# Maxwell's Equations

The Scottish scientist James Clerk Maxwell made a study of both Ampère’s and Faraday’s experiments. From Ampère’s writings Maxwell learned that, to be able to explain the force between current carrying conductors, Ampère proposed the hypothesis that magnetic material consists of microscopic current loops.

Faraday came with another concept. When playing with magnets and iron dust, the iron particles line up. That inspired Faraday to visualize electric and magnetic fields by means of field lines.

Maxwell wondered whether a strictly mathematical description could be given from the newly discovered electromagnetic phenomena. Maxwell focused at that time on fluid mechanics and asked himself if the electromagnetic phenomena were similar to what happens to a non-compressible fluid in media with a certain resistance.

Let us have a closer look at the world of electromagnetism. Electrons are the source of electric field lines. Moving electrons are the source of magnetic field lines. They form a vector field. Vector fields depend on space and time. At each point in space there exists one electric field vector and one magnetic field vector pointing in a certain direction. The length and the direction of the vectors change continuously.

To describe electrical field lines, we can fruitfully make use of Gauss law: the electric flux through any closed surface equals the charge enclosed by that surface:

(1)

To be able to calculate the electric and magnetic field at each and every point in space, Maxwell introduced the mathematical operator’s gradient

(2)

divergence

(3)

and curl

(4)

When we know the electric flux and its direction, we can calculate the electric charge or the charge density.

(5)

Electric field lines travel from a negative charge to a positive charge, so they have a starting and an ending point. Magnetic field lines, however, form a closed path around a current carrying conductor. Electric and magnetic field lines are set perpendicular to another.

What about the origin of magnetic field lines?  Ampère’s law learns that a current through a copper wire creates a magnetic field around that wire. With Ampère’s law we are able to calculate the current in the wire by taking the integral along a closed path around it.

(6)

The magnetic field strength at each point in space can be calculated with the formula:

(7)

This is what we call the curl of the magnetic field. The curl is a vector perpendicular to the surface formed by a closed path around a current carrying conductor.

Faraday discovered that a time varying magnetic field excites an electric field. This is called the curl of the electric field and it is a vector perpendicular to the surface formed by a closed path in space that encloses a time-varying magnetic flux.

(8)

One field excites another and the field vectors are perpendicular to each other.

Knowing these laws, we can return to the question that Maxwell asked himself: Can we describe electromagnetism with mathematical formulas?

Ohms law teaches us that a constant voltage across a resistor results in a constant current through that resistor.

(9)

From Ohm’s point of view a current can not pass through a capacitor because a capacitor consists in fact of two plates with an isolating medium in between. Maxwell reasoned that this was not the case and he introduced the displacement current.

The electric field and the electric flux D are both vectors and for the current in a conductor we can write:

The displacement current can be written as:

(10)

Maxwell described electromagnetism by means of four differential equations. The first equation describes the electric field from a charge or from a charge distribution .

(11)

The second equation learns that in nature no separate north- and south-poles exist. A bar magnet that is broken in two halves, will give two magnets each having a north- and a south-pole.

(12)

The third formula expresses Faraday’s law of induction. A time varying magnetic field excites elsewhere in space an electric field that exerts a force on charges and so it can induce current in a conductor. The minus sign represents Lenz’ law: the induced current creates a magnetic field that opposes the inducing magnetic field.

(13)

Equation number four is known as Ampère’s law and shows the relationship between electric and magnetic phenomena. A flow of moving charges through a conductor creates, in the space around that conductor, a magnetic field. Such a current can be treated as the source for that field, but also a displacement current through a capacitor can be a source.

(14)

Maxwell described the electromagnetic domain by means of a few laws. He published his findings in “A treatise on Electricity and Magnetism”. The two volumes are still for sale!

Albert Einstein called James Clerk Maxwell’s work as “ The most important event in physics since Newton’s time”.