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A particle of mass m moves in a circul...

A particle of mass m moves in a circular orbit in a central potential field
`U(r )=(1)/(2) Kr^(2) .` If Bohr's quantization conditions are applied , radii of possible orbitls and energy levels vary with quantum number n as :

A

`r_(n) prop sqrt(n) E_(n) prop 1/n`

B

`r_(n) prop sqrt(n),E_(n) prop n `

C

`r_(n) prop n^(2),E_(n) prop 1/(n^(2))`

D

`r_(n) prop n, E_(n) prop n `

Text Solution

Verified by Experts

The correct Answer is:
B
`F_(r ) = (-dU)/(dr) = -kr `
For circular motion
`|F_(r )| = kr =(mv^(2))/r " " rArr kr^(2) = mv^(2) " "...(i)`
Bohr.s quantization `rArr mvr = (nh)/(2pi) " "…(ii)`
From (i) and (ii)
`(m^(2)v^(2))/m = kr^(2)`
`rArr 1/m ((nh)/(2pir))^(2)=kr^(2)rArr(n^(2)h^(2))/(4pi^(2)mk)=r^(4)`
`rArr =((h^(2))/(4pi^(2)mk))^(1//4) n^(1//2) rArr r prop sqrt(n)`
From equation (i) `U prop sqrt(n)`
`KE =1/2 mv^(2) , PE = 1/2 kr^(2)`
`E = K + U =1/2 mv^(2) +1/2 kr^(2)=kr^(2) prop n `.
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