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From all the texts I've read, it's always stated that the magnetic field would either curl or warp around the the current flowing within a conductive element, yet, I never was clarified as to why that is. Why would the field "curl" around it? I'm curious if there is an explanation for that nature.

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Amazingly, it seems like a perfect circle too, similar to the diagrams that I've studied.

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  • $\begingroup$ Since this may help answer the question: why do you think the electric field doesn’t curl around a charge? $\endgroup$
    – knzhou
    Commented Jun 8, 2018 at 12:53
  • $\begingroup$ People are giving you mathematical answers having Maxwell's equations as a starting point. That might be what you are after, but it might not. Let us know. The question "Why?" in physics can be interpreted several ways. $\endgroup$
    – garyp
    Commented Jun 8, 2018 at 15:38

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If you ignore all what you learned about the B-field being a "vector" and you treat it as what really is a 3D skew-symmetric tensor then the mystery goes away. In this view the B-field is a bi-vector, a surface-like quantity whose source is the current element from which it radiates outward, so to speak. Unlike the E-field that has lines of force coming out or ending in charges, the B-field has planes of force coming out or ending in currents and these "planes" form surfaces in which the magnetic action, i.e., attraction-repulsion and torque take place. This is analogous to the way the E-field acts along its lines of force. What conventionally is called the lines of force of the B-field are the orthogonal rays to these surfaces. A uniform current generates a uniform set of planes of action whose orthogonal rays are in fact circles, and they stay so approximately even when the source is curled up into a loop. You can see nice pictures of this in Roche: "Axial vectors, skew-symmetric tensors and the nature of the magnetic field", Eur. J. Phys. 22 (2001) 193–203.

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According to the Ampere's Circuital Law (Maxwell's 4th equation):

The magnetic field induced around a closed loop is proportional to the electric current plus displacement current (proportional to the rate of change of electric flux) through an enclosed surface.

and the formula derived for that is: $$∇ \times \vec H = \vec J + \frac{∂\vec D}{∂t}$$ where $\vec H$ is magnetic field strength, and $\vec J + ∂\vec D/∂t$ the enclosed current density.

For linear materials, $\vec H$ and $\vec B$ have the same direction. As it is clear from the equation, vector of $\vec H$ will be created such that it is always perpendicular to the vector $\vec J + ∂\vec D/∂t$, in addition, it is the case in all sides of wire => so curl will be formed.

$\vec J + ∂\vec D/∂t$ is actually vector source of $\vec H$.

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    $\begingroup$ Minor mistake: $H$ and $B$ do not necessarily point to the same direction for linear materials but for isotropic materials. An isotropic non-linear material will not cause $H$ and $B$ to point in different directions, while a linear non-isotropic material will unless $\vec B$ points along an eigenaxis of the permeability tensor. $\endgroup$
    – Sebastian Riese
    Commented Jun 8, 2018 at 15:11

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