What are Maxwell's equations in point (differential) form and how are they derived?

In one line

Maxwell's four equations in point form are \(\nabla\cdot\vec{D} = \rho\), \(\nabla\cdot\vec{B} = 0\), \(\nabla\times\vec{E} = -\partial\vec{B}/\partial t\) and \(\nabla\times\vec{H} = \vec{J} + \partial\vec{D}/\partial t\), describing electric and magnetic fields at every point.

Explained simply

Integral forms describe what happens over a whole surface or loop, like the total traffic through a toll plaza. Point forms describe what happens at each individual point, like the traffic at one lane. They are the same laws, converted using the divergence and Stokes' theorems.

How it works

1. Gauss's law (electric): \(\oint\vec{D}\cdot d\vec{S} = \int\rho\,dV\). Divergence theorem gives \(\int(\nabla\cdot\vec{D})dV = \int\rho\,dV\), so \(\nabla\cdot\vec{D} = \rho\).
2. Gauss's law (magnetic): \(\oint\vec{B}\cdot d\vec{S} = 0\), since magnetic field lines are closed loops. Divergence theorem gives \(\nabla\cdot\vec{B} = 0\).
3. Faraday's law: \(\oint\vec{E}\cdot d\vec{l} = -\int\dfrac{\partial\vec{B}}{\partial t}\cdot d\vec{S}\). Stokes' theorem gives \(\nabla\times\vec{E} = -\dfrac{\partial\vec{B}}{\partial t}\).
4. Ampere-Maxwell law: \(\oint\vec{H}\cdot d\vec{l} = \int\left(\vec{J} + \dfrac{\partial\vec{D}}{\partial t}\right)\cdot d\vec{S}\). Stokes' theorem gives \(\nabla\times\vec{H} = \vec{J} + \dfrac{\partial\vec{D}}{\partial t}\).
5. Constitutive relations (free space): \(\vec{D} = \varepsilon_0\vec{E}\), \(\vec{B} = \mu_0\vec{H}\).
6. In free space with no charge or current (\(\rho = 0, \vec{J} = 0\)): \(\nabla\cdot\vec{E} = 0\), \(\nabla\cdot\vec{B} = 0\), \(\nabla\times\vec{E} = -\partial\vec{B}/\partial t\), \(\nabla\times\vec{B} = \mu_0\varepsilon_0\,\partial\vec{E}/\partial t\).

Key formula

Symbols: \(\vec{D}\) electric displacement (C/m²), \(\rho\) charge density (C/m³), \(\vec{B}\) magnetic flux density (T), \(\vec{E}\) electric field (V/m), \(\vec{H}\) magnetic field intensity (A/m), \(\vec{J}\) conduction current density (A/m²), \(\partial\vec{D}/\partial t\) displacement current density (A/m²).

Quick example

Check a field: is \(\vec{B} = x\hat{i} - y\hat{j}\) a possible magnetic field? \(\nabla\cdot\vec{B} = 1 - 1 = 0\), so yes, it satisfies Maxwell's second equation. \(\vec{B} = x\hat{i} + y\hat{j}\) has divergence 2, so it cannot be a magnetic field.

Common mistakes

• Using the divergence theorem for the curl equations or Stokes' for the divergence equations.
• Leaving out \(\partial\vec{D}/\partial t\) in the fourth equation.
• Writing \(\nabla\cdot\vec{B} = \rho\); it is always zero.

How MU asks it

A very common 5-mark question: "Derive Maxwell's equations in differential form" or "Write Maxwell's equations in integral and differential form". For full marks, list all four with names, show the theorem used for each conversion, and define every symbol. A 3-mark variant asks only to write them for free space.