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My go-to resource for learning Electromagnetism thus far has been Griffiths, which has proven invaluable. On the section on induction, Griffiths makes a very clear distinction between what he refers to as "Faraday's Law", being that:

$\oint \vec E_{induced} \cdot \vec {dl} = -\iint \frac {\partial \vec B}{\partial t}\cdot \vec {dA}$

and what he refers to as "The Flux Law", being that:

$\oint \vec f \cdot\vec {dl} = -\frac {d \phi}{dt}$.

Here, $\vec E_{induced}$ refers to the induced electric field due to exclusively a changing B field (we assume here that the conducting loop is not moving relative to it), whilst I've instead used $\vec f$ for the more general case of the flux law (when we now relax this assumption) to denote what I can only describe as the "Lorentz-force per unit charge". Such is the source of my confusion: what is $\vec f$?

Some texts seem to imply that it's just an electric field and draw an equivalence to what Griffiths would call "Faraday's Law", but this suggests that $\vec f$ is solely the moving-coil generalisation of $\vec E_{induced}$, and that in order to find the total Lorentz-force on a given charge I would need to add the magnetic force separately. My headcanon is contrary to this idea, I understand that $\vec f$ refers to the total Lorentz-force on the charge due to the superposition of any "real" electric fields and induced E-fields (from what Griffiths would call "Faraday's Law"), as well as any forces directly from magnetic fields as well.

Could someone please clarify if I am understanding this correctly?

EDIT: I have an additional question that I'll append to this one because it's related:

I've seen Faraday's law in differential form given as:

$\nabla \times \vec E = -\frac{\partial B}{\partial t}$

Does $\vec E$ here refer to $\vec E_{induced}$ or $\vec f$? If it refers to $\vec f$ then this seems to suggest that whenever the B field is constant (i.e. RHS is zero), the resulting circulation of the Lorentz-force per unit charge ($\vec f$) over the conducting loop will be zero by Stoke's theorem. Thus, despite having a moving coil and therefore a changing flux, if the B field were constant we nonetheless get an emf of zero over the loop, which is wrong. I am therefore inclined to think $\vec E$ here is referring to $\vec E_{induced}$ but this seems inconsistent with what I've argued above about how $\vec E$ should refer to $\vec f$ for the case of the flux law in integral form.

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Yes, in general, you can decompose the time variation of the flux into two parts. The first one is due to time varying field assuming the loop to be fixed, given by Farady’s law. The second one is due to motion of the loop (Lie derivative) assuming the vueld to be time independent, giving rise to an additional term identified as Lorentz’ magnetic force.

For the flux law to work, you therefore need to take into account the full $f$, including the magnetic force: $$ f=E+v\times B $$ with $v$ the velocity of the loop (can depend on the position along the loop).

You can do the formal derivation: $$ \begin{align} \frac{d\phi}{dt} &= \frac{d}{dt}\int d^2x\cdot B\\ &= \int d^2x\cdot \partial_t B+L_v(d^2x\cdot B)\\ &= \int d^2x\cdot (\partial_t B+v\nabla\cdot B)+\int dx\cdot (B\times v)\\ &=-\int dx\cdot (E+v\times B) \end{align} $$ The tricky part is getting the Lie derivative of a 2-form, which has a geometrical representation. Your loop sweeps out a tube. The difference in flux can therefore be see as the flux though the closed surfed comprised of the “caps” fo the beginning and end and the lateral tube, minus the flux through the lateral tube. The first term over the closed surface is given by: $$ \int d^2x\cdot v\nabla\cdot B $$ Which is zero from Maxwell’s equation, while the second term over the lateral tube only is $$ \int dx\cdot (B\times v) $$ which is the emf of the magnetic force.

Btw, you can assume $v$ is perpendicular to the loop’s line element, due to the mixed product. This means that slithering the loop along itself will not generate a magnetic emf as expected. Only an actual motion of the loop matters.

$E$ is always the total electric field. Note that depending on your frame, the contribution due to induction and the contribution due to charges vary so such a separation is frame dependent.

Hope this helps

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