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  • A particle moves in a closed orbit around the origin, due to a force which is directed towards the origin. The de Broglie wavelength of the particle varies cyclically between two values <img

A particle moves in a closed orbit around the origin, due to a force which is directed towards the origin. The de
Broglie wavelength of the particle varies cyclically between two values \lambda _1 and \lambda _2  with \lambda _1>\lambda _2. Which of the
following statements is true?

Option: 1

The particle could be moving in a circular orbit with origin as centre
 


Option: 2

The particle could be moving in an elliptic orbit with origin as its focus
 


Option: 3

 When the de Broglie wavelength is \lambda _1 , the velocity of particle is more than when its value is \lambda _2
 


Option: 4

Both (1) and (3)


Answers (1)

best_answer

The de Broglie wavelength of the particle can be varying cyclically between two values \lambda_1 and \lambda_2, if particle is moving in an elliptical orbit with origin as its one focus.

Let \mathrm{v}_1,\mathrm{v}_2 be the speed of particle at A and B respectively and origin is at focus O. If \lambda_1, \lambda_2 are the de Broglie wavelengths associated with particle while moving at A and B respectively.

Then
\begin{aligned} & \lambda_1=\frac{\mathrm{h}}{\mathrm{mv}_1} \text { and } \lambda_2=\frac{\mathrm{h}}{\mathrm{mv}_2} \\ & \therefore \frac{\lambda_1}{\lambda_2}=\frac{\mathrm{v}_2}{\mathrm{v}_1} \\ & \text { Since } \lambda_1>\lambda_2 \\ & \therefore \quad \mathrm{v}_2>\mathrm{v}_1 \end{aligned}
Thus, option (2) is true.

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