Is it safe to apply Kirchhoff's voltage law to a closed loop containing an inductance with unsteady current? If I have a circuit that is just a battery in series with a resistor and an inductor, can I apply Kirchhoff's voltage law to that loop while the current has not reached its steady state value yet?
[Physics] Is it safe to apply Kirchhoff’s voltage law to a closed loop containing an inductance with unsteady current
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Therefore, current will increase, and voltage will also increase across the entire circuit.
It's not clear exactly what you mean by "across the entire circuit", but think about your model of a battery.
For a simple analysis that is usually used for circuits like this, the battery is considered as a constant voltage source. Therefore, by Kirchoff's Voltage Law,
$$V_b = V_l + V_r$$
where $V_b$, $V_l$ and $V_r$ are the voltages across the battery, the inductor, and the resistor respectively (and when you choose the same sign convention I did for each of them, but since you didn't bother to include a circuit diagram in your question, I don't feel obligated to provide one to indicate the sign conventions in my answer).
So if the voltage across the resistor increases, the voltage across the inductor must decrease.
It is not an issue of the field being conservative or not. Ultimately, Kirchhoff's laws are about the relationship between branch currents and node voltages in a network of lumped circuit elements. If you define three kinds of branch elements denoted by $R,C,L$ using the relationships $v=Ri$, $i=C\frac{dv}{dt}$, and $v=L\frac{di}{dt}$, respectively, then you may freely use Kirchhoff's current and voltage laws. These defining relationships between voltage and current are idealization and simplification not just for an inductor but also for a capacitor and resistor, as well. In the case of the inductor we ignore all fields outside the coil, and if we cannot because we have an inductive transformer then we include that part explicitly by defining a two-port with a pair of equations, such as $v_1=L_{11}\frac{di_1}{dt}+L_{12}\frac{di_2}{dt}$ and $v_2=L_{12}\frac{di_1}{dt}+L_{22}\frac{di_2}{dt}$, and a similar set of equations if you need more ports than two. If the capacitor is physically large then we may encounter problems with the current continuity law and will not be able to neglect the displacement current.
Note too that in no sense one could claim that the fields of a voltage or current generator are "conservative", not even for a battery: electrochemistry is not electrostatics. Somewhere, somehow you must impose a phenomenon that is outside of electricity or magnetism. Instead we postulate that certain node pairs have a predefined voltage history, and a given branch has a predefined current history independently of the rest of the circuit and thus represent a voltage or a current source, resp. In other words sources are time dependent boundary conditions. This way as you go around in a loop you must always get 0 voltage, no conservative field is needed. At the next level of abstraction you only need that in an arbitrary loop at any instant every connecting wire the current must be the same. And assuming linear superposition you can derive that the sum of branch currents at any node must be zero. So then the only questions is whether a loop is physically small enough so that the current uniformity holds. Once you have picked the defining lumped element equations between $v$ and $i$ you may say that KVL and KIL have more to do with network topology than actual physics.
Best Answer
Yes, Kirchhoff's voltage law (KVL):
Sum of voltage drops across all elements connected via perfect conducting wire in series in to a closed circuit is zero.
is valid for lumped element RLC circuits, so also for inductors (for currents that do not change too fast, so voltage can be measured in practice). In practical circuits designed not to radiate, voltage can be measured across any element and KVL can be validated experimentally. It is valid for common frequencies, up to hundreds of MHz and even higher to GHz range if parasitic elements are added to the model.
The whole theory of RLC circuits with harmonic voltage sources is derived from KVL being valid all the time, while currents and voltages change.
Some people say Kirchhoff's law is not valid for a circuit with an inductor, since $\oint \mathbf E \cdot d \mathbf s \neq 0$ if ideal inductor is in the circuit. However, that is actually not a problem for KVL, because KVL is formulated using voltage drops, not integrals of total electric field. Voltage drop across inductor may be non-zero, even if total electric field in the wire is zero, because the drop is defined not by integral of total electric field, but by integral of electrostatic component of that field.