Light of wavelength $λ_{p h}$ falls on a cathode plate inside a vacuum tube as shown in the figure. The work function of the cathode surface is $ϕ$ 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 $λ_{e},$ which of the following statements(s) is (are) true?

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For large potential difference $(V ≫ ϕ / e), λ_{e}$ is approximately halved if $V$ is made four times.

$λ_{e}$ is approximately halved, if $d$ is doubled

$λ_{e}$ increases at the same rate as $λ_{p h}$ for $λ_{p h} < h c / ϕ$

$λ_{e}$ decreases with increase in $ϕ$ and $λ_{p h}$

answer is D.

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Detailed Solution

$K_{\max} = \frac{h c}{λ_{p h}} - ϕ$

Kinetic energy of electron reaching the anode will be
$K = \frac{h c}{λ_{p h}} - ϕ + e V$
Now, $λ_{e} = \frac{h}{\sqrt{2 m K}} = \frac{h}{\sqrt{2 m (\frac{^{h v}}{λ_{p h}} - ϕ + e V)}}$
If $e V > > ϕ$ then,
$λ_{e} = \frac{h}{\sqrt{2 m (\frac{h c}{λ_{p h}} + e V)}}$
If, $V_{f} = 4 V_{i}$ then,
${(λ_{e})}_{f} ≃ \frac{{(λ_{e})}_{i}}{2}$