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Prove. The magnetic field intensity B at a perpendicular distance a from the axis of a long straight wire carrying a current I is
Proof. Let us use the Biot formula
to compute B. We set up an xyz coordinate system as shown in Fig. 1 with the x axis coinciding with the wire and the origin O arbitrarily chosen somewhere on the wire. Let x be the distance from O to a differential element dx, r be the distance from dx to P, and θ be the angle between dx and r. Then formula 1) becomes
The dB vectors at point P arising from all incremental elements dx on the wire coincide. This means we can do an algebraic sum instead of a vector sum and can use regular integration to sum the dB’s. Integrating, we get
To facilitate integration, we make a change of variables. From Fig. 1, we see that
r = a csc θ x = a cot θ
Thus
dx = -a csc2θ dθ
Substituting these expressions into 3) gives
or
