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If an self inductor is placed in a circuit and the current is a function of time then where is the emf induced: across the inductor or across the circuit?

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I think it would me more accurate to say that the EMF is generated across the circuit.

Saying that the EMF is generated across the inductor would imply that it could be measured as a voltage across the inductor, which is not the case, since the voltage across the incuctor would be affected by the inductor's parasitics.

For instance, if we take into account inductor's resistance, R, the voltage across the inductor connected to a battery would be $V=Vbat=L\frac{di}{dt}+iR=EMF+iR$.

If we say that the EMF is generated accross the circuit, we don't have that contradiction: the EMF will be treated as any other voltage source in the circuit.

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We could say either that the emf is induced in the inductor or in the circuit.

My own preference is to talk about the emf as induced in the circuit. Then one can use the very neat definition that the emf is the work done per unit charge on a charge per complete transit of the circuit. This distinguishes an emf from work done by or against forces due to static charge distributions, for the work done by these is zero for a charge transiting a closed loop.

Nonetheless it's very common to talk about the emf induced in an inductor, and in my opinion, not wrong to do so, because almost all the work done arising from the changing magnetic field, will be done in the inductor itself.

If we were talking about the emf that arises when we plunged a magnet into a coil we'd still get an emf around any imaginary closed loop taking the place of the coil. The coil needn't be there for us to talk about induced emfs! But in the case of an inductor you really do have to have the coil, to carry the changing current that produces the changing magnetic field!

One final small point: the preposition… We talk about emfs in, not across. The idea is that a notional charge has to go around the loop. It's not enough to say that it has to get from one terminal to the other: the route matters! [On the other hand it does make sense to talk about the potential difference between two points, or across a resistor, or whatever, because the pd between two points is independent of the route between the points. Pds are due to static or quasi-static charge distributions – which may arise as a result of emfs!]

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