Why is Kirchhoff's second law commonly used for circuits involving inductivity? I have often seen in introductory literature, e.g. for the activation process of a circuit with a power source which delivers the voltage $U$, current $I$, resistance $R$ and an inductivity $L$, that Kirchhoff's second law is frequently used for deducing e.g. $I(t)$ of the process. I cite (translate) from the script of my professor last term:
"We determine the direction of the current and add all the voltages positively and substract all the voltage drops, i.e. $U(t)-RI(t)+U_{ind}(t)=0$. Before turning the circuit on, $U=I=0$, afterwards $U(t)=U_0$, which leads to the differential equation $L\frac{dI(t)}{dt}+RI(t)=U_0$ for $I(t)$."
which basically is Kirchhoff's second law. I'm perfectly fine with the calculations but why may we use Kirchhoff's second law? It follows from the conservativity of the electric field. But when we have an inductivity involved in a circuit, we face, by the law of induction,
$$\oint \vec{E} \cdot d\vec{r} = - \frac{d\Phi}{dt} \neq 0$$
explicitly that $\vec{E}$ is NOT conservative, which actually forbids us to use Kirchhoff's second law, doesn't it?
 A: In circuits with nontrivial inductance, Kirchoff's law of voltages is precisely an expression of Faraday's law,
$$\oint \vec{E} \cdot d\vec{r} +\frac{d\Phi}{dt} =0.$$
The first part are the resistive and capacitive voltages as would be measured in an electrostatic setting, as well as the EMF of any sources in the circuit. The second term, ${d\Phi}/{dt}$, gives the inductive EMF from all the inductors in the circuit. This you can then model appropriately but its origin is directly from the Faraday law.
Thus, for example, you might neglect the inductance of the circuit's overall loop (which is usually very small), and then you can split up the contribution to the flux derivative from the various inductive elements in the circuit. Generally, these flux derivatives are given by a self-inductance term, $$\frac{d\Phi}{dt}=L\frac{dI}{dt},$$ as well as a sum of mutual inductance terms with (in principle all) other circuits in the world, from which you usually neglect all except the important ones. This gives you a simplified expression suitable for use as Kirchoff's voltage law which still emanates from the Faraday law.
A: Inside an ideal conductor, $\vec E=0$ has to hold. The idealized inductor is usually taken to be an ideal conductor, so there is actually no voltage drop in the sense of $\int\mathrm dr\cdot\vec E$ across an ideal inductor. In that sense you are right that Kirchhoff's loop rule $\oint\mathrm dr\cdot\vec E=0$ is not applicable in the presence of inductivities.
What people usually do it to bring the RHS of Faraday's law over to the LHS and interpret the $L\frac{\mathrm dI}{\mathrm dt}$ term as some sort of voltage difference due to the inductor, so they can salvage their beloved Kirchhoff loop rule.
Some people have rather strong opinion on whether or not this is justified. E.g., Walter Lewin of MIT OpenCourseWare fame rejects Kirchhoff's loop rule entirely and stresses this in his lectures quite often ("The hell with Kirchhoff"). I guess, as long as you end up with the correct ODE, it's a matter of philosophy.
