Electricity and magnetism are intricately related, leading to the adoption of the term ​electromagnetism​ to describe associated phenomena. In fact, the extent to which this is true largely eluded scientists until the latter half of the 1800s, when James Clerk Maxwell, building on the work of esteemed physicists before him, produced his famous set of four differential (calculus) equations tying together various properties of magnetic fields and electric fields.

Understanding ​magnetic flux​, or the magnetic field lines passing through a defined geometric plane called a ​vector area​, leads to several important physical phenomena, including ​electromagnetic induction​, or the generation of electromotive force (EMF).

Total magnetic flux is essentially a measure of ​how many magnetic field lines are passing through a given surface area A​ – that is, a measure of the strength of the magnetic field. More formally it is defined as:

where θ is the angle between the magnetic field B and ​the perpendicular to A​ at the defined region.

• The magnetic field B, or the ​magnetic flux density per unit area​, is measured in tesla (T) in SI units, while A is the area the field is passing through in m². The SI unit of magnetic flux is the weber (Wb), where Wb = T⋅m².

If B is not uniform across the surface of A, the calculus definition is that Φ = ∫B⋅dA. This surface integral function means that the flux values through almost infinitely small portions of A are determined independently and summed together to get a composite value.

​Gauss's law:​ ​The net magnetic flux through a closed surface​ ​is 0​. This is the second of Maxwell's equations, and it is consistent with the idea that there are no magnetic monopoles.

No matter how small a volume you choose, a magnetic field can always be described as including a dipole, or a tiny invisible bar magnet. This contrasts with electric fields, which are generated by point charges (or arrays of isolatable point charges).

​Faraday's law of electromagnetism:​ Induced ​electromotive force​ (EMF) in a coil of wire with N turns is N multiplied by the change in in flux with time:

Flux can be changed in time by varying B, altering the cross-sectional area A, or changing the angle between B and A by rotating the coil or field source.

• EMF has units of voltage (potential difference), not force. It's dubbed a "force" because voltage is what induced charges to move, producing current, in the first place.

​Lenz's law:​ Induced electric current flows in a direction that opposes the change that caused it. For example, say you have coil of wire not connected to any power source.

Imagine moving a bar magnet lengthwise into the middle of the coil along its axis, like inserting a rod right down the middle of a long tube without touching the sides of the tube. This increased field in the coil induces current to flow in a direction such that it generates magnetic field opposing the increase.

If you repeat this procedure after swapping the south pole and north pole ends of the magnet, the change produced is equal in magnitude and opposite in direction compared to the first case, and current will flow in the opposite direction as a result.