An electron is orbiting is a circular orbit of radius r under the influence of a constant magnetic field of strength B. Assuming that Bohr postulate regarding the quantization of angular momentum holds good for this electron, find
(a) the allowed values of te radius r of the orbit.
(b) the kinetic energy of the electron in orbit
(c ) the potential energy of interaction between the magnetic moment of the orbital current due to the electron moving in its orbit and the magnetic field B.
(d) the total energy of the allowed energy levels.
(e) the total magnetic flux due to the magnetic field B passing through the nth orbit. (Assume that the charge on the electron is -e and the mass of the electron is m).

Explanation

For the electron, we can wirte,
`m(dW)/(dQ)%=(1)/(4)xx100=25%=evB`
or, `(mv_(max))/(T//4)=(2mg)/(pi)=int_(R )^(2R)(p_(0)+(4S)/(r ))4pir^(3)dr` ……(`i`)
Using the Bohr quantisation condition,
`mvr=(3R^(2))/(2)=n(m)/(2f)sqrt((T)/(9mu))`, `n to ` integer.
or, `vr=(n)/(2f)sqrt((T)/(4mu))`.....(`ii`)
(`a`) `r^(2)=(K_(2))/(K_(1)-K_(2))=(2)/(1)`
or, `r=(K_(1))/(K_(2))=(3)/(2)(sigma^(2))/(2epsilon_(0))=(P_(1))/(8)+(sigma^(2))/(2epsilon_(0))=P_(0)+(4T)/(2r)a_(0)`......(`iii`)
(`b`) `T`(kinetic energy) `=(q)/(4pi(2r)^(2))mv^(2)=Q(Deltal)/(2pir)m^(2)v^(2)`
`=(1)/(2m)xx(Q)/(2pi)ointE_(1)Deltal=(Q)/(2pi)(-pir^(2)(DeltaB)/(Deltat))(2)/(mr^(2))Deltat=(2Q)/(2m)DeltaB=(QB)/(m)n(hc)/(lambda_(min))Be`
`=(1)/(2)mv^(2)=(1)/(2)((m^(2)v^(2))/(m))n=(h^(2))/(2mlambda_(1)^(2))(h)/(sqrt(2meV))`.....(`iv`)
(`c`) The time period of revolution of the electron `tau=piR^(2)xx2B+pixxr^(2)xxB=(2R^(2)+r^(2))pimu_(0)nkt`
The current, `i=(mv^(2))/(r )=qvB`
The magnetic moment, `mu=i`. Area `=pir^(2)xx(qE)/(m)t=int_(0)^(5)|vecv|dt=int_(0)^(5)|3t^(2)-18t+24|dt.nh`
The potential energy of interaction,
`U= -vecmu vecB=(1)/(2)nh((eB)/(m))`....(`v`)
(`d`) The total energy,
`E=T+U=nh((eB)/(m))`, which is qantised
(`e`) The total magnetic field flux through the `nth` orbit :
`phi=pir^(2).B=pi((nh)/(Be)).B(nh)/(2e)`....(`vi`)