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

Magnetic Field Produced Along the Axis of the Current-Carrying Circular Coil

3.8.3

Magnetic Field Produced Along the Axis of the Current-Carrying Circular Coil

For a circular loop of radius $R$ carrying current $I$ , consider the field at a point $P$ on its own axis, distance $z$ from the centre $O$ . Two diametrically opposite elements each of length $dl$ produce fields of equal magnitude $dB = \dfrac{\mu_0}{4\pi}\dfrac{I\,dl}{r^2}$ (with $r=\sqrt{R^2+z^2}$ the distance from element to $P$ , and the angle between $I\,d\vec l$ and $\hat r$ being $90°$ ). By symmetry, the components of $d\vec B$ perpendicular to the axis cancel in pairs around the loop, while the components along the axis, $dB\sin\phi$ (with $\sin\phi = R/r$ ), all add. Integrating around the full loop ( $2\pi R$ of wire) gives, for $N$ turns,

$\boxed{B = \frac{\mu_0 N I R^2}{2\,(R^2+z^2)^{3/2}}}$ …

Figure 3.33Magnetic field due to a current-carrying circular loop

What this figure shows. A circular loop of radius R carries current I, and a field point P sits on its axis at distance z from the centre O. Two diametrically opposite elements dl, at C and D, each produce a field dB along the line from the element to P; each dB is resolved into a component along the axis (dB sin(phi)) and a component perpendicular to the axis (dB cos(phi)); the diagram shows the perpendicular components from C and D cancelling while the axial components add, leaving on …

Misc Example 3.13Field at the centre of a split loop

Worked out. A circular loop carries current I that flows around the upper semicircle in one sense and the lower semicircle in the same overall circulation, entering and leaving through a diameter. Since the field contributions of the upper and lower semicircular halves at the centre O are equal in magnitude but opposite in direction (each half circulates oppositely relative to the centre once the entry and exit points are fixed on a diameter), the net field at the centre of the loop is exactly zero, B=0 -- a useful reminder that symmetry can cancel a field even when cur …

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