core Estimated learning time: 10 h

EC-2.2 Magnetostatics and Maxwell's Equations

You can compute static magnetic fields and inductance for the standard arrangements, and write Maxwell's four equations in both forms while saying what each one asserts.

The magnetic half of the subject mirrors the electric half closely enough that most of the previous topic transfers, with one asymmetry that matters: there is no magnetic charge, so one of the four equations is a statement that a certain quantity is always zero. The four equations only become a theory of waves once the last term is added to them, and that term was a correction on consistency grounds rather than an experimental result. Reading them as four sentences rather than four formulas is what makes the wave in the next topic unsurprising.

Work through these

The Biot-Savart law and Ampere's circuital law, and which to reach for
One adds the contribution of every current element and always works; the other uses a closed path and works quickly wherever the symmetry is high. The choice is the same judgement as between adding charges and enclosing them.
NPTEL: Electromagnetic Fields · Course
Magnetic flux density, magnetic materials, and why there is no magnetic charge
Field lines close on themselves, so the flux out of any closed surface is zero and no isolated pole has ever been found. That absence is the asymmetry between the electric and magnetic halves of the theory.
Inductance, self and mutual, from stored energy or from flux linkage
Inductance can be found either from the flux one circuit links in another or from the energy stored in the field, and the two routes must agree. The energy route is often the shorter one for a shape you can integrate.
MIT OpenCourseWare 8.02: Physics II — Electricity and Magnetism · Course
Faraday's law: a changing magnetic field drives an electric field
A changing flux through a loop produces a voltage around it, and the minus sign says the effect opposes the change that caused it. This is the first equation in which the two fields stop being separate subjects.
NPTEL: Electromagnetic Theory · Course
Displacement current, and the inconsistency it was invented to remove
Ampere's law as first written fails for a charging capacitor, where a current arrives at a plate and none crosses the gap. Adding a term proportional to the changing electric field repairs it, and that term is what allows a wave.
The four equations in both forms, and the boundary conditions that come with them
Each equation has an integral form that describes a region and a differential form that describes a point, connected by the two theorems from the previous topic. At an interface they reduce to rules about which components carry across.
MIT OpenCourseWare 6.013: Electromagnetics and Applications · Course

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