doPhysics · Electromagnetism 2
Eddy Currents, Generators and Inductance
Induction at work. Eddy currents in solid metal brake, heat and waste energy. A coil turning in a field gives an alternating emf. A coil's own changing current induces a back emf, and one coil induces an emf in another. The current in an LR circuit grows and decays with the time constant L/R.
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Why this chapter
Why this chapter exists
The last chapter found that a changing flux induces an emf. This chapter puts that law to work in three places. In a solid piece of metal it drives swirling currents that brake a train, melt scrap iron and cook your dinner. In a coil turning in a field it gives the alternating emf that reaches every socket in India. In a coil whose own current changes, it gives a back emf that makes the current slow to start and slow to stop. That last effect, inductance, stores energy in a magnetic field and sets the pace of every circuit with a coil in it.
You will learn
By the end of this chapter you will
- Explain how eddy currents arise in bulk metal, why they brake a moving conductor, and how laminations cut their losses.
- Derive the emf \(NBA\omega\sin\omega t\) of an AC generator, and say where in a turn it is greatest and where it is zero.
- Define self-inductance by \(N\Phi = LI\) and \(\varepsilon = -L\,dI/dt\), and derive \(L = \mu_0 n^2Al\) for a long solenoid.
- Show that an inductor stores \(\tfrac12 LI^2\), at an energy density \(B^2/2\mu_0\), and use mutual inductance \(M = k\sqrt{L_1L_2}\).
- Solve the LR circuit for the growth and the decay of current, with the time constant \(L/R\).
- Use the rule that an inductor's current cannot jump. Treat an inductor as an open switch at the first instant and as a wire at the end.
Before you begin
Before you begin
From the chapter Faraday's Law, Lenz's Law and Motional EMF you should know magnetic flux, Faraday's law \(\varepsilon = -N\,d\Phi/dt\), Lenz's law and the motional emf \(Blv\). From the chapter Sources of the Magnetic Field you need the field \(\mu_0 nI\) inside a long solenoid. You should know Kirchhoff's loop rule from the chapter Circuits: Kirchhoff's Laws, Networks and RC Transients. You should be able to differentiate \(\sin\omega t\) and \(e^{-t/\tau}\), and to integrate \(1/x\).
Lessons
1
Eddy Currents
A changing flux through a solid piece of metal drives currents round countless closed paths inside it. These eddy currents brake moving metal and heat it. In iron cores they waste energy unless the cores are laminated.
Eddy currents in bulk metalMagnetic brakingUses and losses of eddy currents
2
The AC Generator
A coil turning steadily in a magnetic field has a flux linkage \(NBA\cos\omega t\), so its emf is \(NBA\,\omega\sin\omega t\), an alternating emf. Slip rings connect the turning coil to the circuit. The emf is greatest when the flux is zero, and the peak doubles with \(N\), \(B\), \(A\) or \(\omega\).
Flux of a turning coilThe alternating emf of a generatorSlip rings brushes and real machinesDoubling the peak emf
3
Self-Inductance
A coil's own current sets up a flux through it, \(N\Phi = LI\). When the current changes, that flux changes and induces a back emf \(-L\,dI/dt\) that opposes the change. A long solenoid has \(L = \mu_0 n^2Al\), and breaking a current in a large inductor makes a spark.
Self inductance definedInductance of a solenoidEmf from a graph of currentThe spark at the switch
4
Energy in an Inductor and Mutual Inductance
The work done against the back emf while a current grows is stored in the inductor's field, \(\tfrac12 LI^2\), at an energy density \(B^2/2\mu_0\). Two coils share flux, so a changing current in one induces an emf in the other, \(-M\,dI_1/dt\), with \(M\) the same both ways.
Energy stored in an inductorEnergy density of a magnetic fieldMutual inductanceCoupling and reciprocity
5
The LR Circuit: Growth of Current
Close a switch on a cell, a resistor and an inductor, and the current grows as \((E/R)(1 - e^{-t/\tau})\) with the time constant \(\tau = L/R\). The inductor is an open switch at the first instant and a plain wire at the end, and the energy it stores grows to \(\tfrac12 LI^2\).
