I've been trying to better understand the relativistic origin of magnetism. I tried to do a slight variation on a common derivation and got an unexpected result. I'm hoping someone here can point to an error I made. The problem is to calculate the electric field a test charge would experience in its rest frame due to a relativistic charged particle beam near it. To be definite let's call it a beam of protons traveling in the laboratory frame at 80% the speed of light. Let's have the beam aligned with the x axis and be offset a distance "a" along the +y axis, and aligned with the origin of the z axis. The charges are traveling in the +x direction. I want to calculate the forces on a test charge at the origin (that is, a distance "a" from the line current). First let's consider the electric field around the beam in the rest frame of the beam charges. In this frame (I'll call it the beam frame) the charges in the beam are stationary and uniformly distributed along a line. Let's call the line density of charge sigma_0 in this frame. By standard E&M calculations the electric field is everywhere radial from this line. The magnitude of the electric field on the x-axis (where the test charge will be) is epsilon_0*sigma/2*pi*a and is entirely in the -y direction. It is uniform along x and has no other components, especially along the x axis. In the laboratory frame, the test charge at the origin will experience an electric field from the charges in the beam. For the issue here we don't even need to consider that the test charge is moving, so we only need be concerned with the electric fields as seen in the laboratory frame. Because the charges in the beam are moving at 80% of the speed of light, the linear charge density along the beam is increased by a factor of gamma. This is where I've departed slightly from the more common derivations. The derivations I've seen at this point all calculate a uniform charge density for the beam at the current "present" time. This would undoubtedly be correct for normal wires where the charges are moving very slowly. For reasons that will be clear shortly I don't believe this is correct for this relativistic beam, however. We need to be concerned not just with where the charges are "at present", but with where they were when they were on the past light cone of the test charge. For this next argument I need to describe a three dimensional space time diagram, that is one with 2 space dimensions and one time dimension. We will discard the z direction as it is not important to this problem. If we consider the world lines of the charges in the beam, they will always be on the line y=+a. Consider a slice of this diagram on the y=+a plane. Since the charges are moving in the +x direction, their world lines in a slice of this diagram would look like: +t | / / / | / / / / / / / /| / / / / / / / / | / / / / / / / / | / / / / / / / / | / / / / / / / / |/ / / / ---------------------------------------------> +x / / / /| / / / / / / / / | / / / / / / / ******* / / / / / / ***/ | / ***/ / / / / /** / |/ / ** / / / * / / / * / / / */ / /| / / * / / / */ / / | / / */ / / */ / / | / / /* / / */ / / | / / / * / / */ / / |/ / / */ | -t Each diagonal line represents the world line of one of the charges in the beam. The test charge, at the origin (which is slightly in front of the diagram above) only "sees" the charges on its past light cone. Since the world lines from the beam form a flat sheet parallel to the xt plane, and the past light cone has it's axis on the t axis, the intersection of these two surfaces is a hyperbola. I've tried to represent these on the text mode plot above as *'s. Now it is apparent just looking at this that the charge density on this hyperbola is no longer uniform, but is higher to the right side, where the lines tilt toward the normal of the curve, than on the left side where the lines tilt to be parallel with the curve. The test charge does not see this hyperbola, of course, because it on a space- time diagram. What is does see is the charges on the axis of the beam, but in positions projected up from this hyperbola. That is, the charges appear to be distributed along the beam as shown by the *'s on the x axis: +t | / / / | / / / / / / / /| / / / / / / / / | / / / / / / / / | / / / / / / / / | / / / / / / / / |/ / / / --*---------*-------*-----*----*---*---*--*--> +x | / / | / | /| | / | /| |/ | / |/ / | / |/ | |/ | / | | | / | / | / ******* |/ | /| | / /| / |/ ***/ | / ***| | / | |/ / | / /** / |/ / ** |/ | | | / * / / / * | /| / | / */ / /| / / * |/ | / |/ */ / / | / / */ | / | */ / / | / / /* | / /| */ / / | / / / * |/ / | */ / / |/ / / */ | -t I've done this calculation more rigorously and derived formulas for this distribution. The average charge density along the beam is just gamma times the rest frame (of the beam charges) charge density, but looking in the direction the charges are coming from the density is reduced, and in the direction they are going the density is increased. At large distances the densities are uniform, but lower (to the left) or higher (to the right). The faster the charges are moving, the greater the difference between the charge density on the left and that on the right. If the charges could move at the speed of light, the density on the left would drop to zero. This asymetric charge density it seems to me would create an electric field along the axis of the beam. This is what puzzles me. I have never heard of such a thing. I suspect my analysis above has an error, but I don't see it. I've also tried to do this using EM tensors. For the location of the test charge in the beam rest frame, the only component is the Ey electric field. The "F" tensor therefor only has two non-zero elements corresponding to Ey and -Ey. When this tensor is transformed by the Lorentz transform into the laboratory frame, I get the expected Bz terms, but not the Ex terms predicted by the above analysis. | 0 0 -Ey 0 | | 0 0 0 0 | F(beam) = | Ey 0 0 0 | ==> | 0 0 0 0 | | 0 0 -g*Ey 0 | | 0 0 g*b*Ey 0 | F(lab) = | g*Ey -g*b*Ey 0 0 | | 0 0 0 0 | where "g" is gamma and "b" is beta=v/c, and I'm using c=1 units. So my questions are: -Am I making an error in my analysis of the apparent charge density along the wire? -Are axial electric fields observed around relativistic beams, such as in particle accelerators? I am skeptical because they would induce currents in the beam pipes that would drain energy from the beam, and I haven't heard of such an effect. -Am I doing the tensor transformation correctly? I am taking F in the beam rest frame and calculating Lambda^u_a*F^ab*Lambda^v_b where Lambda is the Lorentz transform for a boost in the x direction and F is the EM tensor above. Any assistance is appreciated, Rich L.