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In the situation above, we have a power line which uses an alternating current. This alternating current causes a change in magnetic flux through the loop below the power line, which induces a current according to Lenz's law/Faraday's law, which eventually goes to the farmer's equipment. This is classed as power theft.

However, I was wondering, does this power tapping by the farmer in some way reduce the power delivered by the power line or cause an energy loss from the power line? If so how?

If it doesn't, then how is this situation exactly consistent with conservation of energy? Does the induced magnetic field in the farmer's loop somehow cause energy to get lost in the power line?

(Where does the energy from changing magnetic flux come from in the first place?)

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    $\begingroup$ That there is a transformer. It's an air-core transformer, I'm not sure how that impacts its performance, but it operates on the same principle as any other transformer. $\endgroup$ Commented May 2, 2023 at 12:08
  • $\begingroup$ Could you elaborate on that please? $\endgroup$
    – Stallmp
    Commented May 2, 2023 at 12:19
  • $\begingroup$ en.wikipedia.org/wiki/Transformer $\endgroup$
    – anna v
    Commented May 2, 2023 at 12:22
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    $\begingroup$ @Stallmp in reality the power line does form a loop - both sides of the loop terminate in the power source (e.g. a power station). From the setup in the drawing it seems that the farmer's coil would be acting as the second coil in a transformer, thereby drawing power from the power line. The amount of power drawn this way is left as an exercise for the reader. $\endgroup$
    – Martin C.
    Commented May 2, 2023 at 13:25
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    $\begingroup$ Real power lines today use three phase systems and the sum of all three currents in such a system is (ideally) zero. In very old systems the return path would have been, literally, the ground. Three phase systems still have a magnetic stray field, but it's much smaller than the shown single wire/ground return path would have been. I am not a lawyer, but if you ask one, I am fairly sure (s)he will tell you that it is considered theft because the power company suffers a financial loss due to the fact that the farmer is not buying energy from them. $\endgroup$ Commented May 3, 2023 at 15:13

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does this power tapping by the farmer in some way reduce the power delivered by the power line or cause an energy loss from the power line? If so how?

Absolutely it does. Here is a circuit diagram for the setup.

Farmer power theft diagram

In this case the mutual inductance, $M$, would be quite low, but non-zero. So on the power company side we have $$v_{power} = i_{power} R_{line} + i_{power} R_{load} - M \frac{di_{farmer}}{dt}+L_1\frac{di_{power}}{dt} $$And on the farmer side we have $$M\frac{di_{power}}{dt}=i_{farmer} R_{farmer}+L_2\frac{di_{farmer}}{dt}$$

On the power company side the $L_1$ is the inductance of the line in the section where the farmer has their loop. That power loss is present irrespective of the farmer, and would normally be lumped into the Line and Load. So that is not the theft.

The theft is in the term with the mutual inductance $M$. Note that this power loss is dependent on the current that the farmer draws in this circuit, which makes sense from a physical standpoint.

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    $\begingroup$ Thanks a lot! So conceptually, the induced current in the farmer's loop takes away some energy of the power line (which can be seen from the mutual inductance term)? Is that a correct statement? $\endgroup$
    – Stallmp
    Commented May 3, 2023 at 20:54
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    $\begingroup$ @Stallmp yes, that is correct $\endgroup$
    – Dale
    Commented May 3, 2023 at 20:58
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    $\begingroup$ Thanks a lot Dale! $\endgroup$
    – Stallmp
    Commented May 3, 2023 at 20:59

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