The loop equation and its solutionVoltages across l and r while the current growsEnergy flow while the current growsThe inductor at t zero and at infinity
6
Decay of Current and the Opened Switch
Take the cell out of an LR circuit and the current does not stop at once. It keeps its direction and dies away as \(I_0e^{-t/\tau}\), turning the stored \(\tfrac12 LI_0^2\) into heat. The current through an inductor can never jump, though the voltage across it can.
Decay of currentThe tangent at t zero meets tauOpening the switch with a parallel pathWhy the current cannot jump
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- AC generator
- A machine in which a coil and a magnetic field turn steadily relative to each other. The induced emf alternates as \(NBA\omega\sin\omega t\).
- alternating emf
- An emf that reverses its direction at regular intervals, such as \(\varepsilon_0\sin\omega t\), which reverses twice in each turn of a generator coil.
- armature
- The coil of a generator or a motor in which the emf is induced or on which the forces act.
- back emf
- The emf induced in a coil by a change of its own current, \(-L\,dI/dt\). It opposes the change of current, not the current.
- brushes
- Carbon blocks that press on the slip rings of a generator and carry the current between the turning rings and the fixed circuit.
- choke
- An inductor on an iron core used in series with a lamp or a motor. The choke of a tube light strikes the tube and then limits its current.
- coupling coefficient
- The fraction \(k = M/\sqrt{L_1L_2}\), between 0 and 1, which says how much of the flux of each of two coils links the other.
- dead-beat
- Said of a meter whose pointer moves to its reading and stops without swinging, because eddy currents in its frame damp the motion.
- decay of current
- The gradual dying away of the current in an inductor closed through a resistor with no source, as \(I_0e^{-t/\tau}\).
- eddy currents
- Currents induced in closed swirls inside a solid piece of conductor whose flux changes. They heat the metal and oppose the change that drives them.
- flux linkage
- The flux through all the turns of a coil together, \(N\Phi\) for \(N\) turns each holding a flux \(\Phi\). It is measured in weber.
- henry
- The SI unit of inductance: \(1\,\text{H} = 1\,\text{Wb/A} = 1\,\text{V s/A}\). One henry gives one volt of back emf for a change of one ampere per second.
- induction furnace
- A furnace that melts metal by the heat of eddy currents, driven by a rapidly alternating current in a coil round the metal.
- inductor
- A coil made for its inductance, drawn in circuits as a row of loops, with two bars beside it when it has an iron core.
- laminations
- Thin iron sheets, insulated from one another, from which the cores of transformers and motors are built so that eddy currents stay small.
- LR circuit
- A circuit of an inductor and a resistor, with or without a cell, in which the current changes gradually with the time constant L/R.
- magnetic braking
- The slowing of a conductor moving through a magnetic field by the forces on its eddy currents. The braking force grows with the speed and vanishes at rest.
- magnetic energy density
- The energy stored per unit volume of a magnetic field, \(u = B^2/2\mu_0\), about \(4\times10^5\,\text{J/m}^3\) at \(1\,\text{T}\).
- mutual inductance
- The flux linkage of one coil per unit current in another, \(M = N_2\Phi_2/I_1\). A changing current in one coil induces \(-M\,dI_1/dt\) in the other.
- peak emf
- The greatest value of an alternating emf. For a generator coil it is \(\varepsilon_0 = NBA\omega\).
- self-inductance
- The flux linkage of a coil per unit of its own current, \(L = N\Phi/I\). It is also the back emf per unit rate of change of current.
- slip rings
- Two metal rings on the shaft of a generator, each joined to one end of the turning coil. They keep the turning coil connected to the fixed circuit.
- terminal speed
- The steady speed at which a braking force that grows with speed balances the driving force. A magnet falls down a copper tube at its terminal speed.
- time constant
- The time \(\tau = L/R\) that sets the pace of an LR circuit. In one time constant a growing current reaches 0.63 of its final value, and a decaying one falls to 0.37 of its start.