Radiation Pulse from an Accelerated Point Charge
Radiation Pulse from an Accelerated Point Charge
J. J. Thomson first suggested a pictorial representation of how an instantaneously accelerated point charge can produce a pulse of electromagnetic radiation. An electron, with charge , moving at a constant speed , even when a significant fraction of the speed of light , produces an electric field of magnitude , (add factor if you cannot live without SI units), where (t) represents the projected position of the source charge at time , assuming that it continues to move at constant speed from its position (t') at the retarded time . This is derived most lucidly in the Feynman Lectures [1]. Thus a uniformly moving point source emits a spherical longitudinal electric field, although its magnitude does vary with direction. This is represented in the graphic by a series of 12 uniformly spaced radial spokes.
e
v
c
E(r,t)=
1
1-/
2
v
2
c
e
r(t)-(t)
r
0
2
|
1/4π
ϵ
0
r
0
t
v
r
0
t'
The electron is assumed to move initially at a speed =c until time , when it reaches the red dot at the center of the figure. The charge is then, in concept, instantaneously accelerated to speed >. For , the charge emits a longitudinal electric field characteristic of the speed . On a sphere of radius , shown in red (a ring of fire?), the field catches up with the field, which still behaves as if the electron were moving at its original speed. Since electric-field lines must be continuous in charge-free space, the two sets of field lines connect with transverse segments along the ring of fire, with an angular intensity proportional to . Transverse magnetic field lines , perpendicular to the lines (not shown on the graphic) are also created by the moving charge. The lengthening of the segments by a factor implies the long-range radial dependence of the radiation fields as , rather than , as for electrostatic fields.
v
1
1
2
t=10
v
2
v
1
t>10
v
2
c(t-10)
v
2
v
1
sinθ
B
E
rsinθ
-1
r
-2
r
Thus, it has been shown that pulses of electromagnetic radiation consist of transverse electric and magnetic fields moving radially outward with the speed of light from the point of instantaneous charge acceleration.