Gauss's Law (Magnetic Fields)

Despite the best efforts of many many scientists, we have never found any evidence for the existence of magnetic charges i.e., magnetic monopoles. This means that magnetic field lines always close on themselves (even if only at infinity) and therefore we may immediately write

$\oint \hskip{-2mm}\oint _A \overset{\rightharpoonup }{B}\cdot d\overset{\rightharpoonup }{S}=0$

where again, the circle in the integral reminds us that the area of integration is closed. Similarly, using the divergence theorem we can write this in differential form

$\overset{\rightharpoonup }{\nabla }\cdot \overset{\rightharpoonup }{B}=0$