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Given a closed 2-dimensional conducting loop, the terminals of a battery are connected at any two points on the loop (but not the same point).

As an example, consider a circle of radius $R$; the two terminals of the battery divide the circle into two arcs. Let one arc subtend an angle $\theta$ at the centroid (center), and let current $i_1$ pass through this arc. The magnitude of the magnetic field at the center due to this arc (from Biot-Savart Law) is

$$ |\vec{B_1}| = \frac{\mu_0 i_1 \theta}{4 \pi R} $$

Similarly for the second arc,

$$ |\vec{B_2}| = \frac{\mu_0 i_2 (2\pi - \theta)}{4 \pi R} $$

Since the two arcs are in parallel, and the resistance is proportional to the length,

$$ \frac{i_1}{i_2} = \frac{L_2}{L_1} = \frac{(2 \pi - \theta)R}{\theta R} = \frac{(2 \pi - \theta)}{\theta} $$

where $L_1$, $L_2$ are the lengths of the arcs. Thus $|\vec{B_1}| = |\vec{B_2}|$, and since $\vec{B_1}$ and $\vec{B_2}$ are in opposite directions the net magnetic field at the center is zero.

I have also found for certain arrangements of a square, rectangle and an equilateral triangle that the net magnetic field comes out to be zero at the centroid. Checking this for asymmetrical shapes seems to be too tedious.

So for any 2-dimensional conducting loop connected to a battery as described in the starting of the question, is the net magnetic field at the centroid always zero? If not, please provide an example of an arrangement in which the field is not zero, else please provide a proof.

I think there may be some way to arrive at the result (if it is true) by relating Biot-Savart law with the definition of the centroid, but I have not had any success in doing so.

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    $\begingroup$ In a non-convex shape the centroid may lie outside. In that case...not zero. $\endgroup$ Jun 14, 2021 at 11:25
  • $\begingroup$ @mikuszefski does it hold convex shapes though? I know this is not a place for extensive discussion but could you provide a convex counterexample? $\endgroup$ Jun 14, 2021 at 18:18
  • $\begingroup$ I am sure about this: If convex there is a point inside where the field is zero. I don't know if that coincides with the centroid, though. Maybe a dive into conformal mapping can help? $\endgroup$ Jun 15, 2021 at 7:48
  • $\begingroup$ In case of a concave polygon, the centroid will still be zero $\endgroup$ Apr 24 at 5:06

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