Magnetic Field due to a Long Straight Wire

As the first, and most basic, application of the Biot-Savart law, consider a long straight wire carrying current $I$, and a point P at perpendicular distance $R$ from the wire (Fig. 10.16). Using the right-hand thumb rule (already familiar from Section 10.1), the field at P is directed into the plane of the paper (for the current direction shown), so only its MAGNITUDE needs to be worked out by integration.

Take a current element of infinitesimal length $d\vec{l}$, situated a distance $r$ from P, making angle $\theta$ with the line joining it to P. By the Biot-Savart law, this element contributes a differential field of magnitude

The total field at P is found by integrating this contribution over the ENTIRE length of the wire -- conveniently split into the upper half (integrated from the point closest to P out to infinity) and the lower half (by symmetry, contributing an identical magnitude, in the same direction, as the upper half), so that

Using the right-triangle geometry relating the perpendicular distance $R$, the along-wire distance $l$ (measured from the foot of the perpendicular from P), and $r=\sqrt{l^2+R^2}$, together with $\sin\theta=R/\sqrt{l^2+R^2}$, this integral can be carried out explicitly (most conveniently via the standard trigonometric substitution $l=R\tan\theta$, $dl=R\sec^2\theta\,d\theta$, with the limits $l=0\to\theta=0$ and $l\to\infty\to\theta=\pi/2$), giving, for the field due to a SEMI-INFINITE wire (one half only, from the foot of the perpendicular out to infinity in one direction),

Adding the equal contribution from the other (semi-infinite) half of an INFINITELY long wire doubles this result, giving the standard, widely-used formula for the field due to a long straight wire, at perpendicular distance $R$:

$B = \frac{\mu_0 I}{2\pi R}.$ …