How can a coil self-induce voltage at $t=0$?

Question

Assume you have a resistor and an inductance in series. When a DC supply is connected at t=0, the coil is inducing an opposing voltage, which is the same as the DC supply's voltage. The current through the circuit at this time is zero.

But how can the coil at t=0 induce an opposing voltage without current? My understanding is that you need a changing magnetic field to induce a voltage, for which you need current. But at t=0 there is no current, so how does coil induce an opposing voltage?
Please help me out as I want to have deep understanding of circuits.

Ján Lalinský · Answer

Coil is not "inducing voltage", rather there is induced EMF being generated. Mobile charge carriers in the coil produce induced electric field, whose net total effect is quantified by so-called EMF (electromotive force).
Magnitude of this EMF is, due to Faraday's law, given by
$$
EMF = -Lfrac{dI}{dt}
$$
The important thing is how quickly current changes, not its value. In other words, if current $I$ is zero but its derivative $dI/dt$ is not (mobile charge carriers are accelerating from zero speed), EMF is present. There is no need for current or magnetic field to be present.
The induced electric field is produced whenever the mobile charge carriers accelerate. Whether they have big speed is irrelevant.

How can a coil self-induce voltage at $t=0$?

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