For a certain metal, when monochromatic light of wavelength $\lambda$ is incident, the stopping potential for photoelectrons is $3 \mathrm{~V}_{0}$. When the same metal is illuminated by light of wavelength $2 \lambda$, then the stopping potential becomes $\mathrm{V}_{0}$. The threshold wavelength for photoelectric emission for the given metal is $\alpha \lambda$. The value of $\alpha$ is ____.

JEE Main · 2026 · 02 Apr 2026 · Shift 1 · Modern Physics

For a certain metal, when monochromatic light of wavelength

λ

is incident, the stopping potential for photoelectrons is

3 V_{0}

. When the same metal is illuminated by light of wavelength

2 λ

, then the stopping potential becomes

V_{0}

. The threshold wavelength for photoelectric emission for the given metal is

α λ

. The value of

α

is ____.

Answer: (B)

Einstein's photoelectric equation gives

\frac{h c}{λ} = ϕ + 3 e V_{0}

and

\frac{h c}{2 λ} = ϕ + e V_{0}

. Subtracting,

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

, so

\frac{h c}{2 λ} = 2 e V_{0}

and

e V_{0} = \frac{h c}{4 λ}

. Then

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

. The threshold wavelength satisfies

\frac{h c}{λ _{0}} = ϕ = \frac{h c}{4 λ}

, giving

λ_{0} = 4 λ

. The value of

α

4

JEE Main · 2026 · 02 Apr 2026 · Shift 1 · Single correct option