Different EMFs around different paths

[Martinius]

Hello everyone.
The definition of EMF, given in "Introduction to Electrodynamics" by David J. Griffiths, is the line integral of force per unit charge around a closed path in the circuit. However, it's pretty clear that this definition is suitable only for infinitely thin wires, in which only one path is possible. For situations in which several (or an infinite number of) paths are possible (e.g. a wire with non-zero diameter), what should be done? Is it some sort of averaging (like integrating EMFs of different paths over the wire's cross section and dividing it by the area)?
Same question arises in case of a Faraday wheel rotating in a non-uniform magnetic field, which would make the EMF integral path-dependent, and this can't be ignored since we know that in this device, the current is somehow distributed throughout the rotating disk, and when the field is uniform, the EMF becomes path-independent.
Any answers would be appreciated.

Regards,
Martinius

EDIT: I understand that as wire thickness increases, Eddy currents resulting from unevenly distributed force per unit charge (and therefore EMF) become more significant, but I'm interested in a more quantitative discussion of these currents and their effect on the overall EMF which generates a current in the entire circuit.

 

[vanhees71]

There's a lot of confusion in the literature concerning Faraday's Law. First of all, the line integral of electromotive force (EMF) is not bound to conductors (wires), but it's a functional of the electric field, via

$$\Phi_{\mathcal{C}}[\vec{E}]=\int_{\mathcal{C}} \mathrm{d} \vec{x} \cdot \vec{E}(t,\vec{x}),$$

where $$\mathcal{C}$$ is an arbitrary closed path, no matter whether it goes through conductors, insulators, or the vacuum.

Secons, the correct equations of motion are Maxwell's equations in differential form. In macroscopic electrodynamics often the constitutive (material) equations are, however, given in a non-relativistic form, and this often leads to confusion in the context of moving media in magnetic fields, particularly Faraday's disk (homopolar generator). Here, a current is induced due to the Lorentz force acting on the electrons of the conducting disk, where to very good approximation one can assume the electron's velocity given by the motion of the whole conductor. This is discussed thoroughly in many articles on this topic in the American Jornal of Physics, e.g., in

Crooks et al, Am. J. Phys. 46, 729 (1978)

 

[Antiphon]

In free space the EMF part of the electric field is due to the time-changing magnetic field H. Since any other fields are the gradient of a scalar, the line integral for EMF will integrate all those to zero. Only the E field due to nonvanishing curl remains and that's the EMF.