Classically, an orbiting charged particle radiates electromagnetic energy, and for an electron in atomic dimensions, it would lead to collapse 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: ω = er-3/2/4пɛ0m. (b) Electromagnetism tells us that a charge whose acceleration is a radiates power P = e2a2/6ɛ0c3. Show that this can also be expressed in terms of the orbit radius, as P = e6/(96п2ɛ03mc3r4). Then calculate the energy lost per orbit in terms of r by multiplying this power by the period T = 2п/ω and using the formula from part (a) to eliminate ω. (c) In such a classical orbit, the total mechanical energy is half the potential energy (see Section 4.4), or Eorbit= -e2/8пɛ0r. Calculate the change in energy per change in r:dEorbit/dr: 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 m. Would the electron's radius change much in a single orbit? (d) Argue that dividing dEorbit/dr by P and multiplying by dr gives the' time required for r to change by dr. Then, sum these times for all radii from rinitial[ to a final radius of 0. Evaluate your result for rinitial = 10-10 m. (One limitation of this estimate is that the electron would eventually be moving relativistically.)
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