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Is there a general way to calculate the magnetic field for a time dependent current of a long thing wire?

For ex:

If the current is

$$ I(t)=I\sin wt, $$ is there a general method to use in order to calculate the magnetic field?

I know for time-independent currents we can use $$ \int \vec B\cdot d\vec l=\mu_0 I_{enc} $$ if the symmetry of the system is nice, or we can use Biot-Savart Law for other cases. Thanks...

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  • $\begingroup$ You could in principle use Jefimenko's equations, but I imagine that's a bit overkill here. $\endgroup$ – David H May 21 '14 at 4:51
  • $\begingroup$ @DavidH Thanks I forgot about that one! Can We use Ampere's law if the symmetry of the system is nice, and and replace $I_{enc}$ by the time dependent current? $\endgroup$ – Jason May 21 '14 at 4:55
  • $\begingroup$ The Ampere law is not obeyed when changing electric field is present, which is what will almost surely be the case when the current changes. The Biot-Savart law is valid even in such cases if electric field is given by gradient of potential. $\endgroup$ – Ján Lalinský May 21 '14 at 4:58
  • $\begingroup$ @JánLalinský Thanks. Thats what I had originally thought. Now you cleared it for me and I understand more. $\endgroup$ – Jason May 21 '14 at 4:59
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You can use the Biot-Savart law, express the magnetic field in terms of current $I$ and then replace $I$ in the formula by $I_0\sin wt$.

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  • $\begingroup$ I want to up vote this also, but I do not have 15 points for a rating. Is there anyway I can still check it? I checked it as the answer. This is what I was looking for Thank you :) $\endgroup$ – Jason May 21 '14 at 4:55
  • $\begingroup$ I have a question, on WIKI it says: The Biot–Savart law is used for computing the resultant magnetic field B at position r generated by a steady current I (for example due to a wire): a continual flow of charges which is constant in time and the charge neither accumulates nor depletes at any point. It seems to say that it can only be used for a steady current. Is that not true, I am confused now. THanks. Because here we have a time varying current $\endgroup$ – Jason May 21 '14 at 5:14
  • $\begingroup$ Biot-Savart law is valid for steady currents and also currents which do not change too violently. If the electric field is potential (like when capacitor is charging slowly), Biot-Savart is fine: A.P. French, J.R. Tessman, Displacement Currents and Magnetic Fields, Am. J. Phys. 31, 201 (1962), dx.doi.org/10.1119/1.1969359 $\endgroup$ – Ján Lalinský May 21 '14 at 5:27
  • $\begingroup$ Thank you for your explanation again and the reference. We use AP French for Optics:) Cool thanks. $\endgroup$ – Jason May 21 '14 at 5:33
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When the current is time dependent, the time retarded current has to be put into the Biot-Savart integral. If the current is in along straight wire so that you can use cylindrical symmetry, then you can use Ampere's law which does not need the retarded time. Then B=I muo/2pi r For any time dependence in I. In any other case, the Biot-Savart law with the retarded time is very complicated. If the time dependence has a single frequency, as in your case, you can treat the Fourier transform of I (t), but this also is a bit complicated. It is done in advanced EM textbooks.

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  • $\begingroup$ Thanks a lot for your help. How do you calculate the time retarded current? Is there a general way? By using the Biot-Savart law I am not quite sure which is the source and test point for the charge. Can we assume cylindrical symmetry since the wire is long and thin? THanks for your answer $\endgroup$ – Jason May 21 '14 at 15:12
  • $\begingroup$ See my edited answer. $\endgroup$ – Jerrold Franklin May 23 '14 at 17:45

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