Put a steel kadhai with a little oil on an induction cooktop and switch it on. In less than a minute the oil begins to smoke. Lift the kadhai away and touch the glass top. It is only warm, and that warmth came from the pan. Hold your palm a few centimetres above the empty glass, and you feel nothing at all. No flame and no hot plate touched the pan, yet it heated from inside.
Currents with no wire to follow
The chapter Faraday's Law, Lenz's Law and Motional EMF found the emf round a loop of wire whose flux changes. Nothing in Faraday's law needs a wire, though. It needs only a closed path in a conductor round which the flux changes. A solid slab of copper or aluminium holds countless such paths, one inside another, in every plane.
Put the slab in a changing magnetic field, or move it through a field that is not the same everywhere. The flux through the closed paths in the metal changes, so an emf acts round each of them. Currents then flow in closed swirls inside the metal, like eddies in a stream. They are called eddy currents.
Eddy currents are real currents. They heat the metal at the rate \(I^2R\), and in a magnetic field they feel forces. By Lenz's law those forces always oppose the change that drives the currents. The resistance of a path through thick metal is tiny, so even a small emf drives a large eddy current.
A copper slab lies across a uniform field that rises at \(0.5\,\text{T/s}\). Picture one circular path in the slab as a ring of radius \(4\,\text{cm}\) and cross-section \(1\,\text{mm}^2\). Find (a) the emf round the ring and (b) its resistance and current. (c) How do these change for a ring of radius \(8\,\text{cm}\)?
(a) The emf is the rate of change of flux: \(\varepsilon = \pi r^2\,dB/dt = \pi\times0.04^2\times0.5 = 2.51\times10^{-3}\,\text{V}\).
(b) The ring is \(2\pi r = 0.251\,\text{m}\) long, so \(R = \rho\ell/a = 1.7\times10^{-8}\times0.251/10^{-6} = 4.27\times10^{-3}\,\Omega\). Then \(I = \varepsilon/R = 0.59\,\text{A}\).
(c) Doubling the radius makes the area four times as large and the path twice as long. So the emf is four times, \(1.01\times10^{-2}\,\text{V}\), and the current twice, \(1.18\,\text{A}\). The heat \(I^2R\) is eight times as large.
Values used: the resistivity of copper, \(\rho = 1.7\times10^{-8}\,\Omega\,\text{m}\), from the chapter Electric Current and Resistance.
The example shows why eddy currents matter most in large pieces of metal. The emf round a path grows with its area, but the resistance grows only with its length. So the widest paths in a slab carry the largest currents and make most of the heat.
In which case do eddy currents flow in a copper plate?
Braking without touching
Hang a metal plate as a pendulum so that it swings through the gap between the poles of a strong magnet. As the plate enters the field, the flux through the paths near its leading edge rises. As it leaves, the flux through the paths near its trailing edge falls. Eddy currents flow in both places. The part of each current inside the field feels a force \(BIl\), and by Lenz's law that force points against the motion.
In Fig 1.1 the solid plate stops within a few swings. Its eddy currents take its kinetic energy and turn it into heat in the metal. The slotted plate swings on, because each slot cuts across the path an eddy current would take. Only narrow loops fit in its fingers, and the example above showed that narrow loops carry small currents. This slowing by eddy currents is called magnetic braking.
The braking force grows with the speed, because a faster plate changes the flux faster. So the force is strongest when the motion is fastest, and it fades to nothing as the motion stops. Magnetic braking takes away speed, but it can never hold a body still.
Why does the slotted plate of Fig 1.1 keep swinging long after the solid plate has stopped?
A magnet down a copper tube
Drop a strong magnet down a vertical copper tube. It takes seconds to fall a length that it falls in air in a blink. Copper is not magnetic, so the tube does not attract the magnet. The tube slows it through eddy currents.
Think of the tube as a stack of copper rings. The ring just below the magnet sees the downward flux through it growing. So its eddy current makes a field upward, which pushes the magnet from below. The ring just above sees its downward flux falling. So its current makes a field downward, along the magnet's own field, and that ring pulls the magnet from above. Both forces point up, against the fall.
