JEE Advanced 2023 Paper 1 · Physics · Atoms, Nuclei and Dual Nature of Radiation · hard

A sample of He^+ ions, all in the ground state, is bombarded with electrons of kinetic energy 51.5 eV. Assume that an electron can transfer any part…

Numerical answer

A sample of

\mathrm{He^+}

ions, all in the ground state, is bombarded with electrons of kinetic energy

51.5\ \mathrm{eV}

. Assume that an electron can transfer any part of its energy to an ion in an inelastic collision. Every ion that is excited then de-excites by emitting photons, and all possible spectral lines (between all pairs of levels reachable) are observed. Take the ground-state energy of hydrogen as

-13.6\ \mathrm{eV}

. The photons from these lines fall on a metal of work function

8.0\ \mathrm{eV}

. The number of distinct spectral lines whose photons can eject photoelectrons from this metal is

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Answer: 4

For

\mathrm{He^+}

(

Z=2

) the levels are

E_n=-54.4/n^2\ \mathrm{eV}

. The excitation energies from

n=1

are

40.8

(

n=2

48.4

(

n=3

51.0

(

n=4

) and

52.2\ \mathrm{eV}

(

n=5

). Electrons of

51.5\ \mathrm{eV}

can therefore excite ions up to

n=4

but not

n=5

. The lines among the levels

n=1

4

are six in number, with energies

40.8

(

2\to1

48.4

(

3\to1

51.0

(

4\to1

7.56

(

3\to2

10.2

(

4\to2

) and

2.64\ \mathrm{eV}

(

4\to3

). Photoemission requires photon energy greater than the work function

8.0\ \mathrm{eV}

. This is satisfied by

40.8

48.4

51.0

and

10.2\ \mathrm{eV}

but not by

7.56

2.64\ \mathrm{eV}

. So

4

lines can eject photoelectrons.

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