Hello everyone
Suppose I have the next example (see figure) : I have two experiments :
My question is : what is the current in coil 1, that would give the exact current i2.
Should I just invert Biot-Savart law, is that doable, is the solution unique?
Thank you in advance
Hi Newgato, I think two limitations apply here:
1. Obviously, you can only compute the AC component of the current in coil 1.
2. It's easy if you are certain that the amount of current is virtually the same at each point of coil 1 (i. e. the shortest wavelength of the current >> length of the conductor constituting coil 1). In the case of current waves traveling around coil 1, I guess that there isn't a unique solution.
An obvious prerequisite is that you know the exact geometry and location of coil 1.
BTW, Biot-Savart isn't enough, in both experiments you have to apply Maxwell, too: rot E = -d B / d t, and you need to know the impedance of coil 2 (and the impedance of the load at coil 2 if coil 2 isn't just shorted).
Hello, Joereg, thank you for the answer,
Sorry for the lack of information in my question.
I assume that current is uniform in each coil (in every experiment) and I am only interested in AC components.
And I know all of these :
- The self-impedance of each coil (i.e resistance +reactance).
- The mutual inductance.
My problem is more of a mathematical problem than a physical one.
Since I can simply use Biot-Savart law to get the magnetic field and from it, Faradays equation would give the induced EMF, and from it I can use the impedance to get the current.
However What I want is this (in simple words).
I have two coils (1 and 2), I have a current in coil 2 (AC of course) , the question is : what is the the current in coil 1, that would induce this current, had coil 1 been an emitter, and coil 2, been a receiver.
Thank you in advance
Hello Newgato, if you know the mutual inductance M (by applying Biot-Savart, Faraday, and Ohm) then for the reverse problem you can dispense with Biot-Savart and Faraday, and just use d I1 / d t = -V2 / M where V2 is the induced EMF in coil 2.
In a real experiment, two questions would arise: What kind of source is driving I1, and what is the purpose of coil 2? If I1 is not driven by an ideal current source and if coil 2 is intended as a sensor, it would be better to measure the voltage at the open ends of coil 2. Otherwise the presence of I2 will affect I1 (M is always the same in both directions).
To answer your original question, I cannot think of any way to invert the Biot-Savart law directly but you don't have to because the geometry of the field never changes, only its strength. So, designating the magnetic flux through coil 2 Phi2, Phi2 = k * I1, and after applying Biot-Savart to an arbitrary current I1 and integrating B over the area enclosed by coil 2, you know the constant k, and you are able to compute d I1 / d t from d Phi2 / d t.
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