Magnetic Field Due to a Straight Current-Carrying Conductor

Setting up the integral. Consider a long straight wire carrying current $I$, and a field

point $P$ at perpendicular distance $r$ from the wire. Take a current element $I\,d\vec{l}$ at some

point along the wire, at a distance $r$ from the line, subtending an angle $\theta$ (measured from

the perpendicular dropped from $P$ to the wire) at $P$. By the Biot-Savart law of Section 4.3, this

element contributes a field of magnitude $dB = (\mu_0/4\pi)\,I\,dl\sin\theta/r'^2$, where $r'$ is the

element's own distance to $P$ (which changes as the element's position along the wire changes); every

element's contribution points in the SAME direction at $P$ (out of the page, or into the page,

consistently, by the right-hand rule), so the individual field magnitudes can simply be added

(integrated) directly, without needing to worry about vector components cancelling.

General result for a finite wire. Carrying out this integration (a standard trigonometric

substitution, expressing $l$ and $r'$ in terms of the angle each element subtends at $P$) for a

straight wire of finite length, seen from $P$ at perpendicular distance $r$ under angles

$\phi_1$ and $\phi_2$ measured from the two ends of the wire to the foot of the perpendicular, gives

The infinitely long wire (the standard, most-used result). For a wire that is effectively

infinite in both directions (very long compared to $r$, or when $P$ is not near either end), both

angles $\phi_1,\phi_2 \to 90^\circ$, so $\sin\phi_1=\sin\phi_2=1$, and the general result above

reduces to the single, compact formula

This is the standard, most frequently used field expression for a long straight current-carrying

wire: the field strength falls off as $1/r$ (one power gentler than a point charge's $1/r^2$

electric field, precisely because the wire is an extended, one-dimensional source rather than a

single point), and, as established already by Oersted's experiment, the field lines form closed

circles around the wire, their direction given by the right-hand thumb rule -- thumb along the

current, curled fingers along the field. …