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So here's the set-up: I have a loop in space and fix a point $p$ outside the loop. Then I turn the current on at $t = 0$ (so the current profile is a step-function). Initially at $t=0$ the E&M fields are obviously zero at $p$ and will continue to be zero until some finite time $t_1$ when it starts to feel the influence of the current loop (in accordance with Jefimenko's retarded solutions). Suppose at $t_2$ the point $p$ has felt the influence of the entire current loop so that at $t > t_2$ the point $p$ is always under the influence of the whole loop (in the sense that every point of the loop contributes to the fields at $p$).

So here's my question: Obviously in $t_1 < t < t_2$ the E&M fields are changing and not steady (since they are yet to feel the influence of the entire current loop), but can we say that at $t > t_2$ the fields become steady at point $p$?

More generally, if we have a bounded current loop configuration (so that a point $p$ will feel the influence of the entire current loop after a finite time) and fix an arbitrary point $p$ can we say that after time $t'$ (time after which the influence of the whole loop is felt) the E&M fields become steady? The issue is that as far as I know there are no exact solutions for any kind of bounded current configuration which have any sort of time dependence (I looked through Jackson, Zangwill, and Griffiths and didn't have any luck).

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According to the formulas for retarded potentials, the fields are steady at $p$ after $t_2$, however, the assumption of a step-function current is dubious / not realistic, as the loop has finite inductance defined by its dimensions (say, the radii of the loop and the wire for a circular loop made of a round wire).

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  • $\begingroup$ Yes now that I look at the equations more carefully one can fully prove this is indeed the case. for each point $p$ you can prove there exist a corresponding time $t$ such that the fields will be static after this (in fact the $E$ field will be zero and $B$ will reduce to the static biot-savart field). However, what puzzles me is that in free space one can still find wave solutions which don't satisfy the aforementioned condition: the sinusoidal solutions satisfy the maxwell equations in free space yet are time dependent for all $t$ even if you fix a point $p$ in space. $\endgroup$
    – Leonid
    yesterday
  • $\begingroup$ the wave solutions seem to imply that $E$ and $B$ fields can sort of reinforce each other without sources which seems to contradict what the retarded solutions say. $\endgroup$
    – Leonid
    yesterday
  • $\begingroup$ @Leonid : yes, you can add free field solutions, but the wording of your problem seems to imply that initially there was no field. $\endgroup$
    – akhmeteli
    yesterday

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