Classical Electrodynamics · Problem 16.2

Radiation Damping, Classical Models of Charged Particles — John David Jackson

Physics — E&M · radiation-damping-circular-orbit

Problem

A nonrelativistic electron of charge

-e

and mass

m

bound in an attractive Coulomb potential

(-Ze^2/r)

moves in a circular orbit in the absence of radiation reaction. (a) Show that both the energy and angular-momentum equations (16.13) and (16.16) lead to the solution for the slowly changing orbit radius,

r^3(t) = r_0^3 - 9Z(c\tau)^3 \frac{t}{\tau}

where

r_0

is the value of

r(t)

t = 0

. (b) For circular orbits in a Bohr atom the orbit radius and the principal quantum number

n

are related by

r = n^2 a_0/Z

. If the transition probability for transitions from

n \to (n-1)

is defined as

-dn/dt

, show that the result of part a agrees with that found in Problem 14.21. (c) From part a calculate the numerical value of the times taken for a mu meson of mass

m = 207 m_e

to fall from a circular orbit with principal quantum number

n_1 = 10

to one with

n_2 = 4

, and

n_2 = 1

. These are reasonable estimates of the time taken for a mu meson to cascade down to its lowest orbit after capture by an isolated atom.

Related theory

Electron Speed In A Circular Orbit

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