Classically, an orbiting charged particle radiates elecIromagnetic energy, and
for an electron in atomic dimensions, if would lead to collupse in
considerably less than the wink of an eye. (a) By equating the centripetal and
Coulomb forces, show that for a classical charge \(-e\) of mass \(m\) held in
circular orbit by its attraction to a fixed charge \(+e\), the following
relationship holds: \(\omega=e r^{-1 / 2} / \sqrt{4 \pi \varepsilon_{0} m}\).
(b) Electromagnetism tells us that a charge whose acceleration is a radiates
power \(P=e^{2} a^{2} / 6 e_{f} f^{\prime}\). Shuw that thiv can also be
expressed in terms of the orbit radius, as \(P=e^{6} /\left(96 m^{2}
\varepsilon_{0}^{3} m c^{3} r^{\lambda}\right)\). Then calculate the energy
lowt per orbit in terms of \(r\) by multiplying this power by the period \(T=2
\pi / \omega\) and using the formula from part (a) to eliminate
\(\omega(\mathrm{c})\) in such a classical orbit. the total mechanical energy is
half the potential energy (see Section 4.4), or \(E_{\text {orbu }}=-e^{2 / 8}
\pi \varepsilon_{0} r .\) Calculate the change in energy per change in \(r, d
E_{\text {obi }} l d r\). From this and the energy lost per orbit from part
(b). determine the change in \(r\) per orbit and evaluate it for a typical orbit
radius of \(10^{-10} \mathrm{~m}\). Would the electron's radius change much in a
single orbit? (d) Argue that dividing \(d E_{\text {arbiu }} / d r\) by \(P\) and
multiplying by dr gives the time required for \(r\) to change by \(d r\). Then.
sum these times for all radii from \(r_{\text {initial }}\) to a final radius of
\(0 .\) Evaluate your result for \(r_{\text {hiniul }}=10^{-10} \mathrm{~m}\).
(One limitation of this extimate is that the electron would eventually be
moving relativistically.)
