KVPY 2020-21 SB SX Exam Previous Papers

Question : 33 of 120

Marks: +1, -0

Instead of angular momentum quantization a sudent posits that energy is quantized as

{E}=-{E}_{0} \/ {n}({E}_{0}>0)

and

{n}

is a positive integer.
Which of the following options is correct?

Solution:

{F}_{{c}}={{mv}^{2}}/{{r}} \⇒ {{mv}^{2}}/{{r}}={{k}({Ze})({e})}/{{r}^{2}}

{1}/{2} {mv}^{2}={{KZe}^{2}}/{2 {r}} \...... .

(i) (Kinetic energy)
Potential energy

={{Kq}_{1} {q}_{2}}/{{r}}={{K}({Ze})(-{e})}/{{r}} \

.......(ii)
Total energy

={KE}+{PE}=-{{KZe}^{2}}/{2 {r}}=-{{E}_{0}}/{{n}}

\∴{r} \∝{n}

As kinetic energy

={{KZe}^{2}}/{2 {r}} \⇒{KE} \∝{1}/{{n}}

{v}^{2} \∝{1}/{{n}} \⇒{v}^{2} \∝{1}/{\√{{n}}}

{L}={mvr}

{L} \∝{vr}

{L} \∝{1}/{\√{{n}}}({n}) \∝\√{{n}}

\⇒{L} \∝\√{{n}}

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