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Energy in the Magnetic field of a moving point charge, simple question!

  1. Oct 16, 2011 #1
    I'd like to know what the energy stored in the magnetic field created by a moving point charge is, as a function of speed. Hopefully, someone knows a relationship like this? or how to get this? I've got a straightforward approach below, but dont know how to integrate it.

    BfieldEnergy = f(|v|)

    For low constant speeds, not worried about direction, so just the magnitude of v... not worried about relativistic effects either, and lets assume its not accelerating.

    My best guess is use u=B^2/2mu0, where B is given by (mu0/4pi)*e(vXr(hat) )/r^2, and integrate that from r = 0 to infinity... but I don't know how to handle the r=0, the integral just explodes... there must be some limit or something. Anyways, I'm guessing the relationship is proportional to |v|^2, but if you know or can figure it out I'd really appreciate it.

    Last edited: Oct 16, 2011
  2. jcsd
  3. Oct 17, 2011 #2
    If you don't care about accelerations or relativistic effect, you can treat the point charge as a piece of current, I dl = v dq. Put this into the Biot-Savart law find the magnetic field due to this "current", than plug this field into u=B^2/2mu0 to find the field energy density.

    Yes, the total field energy is the integral over all space of the field energy density, but you can't really do this integral over all space when using unphysical idealizations like a point charge, because the fields blow up like you said. It is better to define a region not including the singularities and find the total energy just in that region.
  4. Oct 17, 2011 #3
    What would be a reasonable guess for that region approximation? .1, .01, ... 0.0000001? the bohr radius?
  5. Oct 18, 2011 #4


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    This is an interesting question, and I'm not sure of the answer. The classical theory breaks down when you get very close to the particle, and quantum mechanics is the 'correct' theory to use.

    So yes, maybe you should use some lower limit for the values of r to be integrated over. You'll find the energy then depends on this lower limit for r. But you're not interested in that, right? You're interested in the speed dependence.

    The speed dependence would be v^2, as you guessed. This is the speed dependence for the macroscopic energy. For the quantum limit, there is probably some other dependence..
  6. Oct 18, 2011 #5


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    This is a very deep issue and cannot be answered adequately by postings in a forum. The total energy of a charged classical point particle is not well defined, since already for a point particle at rest the total energy in its electric Coulomb field diverges.

    On the other hand, for point particles, classical models do not make any sense. So one has to go to extended objects and define the field energy. However, even this is not so easy in a covariant way. The modern definition is to define the total field energy for the particle at rest (which is electrostatic and/or magnetostatic if there's an intrinsic magnetic momentum, e.g., for a permanent magnet). Then you Lorentz boost the whole system and get the total energy-momentum vector of the electromagnetic field.

    If it comes to acceleration of the particle/body it's even worse. Then you have the problem of radiation damping, i.e., the interaction of the particle with its own electromagnetic radiation field, and that's indeed a very delicate issue. The best treatment of the problem is given in the textbook

    F. Rohrlich, Classical charged particles 3rd ed., World Scientific
  7. Dec 17, 2011 #6
  8. Dec 18, 2011 #7


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    Thanks for that link, it is much advanced beyond my skill of thinking, but sounds as if many questions I have are inline with the general trend of what it implies.
    With much to consider, I have the thought of the charge of the electron is the ether, and that the collapse rate is less than the speed that creates it. Think about the long strand of silk, spun into a cocoon.
    A core-less transformer, that moves energy at different rates because of variation in radius of lines of force, to any reactive particle in the area. No loss of energy, but a redistribution.

    Based on recent study, I'm sure that wiki already covers all this, just in better form. Sigh...

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