The braking force grows with the speed, so the magnet speeds up only until the force equals its weight. In Fig 1.2 that happens at \(10\,\text{cm/s}\), within a hundredth of a second. From then on the magnet falls at this steady terminal speed. Gravity does work on it at the rate \(mgv\), and all that work becomes heat in the copper. For the \(20\,\text{g}\) magnet it is \(0.02\times10\times0.1 = 0.02\,\text{W}\).
A \(50\,\text{g}\) magnet falls at a steady \(0.1\,\text{m/s}\) down a copper tube. Take \(g = 10\,\text{m/s}^2\). At what rate is heat produced in the tube, in watts?
Meters that do not swing
The coil of a moving-coil galvanometer is often wound on a light aluminium frame. When the coil turns in the magnet's field, eddy currents flow round the frame and oppose the turning. The pointer then moves to its reading and stops there without swinging to and fro. Such a meter is called dead-beat. The older electricity meters in Indian homes had an aluminium disc turning between the poles of a small magnet. The magnet's eddy-current brake kept the disc's speed in step with the power being used.
Heat on purpose: the furnace and the cooktop
An induction furnace melts metal with eddy currents alone. A coil carrying a rapidly alternating current surrounds a crucible of scrap iron. The flux through the metal changes many times a second, and eddy currents heat the metal until it melts. No flame touches the metal, which can even be sealed in a vacuum.
An induction cooktop is a small furnace under glass. A flat coil lies under the glass top. The cooktop's electronics turn the \(50\,\text{Hz}\) mains into a current that alternates some tens of thousands of times a second. The coil's changing field passes through the glass into the base of the kadhai. Eddy currents in the steel then heat the pan from inside.
Now the kadhai of the opening can be explained. The glass is an insulator, so no current can circle in it, and it is warmed only by the hot pan. A palm is a far poorer conductor than steel, so it carries almost no eddy current. The fast alternation matters too: a flux that changes faster drives a larger emf, and so a larger current.
Loss by stealth: laminated cores
The iron core of a transformer or a motor sits in a field that alternates \(50\) times a second. A solid core would carry large eddy currents and grow hot, wasting energy all the time. So cores are built from thin sheets of iron, called laminations, each coated with a thin insulating layer. The sheets lie along the field, and the insulation stops any current from crossing between them. Each eddy current is then trapped in one thin sheet, where only narrow loops fit.
For thin sheets, the eddy-current loss in each cubic metre of core goes as the square of the thickness of a sheet. Split a core into sheets one tenth as thick, and the loss falls to one hundredth. The loss also goes as the square of the frequency, which is why the cooktop's fast alternation heats so well.
Why is the iron core of a transformer built from thin insulated sheets instead of one solid block?
The jumping ring
Slide an aluminium ring over the iron core of a coil, and switch on an alternating current. The ring leaps up the core and can fly off the top. At switch-on the flux through the ring grows from zero, so a current is induced in it. By Lenz's law that current runs against the coil's current. Opposite currents repel, so the coil pushes the ring away. A ring with a cut in it carries no current and does not jump.
Which of these depend on eddy currents?
An aluminium ring rests on the vertical iron core of a coil. When the current is switched on, the ring jumps up. Explain why. What would a ring with a cut in it do?
Switching on makes the flux through the ring grow. An emf is induced round the ring and a current flows in it. By Lenz's law this current opposes the growth of flux, so it runs opposite to the coil's current. Opposite currents repel, so the ring is thrown up. A cut ring has no closed path, so no current flows and it stays where it is.
In 1855 the French physicist Léon Foucault turned a copper disc between the poles of an electromagnet. With the magnet off, the disc turned easily. With the magnet on, he had to push much harder, and the disc grew hot. The extra work went into currents circling inside the solid copper. In France eddy currents are still called Foucault currents. Four years earlier, the same Foucault had shown the Earth's rotation with a long pendulum.
- Thinking that eddy currents need a magnetic metal such as iron.Any conductor carries them. Copper and aluminium are not magnetic, yet they brake strongly because they conduct so well.
- Expecting a steady field to brake a plate at rest.Eddy currents need a change of flux: a changing field, or motion through a field that differs from place to place.
- Thinking laminations stop the field.The field runs along the sheets as before. The insulation only stops the currents from crossing between sheets.
- Believing magnetic braking can hold a body at rest.The braking force is proportional to the speed and vanishes when the motion stops.