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Fields due to a moving
charge
Although the fields generated by a uniformly moving charge can be calculated
from the expressions (1525) and (1526) for the potentials, it is simpler to
calculate them from first principles.
Let a charge , whose position vector at time is , move with
uniform velocity in a frame whose axis has been chosen
in the direction of . We require to find the field strengths
and at the event . Let be that frame
in standard configuration with in which the charge is permanently at
rest. In , the field is given by
This field must now be transformed into the frame . The direct method,
using Eqs. (1512)(1515), is somewhat simpler here, but we shall use a somewhat
indirect method because of its intrinsic interest.
In order to express Eq. (1528) and (1529) in tensor form, we need the electromagnetic
field tensor on the lefthand side, and the position 4vector
and the scalar
on the righthand side. (We regard as an invariant for all observers.)
To get a vanishing magnetic field in , we multiply on the right
by the 4velocity
, thus tentatively
arriving at the equation

(1530) 
Recall that and
. However, this equation
cannot be correct, because the antisymmetric tensor
can only be equated to another antisymmetric tensor. Consequently,
let us try

(1531) 
This is found to give the correct field at in , as long as
refers to any event whatsoever at the charge.
It only remains to interpret (1531) in . It is convenient to choose for
that event at the charge at which (not the
retarded event). Thus,

(1532) 
giving

(1533) 
or

(1534) 
Likewise,

(1535) 
or

(1536) 
Lastly, we must find an expression for in terms of quantities
measured in at time . If is the corresponding time in
at the charge, we have

(1537) 
Thus,
Note that acts in line with the point which the charge
occupies at the instant of measurement, despite the fact that,
owing to the finite speed of propagation of all physical effects,
the behaviour of the charge during a finite period before that instant
can no longer affect the measurement. Note also that, unlike
Eqs. (1525) and (1526), the above expressions for the fields are not valid for
an arbitrarily moving charge, not can they be made valid by merely using
retarded values. For whereas acceleration does not affect the potentials,
it does affect the fields, which involve the derivatives of the potential.
For low velocities,
, Eqs. (1538) and (1539) reduce to the
wellknown Coulomb and BiotSavart fields. However, at high
velocities,
, the fields exhibit some interesting behaviour.
The peak electric field, which occurs at the point of closest approach
of the charge to the observation point, becomes equal to times
its nonrelativistic value. However, the duration of appreciable
field strength at the point is decreased. A measure of the time
interval over which the field is appreciable is

(1540) 
where is the distance of closest approach (assuming
). As increases, the
peak field increases in proportion, but its duration goes in the
inverse proportion. The time integral of the field
is independent of . As
, the observer
at sees electric and magnetic fields which are indistinguishable
from the fields of a pulse of plane polarized radiation propagating in
the direction. The direction of polarization is along the radius vector
pointing towards the particle's actual position at the time of observation.
Next: Relativistic particle dynamics
Up: Relativity and electromagnetism
Previous: Potential due to a
Richard Fitzpatrick
20060202