Magnetism

Magnetic Field

A moving charge causes a magetic field, following the right hand rule: Your thumb pointing towards the direction of movement of the positive charge, and your other fingers wrap around to indicate the direction of the magnetic field.

A circulating current forms a magnetic dipole.

Calculate Field

${\vec {B}}({\vec {r}})={\frac {\mu _{0}}{4\pi }}{\frac {I_{1}d{\vec {l}}_{1}\times {\hat {r}}}{r^{2}}}={\frac {\mu _{0}}{4\pi }}{\frac {\left|Idl\right|}{r^{3}}}\rho {\hat {e}}_{\phi }$

Where $\mu _{0}$ is magnetic permeability, and $\rho$ is distance from the wire.

Any component going along the direction of current is cancelled by cross product. Something else. Therefore, magnetic field is circulating.

Unlike electric fields, there is no beginning or end to magnetic fields. It is no surprise, then, that

$\oiint _{S}{\vec {B}}\cdot d{\vec {s}}=0$

$\oint {\vec {B}}\cdot d{\vec {l}}\neq 0$

Recall that a point charge produces an electric field that points radially outward based on Coulomb's Law. This results in Gauss's Law and circulation being zero.