A particle of charge e is moving in nearly uniform nonrelativistic motion. For times near t=t0, its vectorial position can be expanded in a Taylor series with fixed vector coefficients multiplying powers of (t−t0).
(a) Show that, in an inertial frame where the particle is instantaneously at rest at the origin but has a small acceleration a, the Liénard–Wiechert electric field, correct to order 1/c2 inclusive, at that instant is E=Ev+Ea, where the velocity and acceleration fields are
Ev=er2r^+2c2re[a−3r^(r^⋅a)];Ea=−c2re[a−r^(r^⋅a)]
and that the total electric field to this order is
E=er2r^−2c2re[a+r^(r^⋅a)]
The unit vector r^ points from the origin to the observation point and r is the magnitude of the distance. Comment on the r dependences of the velocity and acceleration fields. Where is the expansion likely to be valid?
(b) What is the result for the instantaneous magnetic induction B to the same order? Comment.
(c) Show that the 1/c2 term in the electric field has zero divergence and that the curl of the electric field is ∇×E=e(r^×a)/c2r2. From Faraday's law, find the magnetic induction B at times near t=0. Compare with the familiar elementary expression.