$$E\text{(induced emf)}=-\frac{d\Phi}{dt}\tag{1}$$

Here: $$\Phi$$= magnetic flux across the surface enclosed by the loop.

## Question:

Q1- Is the $$\Phi$$(magnetic flux) in (1) the total flux (net $$\Phi$$)

A situation to clarify Q1
A bar magnet's $$N$$-pole is brought closer to a "CircularLoop(CL)" of wire at a constant velocity. So, $$\Phi$$ across the CL increases. An emf is produced in CL which produces a magnetic field opposing the bar magnet's N-pole. Now their will be some $$\Phi$$(flux) due to this opposing magnetic field and will decrease the (net $$\Phi$$ ) across the CL.

So two things:

1. Bar magnet comes closer, $$\Phi$$ increases across CL
2. Induced emf creates Opposing Magnetic Field(OMF), and $$\Phi$$ due to OMF results in decrease of $$\Phi$$ across the CL

So, in (1) above, do we take net flux (net $$\Phi$$)? So, Faraday's law is that if there is change in net flux ($$\Phi$$ net) then emf is induced?

Q2- If your answer to Q1 is "yes we take net flux", then why do i see that in so many derivations, the $$\Phi$$ due to OMF is ignored and Faraday's law is applied by considering just the change in $$\Phi$$ due to External Magnetic Field?
For eg, this derivation for motional emf:

P.S:

2. I have tried my best to make this post readable by making abbreviations along with their full forms in the text itself.

It is, as I think you have decided, the net flux that is responsible for the net induced emf. However, we can write the net emf as $$\mathscr E = -\frac{d\Phi}{dt}=-\frac{d(\Phi_{ext}+\Phi_{ind})}{dt} = -\frac{d\Phi_{ext}}{dt}+\left(-\frac{d\Phi_{ind}}{dt} \right)= \mathscr E_{ext}+\mathscr E_{ind}$$ Here, $$\Phi_{ext}$$ is the flux from outside the circuit and $$\mathscr E_{ext}$$ the emf that can be attributed to the rate of change of that flux. $$\Phi_{ind}$$ is the flux arising from the induced current at some instant and $$\mathscr E_{ind}$$ is the emf that can be attributed to the rate of change of that flux.
In many textbook treatments, $$\mathscr E_{ind}$$ is not at first mentioned. It is often dealt with later under the heading of self induction, an effect that comes into play only when the circuit is complete.
• (a) Thank you. (b) Wrong? That's a bit strong. The 'partitioning' of flux and emf in the manner I have outlined is so widespread, that I doubt if many would object to calling $Blv$ the motional emf even with $B$ as the external flux density. Sep 9, 2021 at 11:40