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Let's assume that there is a disk of total electric charge $Q$ rotating about its axis with a constant angular velocity $\vec{\omega}$. I know that one can easily compute the magnetic field generated on the axis of the rotation of the disk, which is possible due to the symmetry of the problem.

My question is the following: Is there any way to use the Biot-Savart law (or any other method) in order to compute the magnetic field on an arbitrary point on the plane of the disk itself?

I would, probably naively, think of somehow reducing the problem to a $2D$ one, but fail to implement it.

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Your rotating charged disk can be thought of as many concentric rings of current. Each ring can be thought of a many short elements of current, each of which contributes to the magnet field at any chosen point as predicted by Biot-Savart. Not being a mathematician, if I had to do this I would let a computer do a numeric summation. (Decades ago, I did a similar calculation for selected points within the field of a Helmholtz coil.)

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