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The above Question asks about the Magnetic Flux through the Plane x<0 by the Current Carrying Square Loop.
Now, I came across a trick where it goes as:

The Plane can be replaced by a wire carrying current "I" along y-axis, and the Flux of Magnetic Field of that Wire through the given Loop will be the same as our required one.

So, my question is that is it valid?...
And if Yes, then what's the logic behind that?

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    $\begingroup$ This trick is called $''$Neumann formula$''$ for the mutual inductance : Consider two loops 1 and 2. Whatever the shapes and positions of the loops, the flux through 2 when we run a current $\:i\:$ around 1 is identical to the flux through 1 when we send the same current $\:i\:$ around 2. ($'$Introduction to Electrodynamics$'$, by D.Griffiths, 4th Edition $\S$ 7.2.3 Inductance.) For the application of this trick see @Puk's answer (to be accepted as the best one). $\endgroup$
    – Frobenius
    Apr 18, 2022 at 8:14

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Imagine the shaded area is a very big square loop of wire. The mutual inductance $M$ of the two loops is the magnetic flux $\Phi$ through one loop due to a current $I$ in the other, divided by $I$. $M$ can be calculated from the flux through the bigger loop due to a current in the smaller loop (this is what is being asked), or from the flux through the smaller loop due to a current in the bigger loop: the result will be the same. It is easier to evaluate the latter, however. As a corollary, the flux through the bigger loop due to a current $I$ in the smaller one is the same as the flux through the smaller loop due to a current $I$ in the bigger loop.

When a current $I$ flows in the bigger square loop, the three sides of this square are too far away from the small loop to contribute to the flux through it, and only the closest section along the y-axis matters. Therefore, the flux can be calculated by considering only the magnetic field due to an infinite wire along the y-axis.

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