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It's intuitive, to image a magnetic field wrapping around current flowing within a finite conductor. enter image description here

However, from Biot-Savart law, it seems possible that a point(T) beyond a conductor's length, there is a maagnetic field that is non-zero, even though parallel to that point, there is no current flow.

enter image description here

How can this be the case? If there is no current flow in that region, it seems odd to imagine, even the fact that small segments of $dL$ would contritbute to a magnetic field at various points in various planes beyond current flow within $dL$, how can this be given the fact that the magnetic field due to $I$ wraps around $dL$?

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    $\begingroup$ 1. The current has to go somewhere from the end of the wire. Charge carriers can't just disappear from existence. $\endgroup$
    – The Photon
    Commented Dec 7, 2017 at 5:36
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    $\begingroup$ You can google "monopole antenna radiation pattern" to see what happens in a somewhat similar, but physically realizable scenario with AC currents. $\endgroup$
    – The Photon
    Commented Dec 7, 2017 at 5:42
  • $\begingroup$ @ThePhoton To your first comment, is that just an assumption that ultimately the two ends would connect to a circuit and still produce a magnetic field at point (T)? That makes perfect sense. However if (for the sake of understanding) current did indeed "appear" from the left end of the conduct's length(L), and disappears at the right end, then the magnetic field around point (T) wouldn't be true? Solidifying the fact that the magnetic field produced by a wire will always "wrap" around current flow. $\endgroup$
    – Pupil
    Commented Dec 7, 2017 at 6:08
  • $\begingroup$ Let's imagine a short term scenario in which electrons move from right side to the left side untill voltage=emf caused by compressed electrons to the left. In that very short time current would flow. But if we consider perfect situation, there would be no magnetic field at point T. $\endgroup$
    – MaDrung
    Commented Dec 7, 2017 at 7:02

2 Answers 2

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Your intuitive approach is incorrect and the Biot-Savart law is right. The Biot-Savart law does not tell you that the B-field must circulate around a current. It tells you (via a vector product) that the B-field produced by a current element is perpendicular to that element and to a line joining that element with point in space at which you want to know the B-field. I.e. $$\vec{B}\propto d\vec{l}\times \vec{r}$$

In the diagram you show, there will be a B-field coming out of the diagram at P and at all points T (if P and T are above the wire in the diagram).

The only way to get no magnetic field at T is to put it on the same line as the wire so that the vector product in the Biot-Savart law is zero.

Your intuition is misguided perhaps by the case of the magnetic field due to an infinitely long wire. Only in that case (by symmetry) is it true to say that the B-field must wrap around the current in the way that you suggest.

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Consider that the Biot-Savart law calculates the B field by taking an integral along the entire length of a conducting wire (or, for that matter, over all the wires in all the space around the point being considered). The law itself explicitly tells you that $\vec{B}$ doesn't just depend on the current flowing at the nearest point on the wire.

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