•• The result (11.3) shows that a classical atom would collapse very rapidly. To get a ver...

•• The result (11.3) shows that a classical atom would collapse very rapidly. To get a very rough estimate of the time for a hydrogen atom to collapse completely, do the following: (a) You can find the rate at which the radius r shrinks,

with dr/dE determined from (5.10) and dE/dt = −P. Find dr/dt when r = aB. (b) Making the admittedly crude approximation that dr/dt remains constant, estimate roughly how long the electron takes to spiral in from r = aB to r = 0. (For a more realistic estimate, see Problem 1.)

Problem 1

•• The formula (11.1) for the power P radiated by an accelerating charge q can be derived by the method of dimensional analysis. Since P would be expected to involve k, q, a, and c, we might reasonably guess that it should have the form

where b is some dimensionless number of order 1 (perhaps something like 4π) and where l, m, n, and, p are unknown powers (and k is the Coulomb force constant ≈ 9 × 109 N · m2 · C−2). By inserting their units into the five dimensional quantities in (11.48), you will get an equation that determines the unknown powers l, m, n, and p. Show that you obtain the correct form (11.1), except that the dimensionless number b cannot be determined.