Classical lifetime of hydrogen

Warning: math alert

This problem was inspired by a question in the Astronomy & Cosmology section which asks if classical physics will eventually replace quantum physics.

Picture the hydrogen atom as an electron orbiting a proton. As the electron moves, it will radiate away its energy. The *power* radiated away from a moving charge (for v << c) is given by the *Larmor formula* which is

P = mu<sub>0</sub>q<sup>2</sup>a<sup>2</sup>/(6*pi*c) = dE/dt,

where q = charge,

a = acceleration (centripetal in this case),

c = speed of light,

mu<sub>0</sub> is the permeability of free space, and

E is energy (all in SI units).

For your convenience;

q = e ~ 1.6 * 10<sup>-19</sup> C (C = Coulombs)

c ~ 3 * 10<sup>8</sup> m/s

mu<sub>0</sub> ~ 1.3 * 10<sup>-6</sup> kg m/C<sup>2</sup>

From this, one can calculate the time it would take for the electron to spiral into the proton as it loses its energy, that is, the "*classical lifetime*" of hydrogen.

To solve this, assume all the electron's energy is in the form of kinetic energy, the acceleration (a) is centripetal, and make use of the Coulomb force to find a. There is a minor integration involved but you can get the answer to within an order of magnitude without using calculus. Oh, and you'll have to use the

Bohr radius (a<sub>0</sub> ~ 5 * 10<sup>-11</sup> m), and the

mass of the electron (m<sub>e</sub> ~ 9 * 10<sup>-31</sup> kg)

somewhere along the way....

The challenge then, is for someone to calculate the classical lifetime of hydrogen. (To within an order of magnitude is good enough for me.) It wouldn't hurt to show that v << c through most of the electron's journey for the Larmor formula to apply, but I won't require it. This is a problem I'm sure most of you can do (with or without using calculus)....

Hope you all find this educational.

Cheers,

william

Addendum:

By the way...

you may have guessed that the "classical lifetime" of hydrogen is *very* small. This kept many physicists awake at night before the advent of quantum physics....