# Geometry of magnetic flux in EM induction

Image of surface, shown in blue, through which magnetic flux passes, indicating the boundary in red, in motion at velocity v:

In the Wikipedia article on Magnetic Flux, it is said that an EMF is induced around the boundary of an open surface transiting a magnetic field such that the magnetic flux is normal to that surface. An accompanying diagram shown here explains the various relations, indicating the surface boundary in red and showing that such flux is generally aligned with the B-field in this case.

By Faraday's Law, the effect in which EMF arises is described as the work required to move a charge along that boundary, in combination with the motion of the boundary (including its changes in orientation).

So, in a simple 2-pole scenario of EM induction in which a copper loop is rotated through respective B-fields of the North and South poles, what is the disposition or orientation of such a surface and its boundary with respect to the cylindrical geometry of the copper conductor? This is the main question being asked here. In other words, is the direction of motion of the boundary at velocity v generally that of the rotating loop?

Secondarily, if for instance the surface of that geometrically cylindrical conductor exposed to the 'magnetic flux' arising from one pole is akin to that shown in the diagram so that the boundary is an ovoid (with its long axis in the longitudinal direction of the conductor) around which an electron is moved by the magnetic force implicit in that field, then is that evidently oscillatory motion of the charge/electron around that boundary the basis of AC?

What then is its physical relation to a corresponding surface and boundary simultaneously transiting the other pole? And would it then be generally correct to describe that relation between remote effects in the conductor at the two poles in terms of a phase relation between respective 'electron wave functions', for example by the inference of a phase factor in the exponent of that function?

• Maybe draw a picture of what scenario you propose. Commented Jan 13, 2023 at 16:38
• @jeremiah it would be better if you added a picture rather than requiring readers to follow a link. Commented Jan 13, 2023 at 17:18
• What is a "conduit"? What "N and S poles" -- is this around a magnet of some description and geometry? Commented Jan 13, 2023 at 18:02
• @jeremiah "conduit" has a very different meaning than "conductor". especially in the field of electrical engineering. Commented Jan 13, 2023 at 18:05
• what is the disposition of such a surface and its boundary with respect to the cylindrical geometry of the conduit? What are you asking here, by "disposition" do you mean what outlines the shape? Or where the shape is, how it's shaped in general? (As it happens, the field equations don't care where the surface is, given certain continuity etc. conditions; its perimeter is sufficient information.) What do you mean by "cylindrical geometry"? It's a loop of wire, not a cylinder. Do you mean the (generally cylindrical, albeit bent into a loop) shape of the wire itself? Commented Jan 13, 2023 at 18:09

This is the answer to the main part of the question which was found at the 'Wikipedia' entry for 'Faraday's Law of Induction'. I had suspected but not understood that the boundary in the scenario outlined, of simple EM induction in a single loop transiting a magnetic field (as in a 2-pole generator) is the entire loop itself. That Wikipedia article states;

For a loop of wire in a magnetic field, the magnetic flux ΦB is defined for any surface Σ whose boundary is the given loop. Since the wire loop may be moving, we write Σ(t) for the surface. The magnetic flux is the surface integral:

ΦB=∬Σ(t) [B(t)⋅dA],

where dA is an element of surface area of the moving surface Σ(t), B is the magnetic field, and B · dA is a vector dot product representing the element of flux through dA. In more visual terms, the magnetic flux through the wire loop is proportional to the number of magnetic field lines that pass through the loop.

When the flux changes — because B changes, or because the wire loop is moved or deformed, or both — Faraday's law of induction says that the wire loop acquires an emf, defined as the energy available from a unit charge that has traveled once around the wire loop. (Although some sources state the definition differently, this expression was chosen for compatibility with the equations of special relativity.)

In fact, this expression is taken from Chapter 17 of "Feynman's Lectures on Physics [Vol.2]", which begins as follows;

In the last chapter [Chapter 16] we described many phenomena which show that the effects of induction are quite complicated and interesting. Now we want to discuss the fundamental principles which govern these effects. We have already defined the emf in a conducting circuit as the total accumulated force on the charges throughout the length of the loop. More specifically, it is the tangential component of the force per unit charge, integrated along the wire once around the circuit. This quantity is equal, therefore, to the total work done on a single charge that travels once around the circuit.

We have also given the “flux rule,” which says that the emf is equal to the rate at which the magnetic flux through such a conducting circuit is changing.

Thanks once again to the redoubtable 'Mr Feynman' -- surely he's joking (It's the typically facetious title of one of his books).