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A small particle of mass m moves in such...

A small particle of mass m moves in such a way that the
potential energy `U = ar^2`, where a is constant and r is the distance of the
particle from the origin. Assuming Bhor model of quantization of angular
momentum and circular orbits, find the radius of nth allowed orbit.

Text Solution

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The force at a distance r is ,`F=-(dU)/(dr)=-2ar`
suppose r be the radius of `n^(th)` orbit. Then the necessary centripetal force bt the above force is provided by the aboce force , thus , `mv^(2)/r=2ar` -------(1)
Further,the quantization of angular momentum gives , `mvr=(nh)/(2pi)` ........(ii)
Solving Eqs .`(i)` and `(ii)` for , `r` we get `r=((n^(2)h^(2))/(8ampi^(2)))^(1//4)`
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