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I want to apply the Biot-Savart law to calculate the magnetic field at a point created by current flowing through a square/rectangular conductor. More specifically, a trace on a printed circuit board.

To me, a trace on a circuit board is the summation of many infinitesimal rectangular conductors.

I have seen lots of examples for a wire (circular conductor), but not rectangular.

How should I go about getting started with this problem?

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Since you already know that $ \textbf{dB} =\frac{\mu_{0}}{4\pi} \frac {I\textbf {ds} \times \textbf {r}} {|\textbf{r}|^3}$ you can consider each individual current element $I\textbf{ds}$. If you can write an equation for that, that's typically a very good starting point. Next, of course, is to figure out a position vector for the particular point of interest. If you choose a good coordinate system for your particular system these steps (and the integration) can be much simpler than if you choose arbitrarily.

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