2

2.3. A Non-Radiating, Accelerating Charge.
Reference: Appendix 2.3_1

A rather interesting fact is that a charged particle, subject to a constant force, accelerates but does not radiate! In order to prove this, we begin by noting that the driving agent’s counteraction to F_RadReact, in 1 dimension, is

. (2.3_1)

Clearly no radiation is emitted when F_radiative=0, or when

. (2.3_2)

One obvious solution is when v is constant (and a=0). But a second solution can be computed by imposing initial conditions for v(0) and a(0) and then performing the following algorithm:

t=0

da/dt = -3g²va²/c²

v = v + a dt

a = a + (da/dt) dt

t = t + dt

Loop until v ~ c

Figs. 2.3_1 and 2.3_2 plot v_x vs. t and a_x vs. t respectively. Note that v approaches c asymptotically and a approaches 0 asymptotically.

Figure 2.3_1

vx vs. t

Figure 2.3_2

ax vs. t

Now when F_radiative is zero, the equation of motion (in 1 dimension) simplifies to

. (2.3_4)

Substituting the computed values of v and a (and using some value for m(0)), Fig. 2.3_3 indicates that a plot of F results in a straight line. F is constant. Whence we conclude that a charged particle accelerates, but does not radiate, when subjected to a constant driving force!

Figure 2.3_3

Fx vs. t