Light of wavelength falls on a cathode plate inside a vacuum tube as shown in the figure. The work function of the cathode surface is and the anode is a wire mesh of conducting material kept at a distance from the cathode. [2016]

Question

Light of wavelength $λ_{ph}$ 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 statement(s) is (are) true [2016]

Smart Achievers · Accepted Answer

(3)

de-Broglie wavelength passing through the anode,

$λ_{e} = \frac{h}{p} = \frac{h}{\sqrt{2 m (K . E .)}}$

When $ϕ$ increases, K.E. decreases and therefore $λ_{e}$ increases.

When $λ_{p h}$ increases, $N_{p h}$ decreases, K.E. decreases and therefore $λ_{e}$ increases.

$λ_{e}$ is independent of the distance $d$ .

Also after reaching anode, $\frac{h c}{λ_{p h}} + e V - ϕ = \frac{h^{2}}{2 m λ_{e}^{2}}$

$[λ_{e} = \frac{h}{\sqrt{2 m (K . E .)}}]$ $∴$ $\frac{h c}{e λ_{p h}} + V - \frac{ϕ}{e} = \frac{h^{2}}{2 m e λ_{e}^{2}}$ $\dots (i)$

For $V ≫ \frac{ϕ}{e}, ϕ ≪ e V$

Also $\frac{h c}{e λ_{p h}} ≪ V$ $∴$ $from eq. (i), λ_{e} \propto \frac{1}{\sqrt{V}}$

Hence, if $V$ is made four times, $λ_{e}$ is approximately half.

Solution

1 $λ_{e}$ decreases with increase in $ϕ$ and $λ_{ph}$

2 $λ_{e}$ is approximately halved, if $d$ is doubled

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

4 $λ_{e}$ increases at the same rate as $λ_{ph}$ for $λ_{ph} < \frac{h c}{ϕ}$

Want Us to Email you About Special Offers & Updates?

Solution

1 λe decreases with increase in ϕ and λph

2 λe is approximately halved, if d is doubled

3 For large potential difference (V≫ϕ/e), λe is approximately halved if V is made four times

4 λe increases at the same rate as λph for λph<hcϕ

Want Us to Email you About Special Offers & Updates?

1 $λ_{e}$ decreases with increase in $ϕ$ and $λ_{ph}$

2 $λ_{e}$ is approximately halved, if $d$ is doubled

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

4 $λ_{e}$ increases at the same rate as $λ_{ph}$ for $λ_{ph} < \frac{h c}{ϕ}$