = 12.75 eV

Thus, potential needed to make the transition from n=4 state to n=1 state will be E/q, where q is the charge of electron

Or, V = 12.75 eV/q = 12.75 V

Thus, potential required to make the transition from n=4 to n=1 state is 12.75 V

(b) As per De Broglie relation,

E = hc/......................(1) where

E = Energy required for transition = 12.75 eV

h = Plancks constant = 4.135 X 10-15 eV.s

c= Speed of light = 3 X 108 m/s

= wavelength

On substituting the values in equation (1), we have

= hc/E

Or, =  (4.135 X 10-15 eV.s X 3 X 108 m/s) / 12.75 eV

Or, = 0.97 X 10-7 m = 97 nm

Thus, wavelength og emitted light is 97 nm

(c) As per modified Townes equation

FWHM = h (c2)/ Pout,..............(2) where

= frequency of length emitted from bulb = 3 X 1015 Hz

h = Planck constant = 6.626 X 10-34 Js

c = speed of light

c = bandwidth = 495 nm - 475 nm = 20 nm = 15 X 1015 Hz

On substituting the values in equation (2) we have

Pout = h (c2) / FWHM in frequency

Or, Pout = 3.14 X 6.626 X 10-34 Js X 3 X 1015 Hz X (15 X 1015)2 Hz2 / 60 X 1015 Hz

Or, Pout = 0.023 J/s

Thus, the total radiated power for light in the range of 475 nm to 495 nm is 0.023 J/s