A certain metallic surface is illuminated by monochromatic radiation of wavelength \(\lambda\). The stopping potential for photoelectric current for this radiation is \(3 \mathrm{~V}_{0}\). If the same surface is illuminated with a radiation of wavelength \(2 \lambda\), the stopping potential is \(\mathrm{V}_{0}\). The threshold wavelength of this surface for photoelectric effect is ____ \(\lambda\).

JEE Main · 2021 · 20 Jul 2021 · Shift 2 · Modern Physics

A certain metallic surface is illuminated by monochromatic radiation of wavelength

λ

. The stopping potential for photoelectric current for this radiation is

3 V_{0}

. If the same surface is illuminated with a radiation of wavelength

2 λ

, the stopping potential is

V_{0}

. The threshold wavelength of this surface for photoelectric effect is ____

λ

Answer: 4

Einstein's photoelectric equation gives

e V_{s} = \frac{h c}{λ} - ϕ

. For the two illuminations,

3 e V_{0} = \frac{h c}{λ} - ϕ, e V_{0} = \frac{h c}{2 λ} - ϕ

Subtracting the second from the first,

2 e V_{0} = \frac{h c}{λ} - \frac{h c}{2 λ} = \frac{h c}{2 λ} \Rightarrow e V_{0} = \frac{h c}{4 λ}

Substituting this in the second equation,

ϕ = \frac{h c}{2 λ} - \frac{h c}{4 λ} = \frac{h c}{4 λ}

. The threshold wavelength satisfies

ϕ = \frac{h c}{λ _{0}}

, so the threshold wavelength is

4 λ

JEE Main · 2021 · 20 Jul 2021 · Shift 2 · Integer answer