Deduce Maxwell’s second equation in differential and integral form.

Maxwell's second equation is Gauss's law for magnetism.

Integral form: Magnetic field lines are always closed loops, because isolated magnetic poles (monopoles) do not exist. So every line that enters a closed surface also leaves it, and the net magnetic flux through any closed surface is zero:

\[ \boxed{\oint_S\vec B\cdot d\vec S=0} \]

Differential form: Apply Gauss's divergence theorem to the closed surface S enclosing volume V:

\[ \oint_S\vec B\cdot d\vec S=\int_V(\nabla\cdot\vec B)\,dV=0 \]

This holds for any arbitrary volume, so the integrand must itself vanish:

\[ \boxed{\nabla\cdot\vec B=0} \]

Significance: Magnetic monopoles do not exist, and \(\vec B\) is a solenoidal (divergence-free) field. So \(\vec B\) can be written as the curl of a vector potential: \(\vec B=\nabla\times\vec A\), consistent with \(\nabla\cdot(\nabla\times\vec A)=0\). Compare this with Maxwell's first equation, \(\nabla\cdot\vec D=\rho\), for electric charges.