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Prove. The magnetic field intensity at the center of a flat circular loop of wire of radius r carrying current I is given by
Proof. Let us use Biot’s formula
where θ is the angle between the direction of ds and the line from ds to P. See Fig. 1.
We note that all the dB vectors at point P arising from all incremental elements ds on the wire coincide. This means we can do an algebraic sum instead of a vector sum to sum the dB’s. We also note that
∑ds = circumference of circle = 2πr
sin θ = sin 90o = 1 for all elements ds.
Integrating 2) over the loop C