Electron and Radiation Reaction

The equation of motion for a particle of mass m and charge q in electric and magnetic fields E and B, including the radiation reaction force, is

a) Assuming that the radiative reaction term is very small compared to the Lorentz force and that v << c, find an approximate expression for the radiative reaction force in terms of E and B.

b) A plane electromagnetic wave propagates in the z direction. A free electron is initially at rest in this wave. Under the assumptions of (a), calculate the time-averaged radiative reaction force on the electron (magnitude and direction). What result would you obtain for a positron?

c) Rederive the reaction force by considering the momentum acquired by the electron in the process of forced emission of radiation. Use the Thomson cross section σ

SOLUTION

a) By assuming that v << c, we may write

(1)

Differentiating (1) with respect to time, we obtain

(2)

Substituting for  in (2) results in

where we have disregarded terms first order in v/c. So

(3)

b) Let the E field of the plane wave be polarized in the x direction, so that

The time averages of (3) are

so that

(4)

The radiation reaction force varies with the fourth power of the charge, so a positron would yield the same result.

c) The average power scattered by the charge is

(5)

where σ is the total cross section. The average power is then

(6)

The average incident momentum per unit time p_i is given by

(7)

where we used the relation ε = pc for radiation. Using the Thomson cross section for σ in (7) gives the reaction force

This is the same result as in (4).