Question 391: Dual Nature of Matter and Radiation

Question

Light of wavelength $\lambda$
ph
falls on a cathode plate inside a vacuum tube as shown in the figure. The work function of the cathode surface is $\phi$ and the anode is a wire mesh of conducting material kept at a distance d from the cathode. A potential difference V is maintained between the electrodes. If the minimum de-Broglie wavelength of the electrons passing through the anode is $\lambda$
e
, which of the following statement(s) is (are) true?
![](https://app-content.cdn.examgoal.net/fly/@width/image/1l0pzi04o/da33fc1f-d65e-46bb-901f-2a71dadf6257/0a60da80-a330-11ec-9278-e7cbe8cb9053/file-1l0pzi04p.png?format=png)

Accepted Answer

In photo-electric effect, the maximum kinetic energy of the photo-electron ejected at the cathode, is given by
${K_{\max ,\,c}} = {{hc} \over {{\lambda _{ph}}}} - \phi $,
where $\lambda$
ph
is the wavelength of the incident light and $\phi$ is the work function of the material. The ejected photoelectrons are accelerated from the cathode to the anode by a potential V. Thus, the maximum kinetic energy of the photo-electron at the anode is
${K_{\max ,\,a}} = {K_{\max ,\,c}} + eV = {{hc} \over {{\lambda _{ph}}}} - \phi  + eV$.
Now, minimum de-Broglie wavelength of electrons passing through anode,
${\lambda _e} = {h \over {\sqrt {2m\,{K_{\max ,\,a}}} }} = {h \over {\sqrt {2m\left( {{{hc} \over {{\lambda _{ph}}}} - \phi  + eV} ight)} }}$ ...... (i)
D : For $V >  > \phi e$
${\lambda _e} \approx {h \over {\sqrt {2meV} }} \propto {1 \over {\sqrt V }}$
$	herefore$ If V is made four times, $\lambda$
e
becomes ${1 \over {\sqrt 4 }} = {1 \over 2}$, i.e. halved.
C : When $\phi$ and $\lambda$
ph
increases, $\lambda$
e
also increases.
A : From (i)
${{d{\lambda _e}} \over {dt}} = {{ - h} \over 2}{\left[ {2m\left( {{{hc} \over {{\lambda _{ph}}}} - \phi  + eV} ight)} ight]^{ - 3/2}} 	imes 2m\left( {{{ - hc} \over {\lambda _{ph}^2}}} ight){{d{\lambda _{ph}}} \over {dt}}$
$ \Rightarrow {{d{\lambda _e}} \over {dt}} 
e {{d{\lambda _{ph}}} \over {dt}}$
I : $\lambda$
e
does not depend on d.

Answer

$\lambda$
e
increases at the same rate as $\lambda$
ph
for $\lambda$
ph
< hc/$\phi$

Answer

$\lambda$
e
is approximately halved, if d is doubled

Answer

$\lambda$
e
decreases with increase in $\phi$ and $\lambda$
ph

Answer

For large potential difference (V >> $\phi$/e), $\lambda$
e
is approximately halved if V is made four times

Dual Nature of Matter and Radiation

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