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Biot Savart's law says there should be none as the current element is along the displacement vector, hence as $\sin \theta = 0$, so, the magnetic field must be zero. But, as per Ampere's loop rule there is always some magnetic field even inside and on the conductor. What is the resolution to this conflict? enter image description here

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    $\begingroup$ Where is your Amperian loop supposed to be drawn? And is the current really meant to be only along the 1D straight line at the top of the wire, or throughout the whole surface of any cross-section of the wire whose top passes through dl? (if the latter shouldn't a blue line segment also appear on the bottom of the wire in your diagram?) If the current were present at other points on the surface that don't lie along the 1D straight line you've drawn, wouldn't your argument about $sin \theta = 0$ no longer apply? $\endgroup$
    – Hypnosifl
    Oct 28, 2014 at 4:23
  • $\begingroup$ No, the current is in 3D space of the wire, and the Amperian loop encircles the wire with its center coinciding with the central axis of the wire. So, in total, you mean that there must be some magnetic field at point P by other current elements that lie inside the wire? $\endgroup$
    – Swami
    Oct 28, 2014 at 4:52
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    $\begingroup$ Yes, the r in the Biot-Savart integral has to be taken from every current element to P, and for all the current elements that don't lie on that top line, r should not be parallel to dl, the direction of current for that element, so in that case the cross-product will be nonzero. It might be interesting to consider the case of a 1D wire though--the magnetic field from an infinite straight 1D current should go to infinity as the distance from the wire approaches zero, but for a point along the wire, it seems the cross-product in the Biot-Savart integral should be zero for every current element. $\endgroup$
    – Hypnosifl
    Oct 28, 2014 at 4:59
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    $\begingroup$ (probably the resolution to the 1D problem is that the Biot-Savart integrand also has an r in the denominator, so for the current element that passes through the point where we want to evaluate the field, the integrand becomes undefined, so either you can't use the Biot-Savart law there or the field itself is undefined at that point in classical electromagnetism) $\endgroup$
    – Hypnosifl
    Oct 28, 2014 at 5:01
  • $\begingroup$ Sounds good to me. I need to research more on this. $\endgroup$
    – Swami
    Oct 28, 2014 at 5:04

1 Answer 1

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The r in the Biot-Savart integral has to be taken from every current element to P, and for all the current elements that don't lie on that top line in your diagram (since you mentioned in a comment that the current is supposed to be present throughout the 3-dimensional wire), r should not be parallel to dl, the direction of current for that element, so in that case the cross-product will be nonzero.

It might be interesting to consider the case of a 1D wire though--according to the equation here the magnetic field from an infinite straight 1D current should go to infinity as the distance from the wire approaches zero, but for a point along the wire, it seems the cross-product in the Biot-Savart integral should be zero for every current element, since r and dl are always parallel. Probably the resolution to the 1D problem is that the Biot-Savart integrand also has an $ |r|^3$ in the denominator, so for the current element that passes through the point where we want to evaluate the field, the integrand becomes undefined, so either you can't use the Biot-Savart law there or the field itself is undefined at that point in classical electromagnetism.

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