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 $\varphi$ 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 ${\lambda }_{\theta }$, which of the following statement(s) is (are) true?

• A. For large potential difference $(V>>\varphi /e){\lambda }_e$ is approximately halved if $V$ is made four times
• B. ${\lambda }_e$ increases at the same rate as ${\lambda }_{ph}$ for ${\lambda }_{ph}<hc/\varphi$
• C. ${\lambda }_e$ is approximately halved, if $d$ is doubled
• D. ${\lambda }_e$ decreases with increase in $\varphi$ and ${\lambda }_{ph}$

Answer 1

The correct option is A For large potential difference $(V>>\varphi /e){\lambda }_e$ is approximately halved if $V$ is made four times

light of wavelength ${\lambda }_{ph}$ falls on a cathode plate are true?

$K{E}_{max}$ of $e^{-}$ just after ejection

$K{E}_1=\frac{h_c}{{\lambda }_{ph}}-\varphi$

$K{E}_{max}{e}^{-}$ reaching mode $K{E}_g=[\frac{Ac}{{\lambda }_{ph}}-\varphi ]+eV$

for $V\gg \frac{\varphi }{e}\Rightarrow K{E}_j\approx eV\Rightarrow {\lambda }_e=\frac{h}{\sqrt{2m\left(eV\right)}}\alpha \frac{1}{\sqrt{V}}$

$\Rightarrow \left(A\right)$ is correct

If $\varphi$ & ${\lambda }_{ph}$ increase

$\Rightarrow K{E}_j$ decrease $\Rightarrow {\lambda }_e$ increases

$\left(B\right)$ incorrect

${\lambda }_e=\frac{h}{\sqrt{2m\left(K{E}_j\right)}}\Rightarrow \frac{d{\lambda }_e}{dt}\ne \frac{d{\lambda }_{ph}}{dt}$ $\left(C\right)$ is correct

${\lambda }_e$ is independent of $d$ $\left(D\right)$ in incorrect