In one line
In integral form, Maxwell's equations relate the flux of \(\vec{D}\) and \(\vec{B}\) through closed surfaces, and the circulation of \(\vec{E}\) and \(\vec{H}\) around closed loops, to the charges, currents and changing fields that produce them.
Explained simply
Each equation is a "bookkeeping rule" for a region of space: (1) electric field lines start and end on charges, (2) magnetic field lines never start or end, (3) a changing magnetic field drives a circulating electric field, and (4) currents and changing electric fields drive a circulating magnetic field.
How it works
- Gauss's law for electricity: \(\oint_S\vec{D}\cdot d\vec{S} = Q_{enc} = \int_V\rho\,dV\). The net electric flux out of a closed surface equals the free charge inside.
- Gauss's law for magnetism: \(\oint_S\vec{B}\cdot d\vec{S} = 0\). Net magnetic flux through any closed surface is zero; isolated magnetic poles do not exist.
- Faraday's law: \(\oint_C\vec{E}\cdot d\vec{l} = -\dfrac{d}{dt}\int_S\vec{B}\cdot d\vec{S}\). The emf around a loop equals the negative rate of change of magnetic flux through it.
- Ampere-Maxwell law: \(\oint_C\vec{H}\cdot d\vec{l} = \int_S\left(\vec{J} + \dfrac{\partial\vec{D}}{\partial t}\right)\cdot d\vec{S}\). The magnetomotive force around a loop equals conduction current plus displacement current through it.
| Equation | Integral form tells | Point form |
|---|---|---|
| I | Charges are sources of \(\vec{D}\) | \(\nabla\cdot\vec{D} = \rho\) |
| II | No magnetic monopoles | \(\nabla\cdot\vec{B} = 0\) |
| III | Changing \(\vec{B}\) gives \(\vec{E}\) | \(\nabla\times\vec{E} = -\partial\vec{B}/\partial t\) |
| IV | Current and changing \(\vec{D}\) give \(\vec{H}\) | \(\nabla\times\vec{H} = \vec{J} + \partial\vec{D}/\partial t\) |
Quick example
A transformer is equation III in action: alternating current in the primary creates changing flux in the core, which induces an emf in the secondary. A bar magnet is equation II: however you draw a closed surface around one pole, as many field lines enter as leave.
Common mistakes
- Writing the right side of Gauss's law for magnetism as a pole strength; it is zero.
- Mixing open and closed surfaces: equations I and II use closed surfaces, III and IV use open surfaces bounded by a loop.
- Omitting the physical meaning, which carries marks.
How MU asks it
Asked as "Write Maxwell's equations in integral form and explain their physical significance" (5 marks). Write the four equations with names, one or two sentences of meaning each, and preferably a table linking integral and point forms. Restrict yourself to Cartesian coordinates as per the syllabus.