What does Gauss's law of electrostatics state and how is its point form obtained?

In one line

Gauss's law states that the total electric flux through any closed surface equals the net charge enclosed divided by \(\varepsilon_0\); in point form, \(\nabla\cdot\vec{E} = \rho/\varepsilon_0\).

Explained simply

Think of a balloon around a light bulb: however you squeeze or stretch the balloon, the total light passing through its skin depends only on the bulb inside, not on the balloon's shape. Similarly, the total electric field "flowing out" through a closed surface depends only on the charge inside it. Charges outside contribute equal inflow and outflow, so they add nothing.

How it works

1. Integral form: \(\oint_S\vec{E}\cdot d\vec{S} = \dfrac{Q_{enc}}{\varepsilon_0}\).
2. Write the enclosed charge using volume charge density: \(Q_{enc} = \int_V\rho\,dV\).
3. Apply the divergence theorem to the left side: \(\oint_S\vec{E}\cdot d\vec{S} = \int_V(\nabla\cdot\vec{E})\,dV\).
4. So \(\int_V(\nabla\cdot\vec{E})\,dV = \int_V\dfrac{\rho}{\varepsilon_0}\,dV\) for every volume, which means the integrands are equal.
5. Point (differential) form: \(\nabla\cdot\vec{E} = \rho/\varepsilon_0\), or with \(\vec{D} = \varepsilon_0\vec{E}\), \(\nabla\cdot\vec{D} = \rho\).
6. This is Maxwell's first equation: charges are the sources and sinks of electric field lines.

Key formula

\(\oint\vec{E}\cdot d\vec{S} = \dfrac{Q_{enc}}{\varepsilon_0}\); \(\nabla\cdot\vec{D} = \rho\). \(\vec{E}\): electric field (V/m or N/C), \(Q_{enc}\): enclosed charge (C), \(\varepsilon_0 = 8.85\times10^{-12}\) C²/N m², \(\rho\): charge density (C/m³), \(\vec{D}\): electric displacement (C/m²).

Quick example

Field of a point charge \(q\): take a sphere of radius \(r\) around it. \(E\) is radial and equal everywhere on the sphere, so \(E\times4\pi r^2 = q/\varepsilon_0\), giving \(E = \dfrac{q}{4\pi\varepsilon_0 r^2}\), which is Coulomb's law. For \(q = 1\) nC at \(r = 0.1\) m: \(E = 9\times10^9\times10^{-9}/0.01 = 900\) V/m.

Common mistakes

• Including charges outside the surface in \(Q_{enc}\).
• Thinking \(\vec{E}\) on the surface depends only on enclosed charge; the flux does, the field does not.
• Writing \(\nabla\times\vec{E} = \rho/\varepsilon_0\) instead of divergence.

How MU asks it

Asked as "State Gauss's law and derive its differential form" (5 marks), usually as part of Maxwell's equations. State the law in words, write the integral form, use the divergence theorem step by step, and finish with \(\nabla\cdot\vec{D} = \rho\).