EC-2.2 Magnetostatics and Maxwell's Equations

What this is and why it exists

The magnetic half of the subject mirrors the electric half closely enough that most of the previous topic transfers directly.

One asymmetry matters. There is no magnetic charge. That single absence shapes one of the four equations, and it is worth noticing rather than accepting.

The vocabulary

• Biot-Savart law — the magnetic field contributed by each small piece of current.
• Ampere's circuital law — the field circulating around a closed path equals the current it encloses.
• Magnetic flux density — the field quantity whose flux through a closed surface is always zero.
• Self inductance — the flux a circuit links in itself, per unit of its own current.
• Mutual inductance — the flux one circuit links in another, per unit of the first circuit's current.
• Faraday's law — a changing magnetic flux drives a voltage around a loop.
• Displacement current — the term proportional to a changing electric field, added to Ampere's law.
• Integral form — an equation describing a whole region.
• Differential form — the same equation describing a single point.

The mental model

The first two laws repeat the judgement from the electric half. The Biot-Savart law adds up the contribution of every current element. It always works and it is laborious. Ampere's circuital law relates the field around a closed path to the current threading it. It takes one line, and only where the symmetry is high enough to make the field uniform along that path. Same choice, same reasoning.

Then the asymmetry. Magnetic field lines close on themselves. They have no starting point and no ending point, because no isolated magnetic pole has ever been found. So the flux out of any closed surface is exactly zero, always. That is one of the four equations. It states that a certain quantity is always zero, rather than relating two things to each other.

Inductance can be reached two ways, and they must agree. One route is flux linkage: how much flux a current produces through a circuit, divided by that current. The other route is energy: how much energy is stored in the field for a given current. For a shape you can integrate, the energy route is often shorter, and getting the same answer both ways is a good check.

Faraday's law is where the two fields stop being separate subjects. A changing magnetic flux through a loop produces a voltage around it. The minus sign says the effect opposes the change that caused it, which is why a magnet dropped through a copper tube falls slowly.

Displacement current is the last piece and the most interesting one. Ampere's law as first written fails on a charging capacitor. Take a path around the wire and stretch a surface across it, and the enclosed current is the wire's current. Stretch a different surface across the same path so that it passes between the capacitor plates, and no current crosses it at all. The same path gives two different answers, which cannot be right.

Maxwell added a term proportional to the changing electric field. Between the plates the field is changing exactly fast enough to make the two answers agree. That term was a correction on grounds of consistency, not an experimental result. It is also what makes a wave possible. A changing electric field now produces a magnetic one, and a changing magnetic field produces an electric one.

The four equations then exist in two forms. The integral form describes a region and is the one to reason with physically. The differential form describes a point and is the one to compute with. The two theorems from the previous topic convert between them. At an interface, all four reduce to rules about which components carry across, matching the electric pair you already have.

Read the four as four sentences rather than four formulas. Charge produces electric field. There is no magnetic charge. A changing magnetic field drives an electric one. Current and a changing electric field both drive a magnetic one. Read that way, the wave in the next topic is unsurprising.

What you should now be able to explain or do

Choose between the Biot-Savart law and Ampere's circuital law and say why. Explain why the magnetic flux out of any closed surface is zero. Find self and mutual inductance by flux linkage and by stored energy. State Faraday's law and explain its minus sign physically. Explain the charging-capacitor inconsistency and the term that removes it, and state all four equations as sentences.

Check yourself