JEE Advanced 2024 Paper 2 · Physics · Atoms, Nuclei and Dual Nature of Radiation · medium

Hydrogen-like Li^2+ ions de-excite from the n=3 level to the n=2 level and the emitted photons fall on a metal plate of work function 5.0 eV. The…

Single correct option

Hydrogen-like

\mathrm{Li^{2+}}

ions de-excite from the

n=3

level to the

n=2

level and the emitted photons fall on a metal plate of work function

5.0\ \mathrm{eV}

. The most energetic photoelectrons enter a region of uniform magnetic field

1.0\ \mathrm{mT}

perpendicular to their velocity. The radius of their circular path is closest to (use ground-state energy of hydrogen

13.6\ \mathrm{eV}

m_e=9.11\times10^{-31}\ \mathrm{kg}

e=1.60\times10^{-19}\ \mathrm{C}

)

Show answer and solution

Answer: (A) $1.17\ \mathrm{cm}$

For a hydrogen-like ion

E_n=-13.6\,Z^2/n^2\ \mathrm{eV}

with

Z=3

. The photon energy is

13.6\times9\left(\tfrac14-\tfrac19\right)=17.0\ \mathrm{eV}

. The maximum kinetic energy of photoelectrons is

K=17.0-5.0=12.0\ \mathrm{eV}=1.92\times10^{-18}\ \mathrm{J}

. The momentum is

p=\sqrt{2mK}=\sqrt{2\times9.11\times10^{-31}\times1.92\times10^{-18}}=1.87\times10^{-24}\ \mathrm{kg\,m\,s^{-1}}

. In the magnetic field,

r=\dfrac{p}{eB}=\dfrac{1.87\times10^{-24}}{1.60\times10^{-19}\times10^{-3}}=1.17\times10^{-2}\ \mathrm{m}

. Option (B) uses the photon energy

17\ \mathrm{eV}

as the kinetic energy (forgetting the work function), (C) is half the correct radius (as if

B

were

2\ \mathrm{mT}

), and (D) doubles the radius. The correct option is (A).

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