Why can voltages be summed around a circuit if there is magnetic induction?

Question

So there is this class I'm attending on Electronic Instrumentation which involves mostly circuit analysis. For example, RCL circuits, and circuits that also contain opamps, transistors, diodes, etc. Now when all the currents through these circuits are constant I understand we can talk about potential differences across various electrical components, because the changing of the total magnetic flux is zero, therefore all electric fields are conservative, and consequently a potential function can be defined relative to a chosen ground/reference point of that circuit, in other words, Faraday's law becomes under constant currents: $$\oint \overrightarrow{E}\cdot\overrightarrow{dr}=0$$ Which is exactly Kirchhoff's voltage loop law and the teachers use this law to sum up the voltages/potential differences across all the components, such as resistors, diodes and capacitors. However, the confusion arises when the teachers still sum up over all the 'potential differences' when the magnetic flux change is not zero anymore, for instance, when an inductor is present. I understand they essentially use Faraday's law in the following way and still call it Kirchoff's voltage loop law: $$\oint \overrightarrow{E}\cdot\overrightarrow{dr}+\frac{d}{dt}\iint \overrightarrow{B}\cdot \overrightarrow{dA}=0$$ Which becomes, for electric circuits with inductors: $$\sum\Delta V+\sum LI=0$$ Where the teachers speak of $\Delta V$ as the voltage or 'electric potential difference', but how can this be called a potential difference if in this scenario no potential function can be defined? Furthermore, the frequent expressions for the voltages across resistors and capacitors are still used for computing $\Delta V$, namely, $R=VI$ and $CV=Q$, which I don't understand, since these expressions were defined for electro static or steady state current scenario's. Moreover, the teachers still use the concept of voltage difference between various nodes in a circuit, even though there are changing magnetic fields present, implying the work done per unit charge depends on the path taken, thus the notion of defining work per unit charge between merely two points becomes nonsensical. So in an attempt to encapsulate my confusion, I don't understand how we can still speak of voltages/potential differences between 2 points in a circuit when the electric fields present are non-conservative, additionally, how come we can still use the same mathematical expressions (defined for conservative electric fields) for the voltages across components such as capacitors, resistors, transistors and diodes?

Answer 1

This is a perennial problem in physics education, similar questions appear quite often here. It is caused by lack of understanding of the concept of electromotive force, misunderstanding of Kirchhoff's second circuital law, double meaning for the term "voltage" used in practice and poor understanding of AC circuit theory. Some of these are also in your question.

Faraday's law becomes under constant currents: $$\oint \overrightarrow{E}\cdot\overrightarrow{dr}=0$$ Which is exactly Kirchhoff's voltage loop law and the teachers use this law to sum up the voltages/potential differences across all the components, such as resistors, diodes and capacitors.

No, this is not Kirchhoff's second circuital law, nor Kirchhoff's Voltage Law (KVL) as sometimes called in modern English literature.

The integral above being zero for all paths is a special situation valid in electrostatics.

Kirchhoff's second law is more general, it is valid even when electric field changes in time and is non-conservative and has induced field component. Its original formulation states (paraphrasing):

For any simple closed path (loop) made of conducting elements $k=1..N$, sum of all terms $R_k I_k$, where $I_k$ is current in element $k$, and $R_k$ is its ohmic resistance, equals sum of all electromotive forces $\mathscr{E}_i$, $i=1..M$, acting on current in the loop:

$$ \sum_{k=1}^N R_kI_k = \sum_{i=1}^M \mathscr{E}_i. $$

Notice that this law nowhere refers to potential or voltage. It refers to $currents$ and $emf$'s.

This law can be viewed as generalization of Ohm's law for a circuit element that states potential difference equals current times resistance; the generalization is in going from one simple element to a loop, and in replacing the potential difference by an electromotive force in loop, a more general concept.

It is quite common to rephrase the original Kirchhoff's second law in terms of potential differences: the sum of potential drops when making a full path is zero:

$$ \sum_{k=1}^N \Delta V_k = 0. $$ where $\Delta V_k$ is drop of electric potential $\varphi$ when going from one terminal of element $k$ to another in the chosen positive sense of circling.

Sometimes, the word "voltage" is used instead (in the sense "drop of potential"), and the law is called Kirchhoff's voltage law. Strictly speaking, this formulation is less informative than the original formulation, but it is still useful and it is generally correct.

The reason is that electric potential $\varphi$ is defined (even in situations when there is induced electric field) as the Coulomb potential due to all charges present, i.e. the potential associated with the electrostatic field component $\mathbf E_s$ of total field $\mathbf E$. Such potential is single-valued function of position and sum of drops of this quantity in any closed loop in space has to be zero.

However, the confusion arises when the teachers still sum up over all the 'potential differences' when the magnetic flux change is not zero anymore, for instance, when an inductor is present. I understand they essentially use Faraday's law in the following way and still call it Kirchoff's voltage loop law: $$\oint \overrightarrow{E}\cdot\overrightarrow{dr}+\frac{d}{dt}\iint \overrightarrow{B}\cdot \overrightarrow{dA}=0$$ Which becomes, for electric circuits with inductors: $$\sum\Delta V+\sum LI=0$$ Where the teachers speak of $\Delta V$ as the voltage or 'electric potential difference', but how can this be called a potential difference if in this scenario no potential function can be defined?

There are two things. Yes, the argument is wrong. But not because potential cannot be defined. Potential can be defined: the standard way in both DC and AC circuits is the potential associated with electrostatic part of total field.

The obvious problem with the argument above is that it is inconsistent: it pretends initially that all $\Delta V$'s in a closed circuit are given by integral of total electric field, but then conveniently "forgets" this assumption for the inductor: initially, it would assign zero $\Delta V$ to a perfect inductor, since total electric field in coils of a perfect conductor is zero(due to zero resistance). But this would not produce the correct term $LdI/dt$ in the equation , so the argument then conveniently invents "another" $\Delta V = L\frac{dI}{dt}$ associated with the inductor that does not obey the assumption that $\Delta V$ = integral of electric field.

In a correct analysis, $\Delta V$ for any two-terminal component has only one value, and for perfect inductor, this is indeed $L\frac{dI}{dt}$. This is possible because $\Delta V$ is not given by integral of total electric field (which would produce zero), but by integral of electrostatic component of total field (which produces the correct result). We do not need to "derive" KVL from Faraday's law, because KVL is an independent law. Faraday's law is useful insofar it allows us to find that for perfect inductor, $\Delta V = L\frac{dI}{dt}$.

Furthermore, the frequent expressions for the voltages across resistors and capacitors are still used for computing $\Delta V$, namely, $R=VI$ and $CV=Q$, which I don't understand, since these expressions were defined for electro static or steady state current scenario's. Moreover, the teachers still use the concept of voltage difference between various nodes in a circuit, even though there are changing magnetic fields present, implying the work done per unit charge depends on the path taken, thus the notion of defining work per unit charge between merely two points becomes nonsensical. So in an attempt to encapsulate my confusion, I don't understand how we can still speak of voltages/potential differences between 2 points in a circuit when the electric fields present are non-conservative, additionally, how come we can still use the same mathematical expressions (defined for conservative electric fields) for the voltages across components such as capacitors, resistors, transistors and diodes?

It's simply because in those situations, electric potential is still defined as the potential associated with the electrostatic part of total electric field.