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Q.

A particle of mass m moves in circular orbits with potential energy V(r)=𝐹𝑟, where F is a positive constant and r is its distance from the origin. Its energies are calculated using the Bohr model. If the radius of the particle’s orbit is denoted by R and its speed and energy are denoted by v and E, respectively, then for the nth orbit (here h is the Planck’s constant)

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a

R∝n1/3 and v∝n2/3

b

R∝n2/3 and v∝n1/3

c

E=32n2h2F24π2m1/3

d

E=2n2h2F24π2m1/3

answer is B.

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Detailed Solution

U = Fr[Using U = Potential energy and v = velocity]⇒ Force =−dUdr=−F⇒ Magnitude of force = Constant =F⇒F=mv2R ....(1)⇒mvR=nh2π ...(2)⇒F=mR×n2h24π2×1m2R2⇒R=n2h24π2mF1/3 ……(3)⇒v=nh2πmR⇒v=nh2πm4π2mFn2h21/3⇒v=n1/3h1/3F1/321/3π1/3m2/3 (B) is correct ⇒E=12mv2+U =12mv2+FR⇒E=12mn2/3h2/3F2/322/3π2/3m4/3+F×n2h24π2mF1/3⇒E=n2h2F24π2m1/312+1 =32n2h2F24π2m1/3

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A particle of mass m moves in circular orbits with potential energy V(r)=, where F is a positive constant and r is its distance from the origin. Its energies are calculated using the Bohr model. If the radius of the particle’s orbit is denoted by R and its speed and energy are denoted by v and E, respectively, then for the nth orbit (here h is the Planck’s constant)