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It's well known among physics people that the motional emf generated in a loop of wire is minus the time derivative of the magnetic flux through the loop:

\begin{equation} \varepsilon = -\frac{d\Phi}{dt}. \end{equation}

But if an emf is generated in a conducting loop of wire, then there'd be a current which would result in a magnetic field which would induce a counter emf. Is my understanding correct? If so, how does one keep track of the net emf in a conducting loop of wire when emfs create counter emfs which create counter counter emfs, etc., etc.?

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I assume you are refering to a moving circuit in a fixed magnetic field when you are talking about a 'motional emf'. This emf is induced by the change of magnetic flux through a loop. The induced emf will oppose the change of the magnetic flux that generated it, i.e. it will induce a magnetic field which is opposed to the external field responsible for the change in magnetic flux.

This seems to be the point where you get confused. The induced emf generates a field and not a constantly changing magnetic flux. But as you stated in the formula, only a change in magnetic flux will result in an emf. A constant magnetic field does not induce an additional emf. Your title question is "Does the magnetic field due to the current from a motional emf induce a counter emf?" The answer is "Only a change in magnetic flux can induce an emf."

So for the bookkeeping of the net emf, it is only required to track the net change in the magnetic flux. In most simple examples, you will either have a constant magnetic field or a constant area of the loop. If you have a fixed magnetic field, keep track of the change of the area, and vice versa.

More on the moving circuit and the resulting emfs can be found here. You might also want to read up on 'inductance' and Lenz's law.

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    $\begingroup$ It takes a finite amount of time for the magnetic field due to the induced emf to appear so there's a changing magnetic flux (constant magnetic field + emf magnetic field), no? $\endgroup$ – PiKindOfGuy Apr 17 '18 at 8:34
  • $\begingroup$ Yes, that is true. For the time the field needs to build up, you have a changing magnetic flux. This is a very short time span though but you are correct. It will slow down the build-up of the magnetic field but it will not reverse it. So there is no back and forth that you have to keep track of. Additionally, those flux changes are most likely a lot smaller than the flux change from the external magnetic field and hence negligible. $\endgroup$ – lmr Apr 17 '18 at 8:50

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