Suppose we have a closed loop in a changing magnetic field. By Faraday's law this would induce an emf in the loop. However by the Kirchhoff's law the total emf around a closed loop is zero. It seems like we've reached a contradiction. What is going on here?
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$\begingroup$ However by the Kirchhoff's law the total emf around a closed loop is zero is an incorrect statement. $\endgroup$– FarcherCommented Jun 18 at 22:12
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2 Answers
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Kirchoff's Law is an approximation of Faraday's Law in the specific situation whereby the change in magnetic flux is negligible.
Kirchoff's Law means that voltages are like heights in gravitational potential; at every point in space, there is only one voltage, just like how at every point on the surface of the Earth, there is only one height of the land's surface.
Faraday's Law implies that Kirchoff's Law is wrong, and that there is a new contribution to voltages, confusing that it may be, that you can measure. Of course, for convenience, we will continue to define the scalar electric potential as if this new contribution does not exist, i.e. still has the nice property of having just one value anywhere in space, but there is a new contribution coming from the time derivative of the magnetic vector potential.
answered Dec 10, 2023 at 22:39
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$\begingroup$ It's too strong to say "Kirchoff's Law is wrong" it just requires a slight modification, including an induced EMF term in the KVL expression (e.g. add an equivalent voltage source in the network). Circuit theory is supposed to be practical, just make the adjustment in this special case. IMO instead of "implies Kirchoff's Law is wrong" we should say "Kirchoff's Law is incomplete unless you include induced EMF" $\endgroup$ Commented Dec 11, 2023 at 14:30
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$\begingroup$ @SamGallagher After two earlier statements giving the same idea that you are suggesting, I do not see the point of holding back at that point. It is quite important to hammer in the idea that, contrary to expectations, there is no uniquely positionally defined voltage function whenever magnetic fields are changing. $\endgroup$ Commented Dec 12, 2023 at 1:08
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Just as work and kinetic energy might both be measured in joules, but not be identical concepts, so likewise, there are multiple concepts that are measured in volts. Unfortunately, they often are all designated by the same name, i.e. "voltage".
Probably the most familiar and common use of the term "voltage" is the integral
$$\int_a^b\vec{E}\cdot d\vec{s}$$
If, within a simply connected region of space, $\nabla\times\vec{E}=0$, then the above integral is path independent, and so we can say there is a specific potential difference between any two points.
There are times, however, when $\nabla\times\vec{E}\ne 0$. Specifically this happens when there is a time-varying magnetic field. In that case,
$$\nabla\times\vec{E}=-\frac{\partial\vec{B}}{\partial t}$$
However, despite the fact that the former integral is not longer path-independent, we can still measure something with a "volt"meter. If we fix a voltmeter and it's leads in place, with the leads connected to a circuit, we get some reading. Consider two closed curves, $C$ and $C'$ which both have a section that travels through one of the leads of a voltmeter, through the voltmeter and through the other lead. The remainder of each of curve closes the loop somehow, but through a different path in each case.
If, the voltmeter measures some line-integral, it cannot matter whether that line-integral is of $C$, or of $C'$. The voltmeter only gives us one answer, so the two line integrals must be the same. But since we know that, in the presence of a time-varying magnetic field, the electric field is not necessarily conservative within a non-simply connected region, we know that a voltmeter cannot always be measuring the integral
$$\int_a^b\vec{E}\cdot d\vec{s}$$
But this other thing that is measured, is also measured in volts, and is also called "voltage".
It turns out, that this other thing that is measured by a voltmeter is the electric scalar potential $\phi$ constrained by the equations:
$$\vec{E}=-\nabla\phi + \nabla\times\vec{W}$$ $$\nabla\cdot\vec{W}=0$$
However by the Kirchhoff's law the total emf around a closed loop is zero.
Actually, the law stated by Kirchhoff states (my translation)
- when the wires $1,2,...n$ form a closed figure, $$I_1R_1 + I_2R_2 + ... I_nR_n$$ = the sum of all electromotive forces that are on the way:
And here we have yet another concept that is measured in volts, and is sometimes called a voltage, namely emf.
Kirchhoff does not say that the emf around a loop is 0, but rather, that it is equal to the sum of the IR voltage drops. Of course, many text books present Kirchhoff's Voltage law as if it says that
$$\int_a^b\vec{E}\cdot d\vec{s}=0$$
This latter equation, not only does not hold in the presence of time-varying magnetic fields, but also misrepresents Kirchhoff's own words.
How is emf related to the integral $\int_a^b\vec{E}\cdot d\vec{s}$ and the electric scalar potential $\phi$? Here are some facts that may help one hone one's intuition.
The electric scalar potential $\phi$ does not contribute any net emf to a loop (in steady state conditions).
The emf induced in a section of a conductor by a time varying magnetic field is, in general, not equal to the integral of the electric field $\int_a^b\vec{E}\cdot d\vec{s}$
The electric field $\vec{E}$ in the interior of a conductor follows the microscopic version of Ohm's Law
$$\vec{J}=\sigma\vec{E}$$
regardless of any emf that may be induced in any section of the conductor.
To reconcile the electric field that may be induced outside of a conductor by a time-varying magnetic field, with the electric field within the interior of a conductor that is governed by microscopic Ohm's Law, a surface charge density $\vec{K}$ forms on the surface of the conductor.
The surface charge $\vec{K}$ on the surface of the conductor contributes to the electric scalar potential $\phi$. When charges are intentionally placed in a system, for example via a van de Graaff generator, or in a capacitor, we know they are there. However the surface charges that form on on the surface of a conductor due to the reconciliation between an electric field induced by a changing magnetic field, and between a the Ohm's law controlled field within a conductor appear "spontaneously". However, their effect should not be ignored in calculating $\phi$. These surface charges provide a "record" of the application of the induced emf to the conductor.
The emf $\mathscr{E}_{\partial\vec{B}/\partial t}$ induced in a conductor by time-varying magnetic fields is given by the path integral of the component of the electric field induced by the time-varying magnetic field, $E_{\partial\vec{B}/\partial t}$.
$$E_{\partial\vec{B}/\partial t} =\nabla\times\frac{1}{4\pi}\int_{R^3}\frac{\partial\vec{B}(r')}{\partial t}\frac{1}{|r-r'|}d^3r'$$
$$\mathscr{E}_{\partial\vec{B}/\partial t} =\int_a^b E_{\partial\vec{B}/\partial t} \cdot d\vec{s}$$
- The total electric field is the sum of the negative gradient of $\phi$ and the component of the electric field induced by the time-varying magnetic field, $\vec{E}_{\partial\vec{B}/\partial t}$.
$$\vec{E} = -\nabla\phi + \vec{E}_{\partial\vec{B}/\partial t}$$
- Kirchhoff's Voltage Law, where voltage is defined as the electric scalar potential $\phi$ above, takes the form
$$\sum_C\Delta\phi = \oint_C\nabla\phi\cdot d\vec{s} = 0$$
And this is tautologically true by vector calculus.
- This version of Kirchhoff's Voltage Law conforms to the "voltage" measured by voltmeters, but not to the "voltage" defined by the integral
$$\int_a^b\vec{E}\cdot d\vec{s}$$
