Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Relativistic EM question

  1. Jan 31, 2008 #1
    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.
     
  2. jcsd
  3. Feb 16, 2008 #2
    On Jan 30, 1:34=A0pm, "Rich L." <ralivings...@sbcglobal.net> wrote:
    > I've been trying to better understand the relativistic origin of
    > magnetism. =A0I tried to do a slight variation on a common derivation
    > and got an unexpected result. =A0I'm hoping someone here can point to an
    > error I made.


    The only thing "relativistic" about magnetism is the need to
    consider the finite speed of light where it may affect the
    way charge fields are superposed. Electromagnets work just
    fine with all significant charges moving at a snail's pace.

    I think this is what you are trying, and further caution that
    it is not widely accecpted.
    http://physics.weber.edu/schroeder/mrr/MRRtalk.html

    Electromagnetism operates with volumes of space,
    that don't simplfy to a 2D model easily. I think
    I see several places you've squashed the volume
    to zero where you intended to use mirror symmetry
    to simplify the calculations.

    If you work through a time independent derivation:
    http://en.wikipedia.org/wiki/Multiple_integral#Some_practical_applications
    http://farside.ph.utexas.edu/teaching/em/lectures/node26.html

    =2E..then apply the time dependent extensions for motion:
    http://farside.ph.utexas.edu/teaching/em/lectures/node41.html

    =2E..I think you will develop so apprectiation why the
    Purcell derivation is not widely accepted.

    You may find some work relevant to what you are
    attempting in these subjects:
    http://en.wikipedia.org/wiki/Smith-Purcell_effect
    http://en.wikipedia.org/wiki/Maxwell's_equations#Formulation_of_Maxwell.27=
    s_equations_in_special_relativity

    Sue...

    [...]
    >
    > Any assistance is appreciated,
    >
    > Rich L.
     
  4. Mar 10, 2008 #3
    I couldn't disagree more. So sensitive to changes in inertial frames is electromagnetism that charges having velocity v<<c, develop a field (B) of such magnitude, that relative forces due to electric the and magnetic fields are comparable in strength.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Relativistic EM question
  1. General EM Wave Question (Replies: 15)

  2. EM Wave Phase Question (Replies: 30)

Loading...