doPhysics · Electromagnetism 2
Faraday's Law, Lenz's Law and Motional EMF
How a changing magnetic flux drives a current with no cell in the circuit. Magnetic flux, Faraday's law and Lenz's law, the emf of rods moving and turning in a field, the electric field that a changing magnetic field makes, and rods on rails that slow, reach a terminal speed or act as a motor.
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Why this chapter
Why this chapter exists
Every earlier chapter of electricity needed a cell to push charges round a circuit. This chapter finds a second way. A magnetic field that changes through a loop of wire drives a current round it, with no cell anywhere. The rule is short: the emf equals the rate of change of the flux, and the current it drives opposes the change. From that rule come the bicycle dynamo, the Metro card that needs no battery, and the brakes that turn a train's motion back into electricity.
You will learn
By the end of this chapter you will
- Find the magnetic flux \(BA\cos\theta\) through a surface and the flux linkage of a coil. State why the net flux through a closed surface is zero.
- Use Faraday's law \(\varepsilon = -N\,d\Phi/dt\), read the emf from the slope of a flux graph, and find the charge \(N\Delta\Phi/R\) that a change of flux drives.
- Find the direction of any induced current with Lenz's law, and explain the law from the conservation of energy.
- Derive the motional emf \(Blv\) from the force on free electrons and find which end is positive. Apply it to aircraft and trains in the Earth's field.
- Find the emf \(\tfrac12 B\omega L^2\) of a rotating rod and a Faraday disc, and the induced electric field of a changing field inside and outside it.
- Solve rail problems: the drag \(B^2l^2v/R\), the exponential slowing and the terminal speed. Explain the cell-driven rod and its back emf, and rails closed by a capacitor or an inductor.
Before you begin
Before you begin
From the chapter Sources of the Magnetic Field you need the field of a long wire and of a solenoid, \(B = \mu_0 nI\). From the chapter Magnetic Force on Currents and the Magnetic Dipole you need the force \(I\vec L\times\vec B\). You also need the idea that a current loop is a small magnet. From the chapter Electric Current and Resistance you need emf and \(I = \varepsilon/R\). From Mechanics you need \(v = \omega r\), power as \(Fv\), and Newton's second law.
Lessons
1
Magnetic Flux
How much magnetic field passes through a surface. The flux \(BA\cos\theta\), its unit the weber, the flux linkage of a coil, the flux through a solenoid, the three ways to change a flux, and why the net flux through any closed surface is zero.
Magnetic flux through a surfaceFlux through a solenoid turnThree ways to change the fluxClosed field lines and zero net flux
2
Faraday's Law
A changing flux drives an emf round a circuit, \(\varepsilon = -N\,d\Phi/dt\). The emf is the slope of the flux graph, the charge that flows depends only on the change of flux, and a coil flipped or turned in a field gives an emf of its own.
Faradays law of inductionEmf from a graph of b against tThe charge that flows does not depend on speedTurning a coil in a field
3
Lenz's Law
The induced current always opposes the change that produces it. Why energy conservation demands this, a two-step recipe for the direction of any induced current, and the drag on a loop pulled out of a field.
Lenzs lawLenzs law from energy conservationFinding the direction of an induced currentA loop leaving a field region
4
Motional EMF
A rod moving across a magnetic field carries an emf Blv, made by the magnetic force on its free electrons. Which end is positive, the emf across an aircraft's wings and a train's axle, and why the answer agrees with Faraday's law.
Emf of a moving rodWhich end is positiveThe aircraft and the train axleMotional emf agrees with faradays law
5
Rotating Rods, Discs and the Induced Electric Field
A rod turning in a field has an emf \(\tfrac12 B\omega L^2\), because its points move faster the further they are from the pivot. The Faraday disc, a rod turned about its middle, and the electric field with closed loops that a changing magnetic field makes even where there is no wire.
Emf of a rotating rodThe faraday discA rod turning about its midpointThe induced electric field
6
Rails: Drag, Terminal Speed and Energy
A rod on rails closed through a resistance feels a drag \(B^2l^2v/R\). Where the pulling work goes, how a rod let go slows exponentially, how a steady pull or a cell brings it to a terminal speed, and what a capacitor or an inductor across the rails does instead.
The rail circuitA rod let go slows exponentiallyTerminal velocity under a constant forceThe rod driven by a batteryRails closed by a capacitor or an inductor
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.
- area vector
- A vector \(\vec A\) whose size is the area of a flat surface and whose direction is along the normal to the surface.
- back emf
- The emf that a moving motor makes by induction. It opposes the supply, and it limits the current once the motor is running.
- eddy currents
- Currents induced in a solid piece of metal, rather than in a wire, by a changing flux through it. They circulate inside the metal and heat it.
- electromagnetic induction
- The production of an emf, and of a current in a closed circuit, by a change of magnetic flux.
- flux linkage
- The flux through one turn of a coil multiplied by the number of turns, \(N\Phi\). Faraday's law uses its rate of change.
- galvanometer
- A sensitive meter for small currents. A centre-zero galvanometer swings one way or the other according to the direction of the current.
- induced electric field
- The electric field made by a changing magnetic field. Its lines are closed loops, and it drives currents in loops held still.
- induced emf
- The emf that appears in a circuit because the magnetic flux through it changes, \(\varepsilon = -N\,d\Phi/dt\).
- inductor
- A coil used in a circuit for its self-inductance. It opposes changes of the current through it and stores energy in its magnetic field.
- magnetic flux
- The amount of magnetic field passing through a surface. For a flat surface in a uniform field it is \(\Phi = \vec B\cdot\vec A = BA\cos\theta\), with \(\theta\) the angle between the field and the normal. It is a scalar, measured in webers.
- magnetic monopole
- An isolated north or south magnetic pole. None has ever been found, which is why magnetic field lines have no ends.
- motional emf
- The emf across a conductor moving through a magnetic field, made by the magnetic force on its free charges. For a straight rod across the field it is \(Blv\).
- non-conservative
- Said of a force or field whose work round a closed path is not zero. Such a field has no potential. The induced electric field is non-conservative.
- search coil
- A small coil of many turns used to measure a magnetic field. Flipped over or pulled out of the field, it sends a charge \(N\Delta\Phi/R\) through a meter, and that charge gives the flux.
- terminal velocity
- The steady speed reached when a drag that grows with speed becomes equal to the driving force. On rails under a steady pull \(F\), \(v_t = FR/B^2l^2\).
- time constant
- The time \(\tau\) in which a quantity that decays exponentially falls to \(1/e\), about 0.37, of its value. For a rod let go on rails, \(\tau = mR/B^2l^2\).
- weber
- The SI unit of magnetic flux, symbol Wb. One weber is one tesla square metre, so a tesla is a weber per square metre.