Loops and Arcs

The easiest sum of all

Bend a wire into a circle of radius \(R\) and pass a current \(I\). Look at the field at the centre C. Every element of the loop is at the same distance \(R\) from C. Every element is also at right angles to the radius that joins it to C, so \(\sin\theta = 1\) for all of them.

The direction is the same for every element too. For a loop in the plane of the page carrying current anticlockwise, each \(d\vec l\times\hat r\) points out of the page. So the fields of all the elements add as plain numbers. Here the Biot-Savart law needs no integral, only a sum of lengths.

A flat coil of \(N\) turns, wound close together, is \(N\) such loops at the same place. Each turn adds the same field. The right-hand rule of the loop is the thumb rule turned round. Curl the right fingers along the current, and the thumb points along the field at the centre.

Part of a loop

An arc of a circle is part of a loop. Every element of it is still at distance \(R\) from the centre and at right angles to its radius. An arc that subtends the angle \(\theta\) at its centre, in radians, has the length \(R\theta\). It gives the fraction \(\theta/2\pi\) of the full-loop field.

The current reaches an arc along straight leads. When a lead lies along a radius, its line passes through the centre. By the last lesson, it gives no field there. So an arc fed by radial leads is the only source that counts at its centre.

A quarter circle of radius \(0.1\,\text{m}\) carrying \(8\,\text{A}\) has \(\theta = \pi/2\). It gives \(10^{-7}\times 8\times(\pi/2)/0.1 = 1.26\times10^{-5}\,\text{T}\). A \(60°\) arc with \(6\,\text{A}\) has \(\theta = \pi/3\) and gives \(6.28\times10^{-6}\,\text{T}\). A semicircle with \(4\,\text{A}\), fed by leads along its diameter, gives \(\mu_0 I/4R = 1.26\times10^{-5}\,\text{T}\). The sense comes from the right hand: an arc run clockwise, as seen, gives a field into the page.

Along the axis

Now move P out along the axis of the loop, to a distance \(x\) from its centre. Every element is at the same distance \(\sqrt{R^2 + x^2}\) from P and still at right angles to the line to P. So every element gives a field of the same size. But the fields now point in different directions, each tilted away from the axis.

Pair each element with the one opposite it on the loop. Their tilts are equal and opposite, so the parts across the axis cancel. Only the parts along the axis survive, each a fraction \(R/\sqrt{R^2 + x^2}\) of \(dB\).

For the loop in the figure, the centre field is \(3.14\times10^{-5}\,\text{T}\). At \(x = \sqrt3 R = 17.3\,\text{cm}\), the bracket \(R^2 + x^2\) is \(4R^2\), and its power \(3/2\) is \(8R^3\). So the field is one eighth of the centre value, \(3.9\times10^{-6}\,\text{T}\). On both sides of the loop the field points the same way along the axis.

Far from the loop, \(x \gg R\), the bracket is nearly \(x^2\). The field becomes \(\mu_0 I R^2/2x^3\), which falls as \(1/x^3\). In the chapter Magnetic Force on Currents and the Magnetic Dipole, a loop was a magnetic dipole of moment \(M = IA\). The far field is the field of that dipole, \((\mu_0/4\pi)(2M/x^3)\). A long wire falls only as \(1/d\), because it is endless. The near and far sides of a loop carry opposite currents, and far away they almost cancel.

Squares, polygons and pairs

A square loop of side \(a\) is four finite wires. The centre is \(a/2\) from each side, and each side subtends \(45°\) on either side of the perpendicular. So one side gives \((\mu_0 I/4\pi(a/2))(2\sin 45°)\). All four sides circulate the same way round the centre, so their fields add.

A regular polygon of \(n\) sides works in the same way. Each side subtends \(180°/n\) on either side of its perpendicular. A hexagon of side \(a\) has its centre \(\tfrac{\sqrt3}{2}a\) from each side, and each side subtends \(30°\) each way.

The same wire bent into a square gives more field at the centre than bent into a circle. For a wire of length \(L\), the circle has \(R = L/2\pi\) and \(B = \pi\mu_0 I/L\). The square has \(a = L/4\) and \(B = 8\sqrt2\,\mu_0 I/\pi L\). Their ratio is \(8\sqrt2/\pi^2 = 1.15\). The sides of the square pass closer to the centre than the circle does, \(L/8\) against \(L/2\pi\), and near parts count most.

Pairs of loops add in the same way. Two concentric loops in one plane, of radii \(0.1\,\text{m}\) and \(0.2\,\text{m}\), carrying \(5\,\text{A}\) in opposite senses, give \(\tfrac12\mu_0 I(1/0.1 - 1/0.2) = 1.57\times10^{-5}\,\text{T}\). This is in the sense of the smaller loop. Two coaxial loops of radius \(0.1\,\text{m}\), \(0.2\,\text{m}\) apart, with \(5\,\text{A}\) the same way, each give \(B_{centre}/2\sqrt2\) at the midpoint. Together they give \(2.2\times10^{-5}\,\text{T}\).

A straight wire and a loop add as well. A long wire carries \(10\,\text{A}\), and a loop of radius \(0.05\,\text{m}\) carrying \(2\,\text{A}\) lies in the same plane, its centre \(0.1\,\text{m}\) from the wire. At the centre the loop gives \(2.51\times10^{-5}\,\text{T}\) and the wire \(2\times10^{-5}\,\text{T}\). If they are opposite, the net field is \(5.1\times10^{-6}\,\text{T}\), in the sense of the loop's own field.