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Field Due to a Moving Charge

Although the fields generated by a uniformly moving charge can be calculated from the expressions (1822) and (1823) 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

 (1823) (1824)

This field must now be transformed into the frame . The direct method, using Equations (1809)-(1812), is somewhat simpler here, but we shall use a somewhat indirect method because of its intrinsic interest.

In order to express Equations (1825) and (1826) in tensor form, we need the electromagnetic field tensor on the left-hand side, and the position 4-vector and the scalar on the right-hand side. (We regard as an invariant for all observers.) To get a vanishing magnetic field in , we multiply on the right by the 4-velocity , thus tentatively arriving at the equation

 (1825)

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

 (1826)

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 Equation (1828) in . It is convenient to choose for that event at the charge at which (not the retarded event). Thus,

 (1827)

giving

 (1828)

or

 (1829)

Likewise,

 (1830)

or

 (1831)

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

 (1832)

Thus,

 (1833) (1834)

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 behavior of the charge during a finite period before that instant can no longer affect the measurement. Note also that, unlike Equations (1822) and (1823), the previous expressions for the fields are not valid for an arbitrarily moving charge, nor 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, , Equations (1835) and (1836) reduce to the well-known Coulomb and Biot-Savart fields. However, at high velocities, , the fields exhibit some interesting behavior. The peak electric field, which occurs at the point of closest approach of the charge to the observation point, becomes equal to times its non-relativistic 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

 (1835)

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 that 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 2014-06-27