First, consider a simple circuit comprising of an ideal battery of EMF: E and a single resistor of resistance R.
Imagine that the switch is opened in the circuit. In the battery, some internal mechanism drives (by applying a force) negative charges to one terminal, leaving positive charge on other terminal, at equilibrium an opposing electric field is generated inside battery. No further accumulation of charges takes place.
As soon as the switch is closed, electrons get another 'path' to go from terminal having higher potential to that having lower relative potential through the resistor. An electric field (conservative in nature) is set up across resistor.
Now, the work done by conservative fields (present inside the battery and across the resistor also) on any charge over a closed (or cyclic loop) path is ZERO. That's why I think Kirchhoff's Voltage Law works.
My problem is that why do we apply Kirchhoff's Law for Inductors as well?
In an inductor, an EMF is induced by changing magnetic flux. The EMF is simply the work done by the NON CONSERVATIVE electric field,(produced by changing magnetic flux in the inductor coil) on moving across the a particular path i.e across the inductor coil.
Since this field is non-conservative, according to my analogy given at the starting of the question, this work done across closed path on the charge shouldn't be zero. But in books I see that Kirchhoff's Law is applied here! How can this happen?
Any help will be appreciated.
Best Answer
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.