Lay a small compass on the engine of a car. Put it a few centimetres from the thick cable that runs from the battery to the starter motor. With the engine off, the needle points north as usual. Turn the key. For the second or two that the starter turns, the needle swings round hard. When the engine catches, the needle swings back.
Where magnetic fields come from
In the chapter Magnetic Force on Moving Charges, a magnetic field \(\vec B\) pushed on a moving charge with the force \(q\vec v\times\vec B\). Its unit is the tesla, T. That chapter took the field as given, made by a magnet or a coil. This chapter asks where the field comes from.
The answer is short: moving charge makes a magnetic field. A current in a wire is moving charge, so every current is surrounded by a magnetic field. A magnet makes its field in the same way, from the motion of the electrons inside its atoms. The chapter Magnetism and Matter tells that part of the story.
The field of the starter cable has no connection with the battery's voltage. It appears only while a current flows. The starter draws a large current, well over a hundred amperes, for a moment. That current, not the battery, turns the needle.
The field lines of a straight current
Stand a long straight wire upright through a horizontal card and pass a large current. Sprinkle iron filings on the card and tap it. The filings line up in circles round the wire. A compass set on any circle points along it. The field lines of a straight current are closed circles centred on the wire.
The circles have a sense, which reverses when the current reverses. It is given by a rule of the right hand.
Grip the wire with the right hand so that the thumb points along the current. The curled fingers then point the way the magnetic field lines circle round the wire.
Seen from above, a current coming up out of the page makes anticlockwise circles. A current going down into the page makes clockwise circles. In figures a current out of the page is drawn as a dot, the tip of an arrow coming at you. A current into the page is a cross, the tail feathers of an arrow going away.
A long vertical wire carries a current straight down, into the floor. A compass is placed on the floor just east of the wire. Which way does the wire's field point at the compass?
The field of one small piece of wire
To find the field of a wire of any shape, cut it into very short pieces. Each piece of length \(dl\), carrying the current \(I\), is a current element \(I\,d\vec l\). Its direction is that of the current. Find the field of one element at a point P. Then add the fields of all the elements.
In the autumn of 1820 Jean-Baptiste Biot and Félix Savart measured the field near a long wire. Pierre-Simon Laplace worked out the field of one element that fits their results. It is called the Biot-Savart law.
A current element \(I\,d\vec l\) produces at a point P, at a distance \(r\) from it, a field of size \(dB = \dfrac{\mu_0}{4\pi}\dfrac{I\,dl\sin\theta}{r^2}\). Here \(\theta\) is the angle between \(d\vec l\) and the line from the element to P. The field is along \(d\vec l\times\hat r\), perpendicular to both.
The constant \(\mu_0\) is the permeability of free space, \(4\pi\times10^{-7}\,\text{T m/A}\). The factor \(\mu_0/4\pi = 10^{-7}\) is the one used in almost every calculation of this chapter. The direction comes from a cross product, as the torque \(\vec r\times\vec F\) did in the chapter Torque and Equilibrium. Point the fingers of the right hand along \(d\vec l\) and curl them towards \(\hat r\). The thumb gives \(d\vec B\).
Move P round the circle in the figure above. Broadside to the element, at \(\theta = 90°\), the field is largest. At \(30°\) it is half as large, because \(\sin 30° = 0.5\). When P lies on the line of the element itself, at \(0°\) or \(180°\), the element gives no field at all. For every position on the right, the field is into the page. There \(d\vec l\) points up and \(\hat r\) points right, and up \(\times\) right is into the page.
The Biot-Savart law looks like Coulomb's law for the field of a small charge, from the chapter Electric Charge, Coulomb's Law and the Electric Field. Both fall as \(1/r^2\). The differences matter more.
A wire carries \(10\,\text{A}\). Find the field of a \(1\,\text{cm}\) piece of it at a point \(0.1\,\text{m}\) away. Take the point (a) broadside to the piece, (b) on a line at \(30°\) to it, (c) broadside but twice as far.
(a) \(dB = 10^{-7}\times 10\times 0.01\times\sin 90°/(0.1)^2 = 10^{-6}\,\text{T}\). Convert \(1\,\text{cm}\) to \(0.01\,\text{m}\) first.
(b) \(\sin 30° = 0.5\), so \(dB = 5\times10^{-7}\,\text{T}\).
(c) The distance doubles, so the field falls four times, to \(2.5\times10^{-7}\,\text{T}\).
Values used: \(\mu_0/4\pi = 10^{-7}\,\text{T m/A}\).
A short element of a wire lies along the \(x\) axis at the origin, carrying current in the \(+x\) direction. What is the direction of its field at the point \((0, 0, a)\) on the \(z\) axis?
Adding up a long straight wire
Take a long straight wire along the \(y\) axis, carrying \(I\) up the page. P is at a distance \(d\) to its right. Every element of the wire gives a field into the page at P. So the fields add as plain numbers, with no cancelling. The only work is to add \(dB\) over the whole wire.
