Editing Classical field theory (section)

==== Magnetostatics ====
{{Main|Magnetostatics}}

A steady current ''I'' flowing along a path ''ℓ'' will exert a force on nearby charged particles that is quantitatively different from the electric field force described above. The force exerted by ''I'' on a nearby charge ''q'' with velocity '''v''' is
<math display="block">\mathbf{F}(\mathbf{r}) = q\mathbf{v} \times \mathbf{B}(\mathbf{r}),</math>
where '''B'''('''r''') is the [[magnetic field]], which is determined from ''I'' by the [[Biot–Savart law]]:
<math display="block">\mathbf{B}(\mathbf{r}) = \frac{\mu_0 I}{4\pi} \int \frac{d\boldsymbol{\ell} \times d\hat{\mathbf{r}}}{r^2}.</math>

The magnetic field is not conservative in general, and hence cannot usually be written in terms of a scalar potential. However, it can be written in terms of a [[magnetic vector potential|vector potential]], '''A'''('''r'''):
<math display="block"> \mathbf{B}(\mathbf{r}) = \nabla \times \mathbf{A}(\mathbf{r}) </math>

[[Gauss's law]] for magnetism in integral form is
<math display="block">\iint\mathbf{B}\cdot d\mathbf{S} = 0, </math>
while in differential form it is
<math display="block">\nabla \cdot\mathbf{B} = 0. </math>

The physical interpretation is that there are no [[magnetic monopole]]s.