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I understand that a moving charge will produce a magnetic field, but what causes them to appear circular and perpendicular to the flow of current, as they are?

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First, do keep in mind that a moving charge also produces an electric field. And Maxwell alone doesn't specifically attribute an electric or magnetic field due to any specific charge. But an example of a solution to Maxwell can be provided if both the electric and magnetic field are computed as the electric and magnetic parts of the electromagnetic field given by Jefimenko's equations:

$$\vec E(\vec r,t)=\frac{1}{4\pi\epsilon_0}\int\left[\frac{\rho(\vec r',t_r)}{|\vec r -\vec r'|}+\frac{\partial \rho(\vec r',t_r)}{c\partial t}\right]\frac{\vec r -\vec r'}{|\vec r -\vec r'|^2}\; \mathrm{d}^3\vec{r}' -\frac{1}{4\pi\epsilon_0c^2}\int\frac{1}{|\vec r-\vec r'|}\frac{\partial \vec J(\vec r',t_r)}{\partial t}\mathbb{d}^3\vec r'$$ and $$\vec B(\vec r,t)=\frac{\mu_0}{4\pi}\int\left[\frac{\vec J(\vec r',t_r)}{|\vec r -\vec r'|^3}+\frac{1}{|\vec r -\vec r'|^2}\frac{\partial \vec J(\vec r',t_r)}{c\partial t}\right]\times(\vec r -\vec r')\mathbb{d}^3\vec r'$$ where $t_r$ is actually a function of $\vec r'$, specifically $t_r=t-\frac{|\vec r-\vec r'|}{c}.$

These reduce to Coulomb and Biot-Savart only when those time derivatives are exactly zero, which is statics. So Jefimenko is an example of proper time dependent laws for the electromagnetic field. Note that both the electric and the magnetic part of the electromagnetic field have parts that depend on the time variation of current. But only the magnetic field depends on the current.

When the time variation of the current is zero, the magnetic field is solely determined by the current. Full stop. And if the wires are perfectly neutral (and stays that way) and the time variation of the current is zero, then the electric field is zero.

As for why it is circular. The field due to the current always points orthogonally to the current and the vector from the current to where the field is. To join fields lines together you note that the points that are equally far from a piece of current naturally join in a circle.

But the real issue is that you want the field due to other current elements to also be circular. A straight wire can do this.

But there is another deeper issue. Because there are no magnetic charges the fields line can only loop together or go out to or from infinity.

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