Label an element by the angle \(\phi\) at P between the perpendicular to the wire and the line to the element. As the element runs from far below to far above, \(\phi\) runs from \(-90°\) to \(+90°\). This angle makes the sum easy.
\[ \begin{gathered} y = d\tan\phi, \quad dy = d\sec^2\phi\,d\phi \\ r = d\sec\phi \end{gathered} \]
Why. the element at height \(y\) is seen from P at the angle \(\phi\) from the perpendicular, and its distance from P is \(r\).
\[ \sin\theta = \cos\phi \]
Why. \(\theta\) is the angle between the wire and the line to P, and \(\theta = 90° - \phi\).
\[ \begin{gathered} dB = \dfrac{\mu_0}{4\pi}\,\dfrac{I\,dy\cos\phi}{r^2} \\ dB = \dfrac{\mu_0 I}{4\pi d}\cos\phi\,d\phi \end{gathered} \]
Why. put the three results into the Biot-Savart law; \(\sec^2\phi\) cancels.
\[ \begin{gathered} B = \dfrac{\mu_0 I}{4\pi d}\displaystyle\int_{-\pi/2}^{\pi/2}\cos\phi\,d\phi \\ B = \dfrac{\mu_0 I}{4\pi d}\times 2 \end{gathered} \]
Why. every element adds in the same direction, so the fields add as numbers.
\[ B = \dfrac{\mu_0 I}{2\pi d} \]
Read it. simplify.
Watch the bar in Fig 1.1. It fills fastest while the element is close to P and hardly moves while the element is far away. The part of the wire within \(d\) of the foot of the perpendicular gives \(\sin 45° = 71\) % of the whole field. Each half of the wire, above and below the foot, gives exactly half. The next lesson uses that half.
Put in numbers. A wire carrying \(10\,\text{A}\) gives, at \(0.05\,\text{m}\), \(B = 2\times10^{-7}\times 10/0.05 = 4\times10^{-5}\,\text{T}\). At \(0.1\,\text{m}\) the field is half as large. The field falls as \(1/d\), much more slowly than the \(1/r^2\) of one element, because a longer stretch of wire counts as P moves away.
The starter cable of a car carries \(100\,\text{A}\) for a moment. A compass sits \(5\,\text{cm}\) from it, where the cable's field is horizontal and points east. Delhi's horizontal field is \(3.6\times10^{-5}\,\text{T}\), towards the north. Find the cable's field and the angle through which the needle turns. Repeat for a power cable carrying \(100\,\text{A}\), \(1\,\text{m}\) away.
The cable is long and straight near the compass, so \(B = \mu_0 I/2\pi d\). The two fields are at right angles, and the needle lies along their sum.
\(B = 2\times10^{-7}\times 100/0.05\), and \(\tan\delta = B/B_H\).
\(B = 4\times10^{-4}\,\text{T}\), about \(11\) times \(B_H\). So \(\tan\delta = 11.1\) and \(\delta = 85°\). At \(1\,\text{m}\), \(B = 2\times10^{-5}\,\text{T}\), \(0.56\) of \(B_H\), and \(\delta = 29°\).
Near the starter cable the needle swings almost to east. Even the power cable a metre away turns it by \(29°\), which is why a surveyor keeps a compass well clear of cables.
Values used: \(\mu_0/4\pi = 10^{-7}\,\text{T m/A}\), and at Delhi \(B_H = 3.6\times10^{-5}\,\text{T}\) with a dip of \(42°\), so the whole field is \(4.84\times10^{-5}\,\text{T}\). The compass needle swings only in a horizontal plane, so it feels \(B_H\) alone. That is why the example compares the cable with \(B_H\), not with the whole field.
Drive the figure above. The circles are drawn wider apart further out, because the field weakens as \(1/d\). Doubling the distance halves the reading. Reversing the current reverses every circle and turns the needle round. With the Earth's field added, \(10\,\text{A}\) at \(5\,\text{cm}\) gives \(40\,\mu\text{T}\) east against \(36\,\mu\text{T}\) north. The needle settles \(48°\) east of north.
The phenomenon is now explained. The starter cable carries about \(100\,\text{A}\), and \(5\,\text{cm}\) from it that makes \(4\times10^{-4}\,\text{T}\). That is eleven times the Earth's horizontal field, so the needle turns by \(85°\), nearly a right angle. When the starter stops, the current stops, and the Earth's field alone swings the needle back.
A long straight wire carries \(5\,\text{A}\). What is the magnetic field at a distance of \(2\,\text{cm}\) from it, in microtesla?
The field of a long straight wire is \(80\,\mu\text{T}\) at \(2\,\text{cm}\). What is it at \(4\,\text{cm}\)?
A phone charger cable carries about \(2\,\text{A}\) in each of its two wires, one going and one returning. Consider one wire alone and estimate its field \(1\,\text{cm}\) away, in microtesla.
In the spring of 1820 Hans Christian Oersted was lecturing in Copenhagen with a battery, a wire and a compass on the bench. When the current flowed in a wire held above the needle, the needle turned. The swing was small, but it was the first sign that electricity and magnetism are linked. Oersted published his result in July 1820. Within months Ampère in Paris, and Biot and Savart, had turned it into laws. The right-hand thumb rule of this lesson describes what Oersted saw.
- Drawing the field of a straight current pointing away from the wire, like an electric field.The field lines of a straight current are circles round it. The field is along the circle, never along the radius.
- Using the left hand, or curling the fingers the wrong way.Right thumb along the current; the fingers give the sense of the circles. A dot (out of the page) means anticlockwise circles.
- Using \(1/d^2\) for a long wire.One element gives \(1/r^2\); a long wire gives \(B = \mu_0 I/2\pi d\), which falls as \(1/d\).
- Forgetting to convert centimetres to metres before using \(2\times10^{-7}\,I/d\).Put \(d\) in metres to get \(B\) in tesla.