Does there exist a rigorous general mathematical proof for $M <\sqrt{L_1L_2}$? Here $M$ is the mutual inductance between two conductors and $L_1$ and $L_2$ are their respective self-inductances. (The proof must not assume that the two conductors are solenoids)
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The total magnetic energy of a transformer carrying currents $I_1$ and $I_2$ is $$W= \frac12 L_1I_1^2 + \frac12 L_2I_2^2 +M I_1I_2$$ or $$W=\frac12 (\sqrt{L_1}I_1-\sqrt{L_2}I_2)^2 + (M+\sqrt{L_1L_2})I_1I_2.$$For this to be nonnegative $W\ge 0$ for any $I_1$ and $I_2$ one must have $M^2 \le L_1L_2$.
answered May 10, 2017 at 14:30
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$\begingroup$ I have not much knowledge here but I'm just interested. Why would the total energy be nonnegative ? Is this equivalent to stating that the energy in your system cannot decrease by applying a current ? $\endgroup$– gertianCommented May 10, 2017 at 14:47
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1$\begingroup$ @gertian No - if $W$ were negative for some combination of $I_1$ and $I_2$, it would be unbounded from below, i.e. it would spontaneously go to higher and higher currents, performing an arbitrarily large amount of work on an outside system if so desired. $\endgroup$ Commented May 10, 2017 at 15:23
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1$\begingroup$ In other words this is a condition of passivity, if it were negative it would be a source of energy. $\endgroup$ Commented May 10, 2017 at 15:24