Derive Gauss law for static electric and magnetic field in differential and integral form

Answer: for the electric field, \(\oint\vec E\cdot d\vec A = q/\varepsilon_0\) (integral form) and \(\nabla\cdot\vec E = \rho/\varepsilon_0\) (differential form). For the magnetic field, \(\oint\vec B\cdot d\vec A = 0\) and \(\nabla\cdot\vec B = 0\).

1. Gauss's law for the electrostatic field

Integral form. Consider a point charge \(q\) inside a closed surface S. The field at distance \(r\) is \(\vec E = \dfrac{q}{4\pi\varepsilon_0 r^{2}}\hat r\). The flux through a small area \(d\vec A\) is

\[d\phi = \vec E\cdot d\vec A = \frac{q}{4\pi\varepsilon_0}\cdot\frac{dA\cos\theta}{r^{2}} = \frac{q}{4\pi\varepsilon_0}\,d\Omega\]

where \(d\Omega\) is the solid angle subtended by \(dA\) at the charge. The total solid angle around a point inside a closed surface is \(4\pi\), so

\[\oint\vec E\cdot d\vec A = \frac{q}{4\pi\varepsilon_0}\times4\pi = \frac{q}{\varepsilon_0}\]

For a charge outside the surface, the flux entering equals the flux leaving, so it contributes zero. By superposition, for any charges, \(q\) is the total enclosed charge.

Differential form. Write the enclosed charge as \(q = \int_V\rho\,dV\), and convert the surface integral by the Gauss divergence theorem:

\[\oint\vec E\cdot d\vec A = \int_V(\nabla\cdot\vec E)\,dV = \frac{1}{\varepsilon_0}\int_V\rho\,dV\]

This holds for every volume, so the integrands are equal:

\[\nabla\cdot\vec E = \frac{\rho}{\varepsilon_0}\quad\text{or}\quad\nabla\cdot\vec D = \rho\]

2. Gauss's law for the magnetic field

Integral form. Isolated magnetic poles (monopoles) do not exist; every magnet has both N and S poles, and magnetic field lines form closed loops. Hence every line that enters a closed surface also leaves it, and the net magnetic flux is zero:

\[\oint\vec B\cdot d\vec A = 0\]

Differential form. By the divergence theorem, \(\int_V(\nabla\cdot\vec B)\,dV = 0\) for every volume, so

\[\nabla\cdot\vec B = 0\]

The magnetic field is solenoidal: it has no sources or sinks.