A classical model for the hydrogen atom consists of a singal electron of mass `m_(e)` in circular motion of radius r around the nucleous (proton). Since the electron is accelerated, the atom continuously radiates electromagnetic waves. The total power P radiated by the atom is given by `P = P_(0)//r^(4)` where `P_(0) = (epsi^(6))/(96pi^(3)epsi_(0)^(3)c^(3)m_(epsi)^(2))` (c = velocity of light) (i) Find the total energy of the atom. (ii) Calculate an expression for the radius r(t) as a function of time. Assume that at `t = 0`, the radiys is `r_(0) = 10^(-10)m`. (iii) Hence or otherwise find the time `t_(0)` when the atom collapses in a classical model of the hydrogen atom. Take: `[2/(sqrt(3))(e^(2))/(4piepsi_(0))1/(m_(epsi)c^(2)) = r_(e) ~~ 3 xx 10^(-15) m]`

Answer

(i) `- (1)/(8piepsi_(0)) (e^(2))/(r)` (ii) `r_() (1-(3cr_(e)^(2)t)/(r_(0)^(3)))^(1//3)` (iii) `10^(-10) xx (100)/(81) sec`

Explanation

(i) `(mv^(2))/r=1/(4pi in_(0)) e^(2)/r^(2) implies mvr=(h)/(2pi) implies E_("total")=e^(2)/(8pi in_(0) r)`
(ii) `(dE)/(dt)=-P` (loss of energy per sec)
`implies d/(dt) (- e^(2)/(8pi in_(0) r))= - P_(0)/r^(4) implies (e^(2)/(8pi in_(0) r^(2))) (dr)/(dt)= - P_(0)/r^(4)`
`implies e^(2) underset(r_(0))overset(r)(int) r^(2) dr= -8 pi in_(0) P_(0) underset(0)overset(t) (int) dt`
`implies r^(3)=r_(0)^(3) - (6P_(0) (4pi in_(0))t)/e^(2) implies r=r_(0) [1-(3cr_(e)^(2) t)/r_(0)^(3)]^(1//3)`
(iii) For `r=0`, (to collapse and fall into nucleus)
`implies 1- (3cr_(e)^(2) t)/r_(0)^(3)=0`
`implies t=r_(0)^(3)/(3cr_(e)^(2))=10^(-30)/(3xx3xx10^(8)xx9xx10^(-30))=(10^(-10)xx100)/81 sec`