Electric Fields around a Moving Line Charge

Consider a line of charges along the z-axis with charge density rho_0 (in the observers frame) and moving at velocity beta = v/c in the +z direction.  We have a test charge q_t located a distance r from the line charge.  Based on the derivation in the last section one might think that the charges on the –z axis that are approaching the test charge would have a stronger effect than those on the +z axis that are receding, resulting in a component of the electric field that is parallel to the z axis.  (The author has seen this claim made on several occasions on the web, such as the “Relativistic Electromagnetism” page on Wikipedia.)  This does not happen due to a compensating effect that we will calculate now.

To do this calculation correctly we need the correct picture of the retarded positions of the line charges.  First we will look at a simple special case, where the test charge is very close to the line charge.  The space-time diagram below shows this case:

           Space-Time diagram in q_t frame

The test charge “sees” the charges in the line on its own past light cone.  These charges are indicated by the red dots on the diagonal lines above.  The black arrows are the world lines for each of these charges.  The tails of these arrows I've set on the position of the wire at some time in the past.  Notice the spacing of the line charges on the z_t axis, and also the spacing on the past light cone.  At any specific time, the spacing of the charges on the wire is uniform.  But since the test charge will be responding to charge on its past light cone, it is the spacing on the light cone that is important.  

The charges approaching q_t, i.e. those to the left of q_t, have a wide spacing along the light cone, whereas the  charges receding q_t, to the right of q_t, have a smaller spacing.  The charges to the left, which are approaching q_t and thus have an augmented gradient to their potential, are spaced further apart on the past light cone due to the angle their world line is making with the light cone.  Conversely the charges to the right of q_t, which are receding from q_t and thus have a reduced gradient to their potential, are more closely spaced.  The apparent line charge density,  as observed in the q_t frame, corresponds to the spacing of these charges on the past light cone, but projected onto the z_t axis.

We use the triangle to the right in the figure to calculate the charge density along the light cone as seen by the test charge.  The spacing between the charges corresponds to the distance along the base of this triangle minus the short horizontal line at the top left:

A little algebra then gives:

Thus the charge density for the charges receding from the test charge is enhanced by the factor 1+beta.  Note however that from the derivation in the previous section that the electromagnetic forces for these charges is divided by  this same factor,  The result is that the electromagnetic force of the charges receding from the test charge is exactly the same as would be calculated for the normal, rho_0 , charge density.

For the charges approaching the test charge, the density factor is just 1-beta because the light cone slopes in the other direction.  Also the enhancement of the electromagnetic force is by this same factor so that again the net effect is that the force is exactly the same as if the charges were stationary with the linear charge density rho_0.  As a result we conclude that despite the so called “Lienard-Wiechert potentials”, that the electric fields for an infinite line charge are always normal to the line charge even though the charges are moving.

Taking this charge density and standard electrostatic theory, we calculate the 4-potential at the point r from the wire.  Note that in the derivation in the last section, rho_0 was the charge density of the line charge in the observers frame.  If we now take rho_0 as the linear charge density in the rest frame of the line charge, the observers charge density becomes gamma rho_0.  If we integrate the potential contribution of all the charges in the line, and apply the Lorentz transform as described in section I.6, we get:

(Actually this is an approximation.  The potential near an infinitely long line charge is infinite.  This should not be too surprising since there would have to be an infinite charge on the line.  For a finite line charge, the above expression is correct to within a constant.  See a later post for a more complete derivation.)

The force on the test charge, assumed stationary in the observers frame, is then:

Since the test charge is not moving towards or away from the source wire, there is no rate of change with time:

Since the charge density along the past light cone is proportional to 1+beta, we get:

Using the expression for the potential near a wire, we can evaluate the electrostatic force around a line charge:

Where gamma rho_0 is the linear charge density on the wire in the observers frame, in terms of the charge density in the source charge rest frame and the Lorentz Transform  factor.  This gives the “electric field” around the charge line:

Because the charges are moving, there is also a vector potential.  It will also cause a force if the test charge is moving.  That will be calculated in the next couple of posts.