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Question:

Light of wavelength λph 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 λe, which of the following statement(s) is(are) true?

λe decreases with increase in Φ and λph

For large potential difference (V>>Φ/e) λe is approximately halved if V is made four times

λe increases at the same rate as λph for λph<hc/Φ

λe is approximately halved, if d is doubled

Solution:

λe=minimum be broglie wavelength.
hc/λph = hc/λe + Φ + eV ⇒ hc(1/λph - 1/λe) = Φ + eV
This denies option A, B and C.
V>>Φ/e
Energy of electron, E = p²/2m
λe = h/p = h/√(2mE) = h/√(2meV)
This supports option D.