A particle of mass `m` moves along a circular orbit in centrosymmetrical potential field `U(r )=kr^(2)//2`. Using the Bohr quantization condition, find the permissible orbital radii and energy levels to that particle.

Options

A. `(nh)/(2pi) sqrt((K)/(m))`
B. `(2nh)/(pi)sqrt((K)/(m))`
C. `(nh)/(2)sqrt((K)/(m))`
D. None of these

Answer

A

Explanation

`F = (d)/(dr) u(r) = Kr (mv^(2))/(r)`
or `mv = sqrt(Kr^(2)m)` and `mvr = (nh)/(2pi)`
`:. rsqrt(kr^(2)m) = (nh)/(2pi)` or `r = ((nh)/(2pisqrt(Km)))^(1/2)`
and `mv^(2) = Kr^(2) = K ((nh)/(sqrt(Km)))`
`= sqrt(mKr) = ((nh)/(2pi//mk))^(1/2)`
`TE = KE +PE`
`= (1)/(2pi)K [(nh)/(sqrt(Km))] +(K)/(2pi) [(nh)/(sqrt(Km))] = (nh)/(2pi) sqrt((K)/(m))`