By attempting to ask this question ...

Has it ever been proven, or disproven, that a coincident set of mutual inductances are always conserved?

... at Quora, I got this first response ...


... suggesting to me that I must have violated something sacred and/or fundamental to inspire acts of destruction in the mind of my reader as being the only solution to my question.

Yet, all I wish to know concerns a hole in my awareness regarding conservability within the domain of multiple, interactive, mutual inductances? Because the manner in which Kirchhoff's Current Law is phrased, could suggest that anything outside of a circuit's nodal connections are outside of the Conservation of Energy.

Usually, regarding manmade laws, anything not explicitly prohibited is always allowed. So, I applied that standard to this situation to conclude that Conservation does not apply to mutual inductance of the type which I am suggesting (not the commonplace type of a normal transformer).

  • $\begingroup$ For 99% of the people learning circuits, inductive and capacitive interactions between components will not matter. For rf and microwave gurus, sure, they need to deal with it, but it will be a full 3D E&M code, not simple nodal analysis. $\endgroup$
    – Jon Custer
    Dec 16, 2022 at 16:02
  • $\begingroup$ @JonCuster Thank you. $\endgroup$
    – Vinyasi
    Dec 16, 2022 at 16:09

1 Answer 1


Before calculus was used for modeling physics observations , there were a number of "laws" that can now be defined within the mathematical theories . If you read the introduction to Kirchhoff's circuit laws in wikipedia you will see that these laws are mathematically explainable in the strict theory of Maxwell's equations.

Both of Kirchhoff's laws can be understood as corollaries of Maxwell's equations in the low-frequency limit. They are accurate for DC circuits, and for AC circuits at frequencies where the wavelengths of electromagnetic radiation are very large compared to the circuits.

Italic mine for emphasis.

As Maxwell's equation for electromagnetism have been validated at the classical level there is no need to bother about conservation of energy. Even in a complicated set up of circuits if one took the trouble one would get back to the conservation laws of Maxwell's equation.

  • $\begingroup$ Am I to understand that "low frequencies" and DC are the only stipulations which are considered? What does this imply about all of the other factors of electrical reactance have upon conservation, such as a high frequency and an extreme (180 degree) phase shift between current and voltage? $\endgroup$
    – Vinyasi
    Dec 16, 2022 at 16:20
  • $\begingroup$ My simulated example may not qualify for consideration of conservability since it operates at 1MegaHz! = commons.wikimedia.org/wiki/… $\endgroup$
    – Vinyasi
    Dec 16, 2022 at 16:31
  • $\begingroup$ I am sorry. I misunderstood your answer. So, there's no need to bother with my question, yet, you did, anyway. Thanks. $\endgroup$
    – Vinyasi
    Dec 16, 2022 at 17:01
  • $\begingroup$ Where can I go to learn about the "validation" to which you refer? $\endgroup$
    – Vinyasi
    Dec 16, 2022 at 17:03
  • $\begingroup$ you can find the history of maxwell equations in wikipedia , otherwise in a course of electricity and magnetism and electrodynamics.? I have absorbed this through the years , I never found the Maxwell equations invalid. $\endgroup$
    – anna v
    Dec 16, 2022 at 18:46

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