Physics · Ch 3 — Magnetism and Magnetic Effects of Electric Current

Magnetic Field Due to Long Straight Conductor Carrying Current

3.8.2

Magnetic Field Due to Long Straight Conductor Carrying Current

For an infinitely long straight conductor YY' carrying current $I$ , consider a field point $P$ at perpendicular distance $a$ from the wire, and a current element $dl$ on the wire making angle $\theta$ with the line joining it to $P$ . Using geometry (dropping a perpendicular from one end of the element and relating angles to the small subtended angle $d\phi$ at $P$ ), the Biot-Savart contribution reduces to $dB = \dfrac{\mu_0 I}{4\pi a}\cos\phi\,d\phi$ (with $\phi$ the angle between $OP$ and the line to the element). Integrating from $\phi_1$ to $\phi_2$ (the extreme angles subtended by the wire at $P$ ):

B = \frac{\mu_0 I}{4\pi a}(\sin\phi_1+\sin\phi_2)

For an infinitely long wire, $\phi_1=\phi_2=90°$ , giving the standard result …

Figure 3.32Magnetic field due to a long straight current-carrying conductor

What this figure shows. An infinitely long straight wire YY' carries current I. A field point P sits at perpendicular distance a from the wire, and a small element AB=dl is marked on the wire at distance r from P, with a perpendicular AC dropped from A onto line BP, and angles phi1, phi2 marked at P showing the extreme angles subtended by the infinite wire -- the geometry used to convert the Biot-Savart integral over dl into an integral over the angl …

Misc Example 3.15Field of a 1 A wire compared to Earth's field

Worked out. A long straight wire carrying I=1 A produces, at a perpendicular distance of r=1 m, a field B = mu0 I/(2 pi r) = (2x10^-7)(1)/(1) = 2x10^-7 T. Since the Earth's own field is of order 10^-5 T, the wire's field at this modest distance is about one hundred times smaller than the Earth's field -- a useful sense check for why a single wire carrying an ordinary current does not visibly overpower a compass unless the compass is held v …

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