The ratio of wavelengths of proton and deuteron accelerated by potential $\mathrm{V}_{\mathrm{p}}$ and $\mathrm{V}_{\mathrm{d}}$ is $1: \sqrt{2}$. Then, the ratio of $\mathrm{V}_{\mathrm{p}}$ to $\mathrm{V}_{\mathrm{d}}$ will be

JEE Main · 2022 · 25 Jul 2022 · Shift 2 · Modern Physics

The ratio of wavelengths of proton and deuteron accelerated by potential

V_{p}

and

V_{d}

1 : 2

. Then, the ratio of

V_{p}

V_{d}

will be

(A) $1 : 1$
(B) $2 : 1$
(C) $2 : 1$
(D) $4 : 1$ ✓ Correct

Answer: (D)

Solution

The de Broglie wavelength of a particle accelerated through potential V is:

λ = \frac{h}{2 m q V}

For proton and deuteron (with

m_{d} = 2 m_{p}

and same charge):

\frac{λ _{p}}{λ _{d}} = \frac{m _{d} V _{d}}{m _{p} V _{p}} = \frac{2 m _{p} V _{d}}{m _{p} V _{p}} = \frac{2 V _{d}}{V _{p}}

Given

\frac{λ _{p}}{λ _{d}} = \frac{1}{2}

\frac{1}{2} = \frac{2 V _{d}}{V _{p}} ⟹ \frac{1}{2} = \frac{2 V _{d}}{V _{p}} ⟹ \frac{V _{p}}{V _{d}} = 4

JEE Main · 2022 · 25 Jul 2022 · Shift 2 · Single correct option

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