D.4 Induction · IB DP Physics

      
      IB DP Physics · Theme D · Fields
Induction.D.4 · Magnetic flux, Faraday’s law, Lenz’s law and the rotating coil
Higher level only: there is no standard level content in D.4

              Additional HL · 6 hours
              D.3 Motion in electromagnetic fields ↗
            
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      The journey
Three parts.
Part 01
Flux and Faraday’s lawA changing flux induces an emf
Part 02
Lenz’s law and moving conductorsThe direction of the emf, and \(\varepsilon = BvL\)
Part 03
The rotating coilA sinusoidal emf, and the effect of frequency
In D.3 a current in a field gave a force. Here motion in a field gives an emf.
D.4 INDUCTION · HL ONLY


      
      Part 01 · HL only
Flux and Faraday’s law.What has to change for an emf to appear, and how large the emf is
D.4 INDUCTION · HL ONLY


      The story · London, 1831
Background story · not assessed
Faraday makes electricity from magnetism.
            Michael Faraday, painted by Thomas Phillips in 1842
            His disk generator of 1831 (engraving of 1884)
          
A bookbinder’s apprentice at 14, with little schooling
Ørsted got magnetism from electricity. Faraday wanted the reverse
1831: two coils on an iron ring. Switching one gave a brief current in the other
Then a magnet into a coil, and a spinning copper disk
A magnet held still beside a coil does nothing. Only a change in the flux through the coil induces a current.
Portrait: Thomas Phillips, 1842, public domain. Engraving: Émile Alglave, 1884, public domain.
D.4 INDUCTION · HL ONLY


      The basic observation
Inducing an emf.Magnet moving in: the meter deflects
Magnet at rest: nothing
Magnet moving out: deflects the other way
Faster, a stronger magnet or more turns: a larger emf
A MAGNET AND A COIL
D.4 INDUCTION · HL ONLY
Only relative motion mattersHold the magnet still and move the coil over it, and the meter deflects in exactly the same way. An emf is induced whenever the magnetic field through the coil is changing, whatever the cause.
Why “emf” and not “current”The change always induces an emf: energy per unit charge made available to drive charge round. A current flows only if the circuit is complete. With the meter disconnected there is still an emf across the ends of the coil, and you could measure it with a voltmeter.
Watch the needle and the magnet together. The deflection is largest when the magnet is moving fastest, in the middle of its swing, and zero at each end, where it stops for an instant.


      How much field passes through
Magnetic flux.\(\Phi = BA \cos \theta\)
\(\theta\): the angle between the field and the normal to the area
Unit: the weber, \(1 \,\mathrm{Wb} = 1 \,\mathrm{T} \,\mathrm{m}^{2}\)
For a coil of \(N\) turns, the flux linkage is \(N \Phi\)
A LOOP IN A UNIFORM FIELD
D.4 INDUCTION · HL ONLY
A picture for fluxThink of the field lines as rain falling straight down and the loop as the mouth of a bucket. Held level, the bucket catches the most rain. Tilted, it catches less. Held on its side, none goes in at all. Flux measures how much field passes through the loop: the number of field lines it catches.
Why the angle is to the normalThe flux is greatest when the loop faces the field squarely. In that position the field is at right angles to the plane of the loop but along its normal, so \(\theta = 0\) and \(\cos \theta = 1\). When the plane of the loop lies along the field, \(\theta = 90^{\circ}\) and the flux is zero.
This is why \(B\) is called flux densityRearranged for a loop facing the field, \(B = \frac{\Phi}{A}\): the flux through each square meter.
Using the angle between the field and the plane of the coil. If a question gives that angle, subtract it from 90° first.


      Worked example
The flux through a coil.
            \(B = 0.20 \,\mathrm{T}\) · area \(A = 0.030 \,\mathrm{m}^{2}\)
            \(\Phi = BA \cos \theta\)
            Facing the field: \(\theta = 0^{\circ}\)
            \(\Phi = 0.20 \times 0.030 \times \cos 0^{\circ} = \class{res}{6.0 \times 10^{-3} \,\mathrm{Wb}}\)
            Normal at \(60^{\circ}\) to the field
            \(\Phi = 0.20 \times 0.030 \times \cos 60^{\circ} = \class{res}{3.0 \times 10^{-3} \,\mathrm{Wb}}\)
          

            The coil has 50 turns
            Flux linkage \(= N \Phi\)
            \(= 50 \times 6.0 \times 10^{-3} = \class{res}{0.30 \,\mathrm{Wb}}\)
            when facing the field
          
D.4 INDUCTION · HL ONLY
What the numbers sayTurning the coil through 60° halves the flux, although nothing about the field or the coil itself has changed. Turn it a further 30° and the flux is zero. Rotating a coil is therefore one way of changing the flux through it, and it is how every generator works.
Why flux linkageEach turn of the coil has the same flux through it and the same emf induced in it. The turns are joined in series, so the emfs add. A coil of 50 turns gives 50 times the emf of a single loop for the same change of flux.
Areas are often given in cm². \(1 \,\mathrm{cm}^{2} = 10^{-4} \,\mathrm{m}^{2}\), so 300 cm² is 0.030 m².


      The size of the induced emf
Faraday’s law of induction.\(\varepsilon = -N \dfrac{\Delta \Phi}{\Delta t}\)
The induced emf equals the rate of change of flux linkage
The minus sign: the emf opposes the change (Part 2)
Change \(B\)
A time-varying fieldMove a magnet, or change the current in a nearby coil
Change \(A\)
A moving conductorThe area of the circuit in the field grows or shrinks
Change \(\theta\)
A rotating coilThe coil turns to face the field, then away
D.4 INDUCTION · HL ONLY
It is the rate that mattersA huge flux that stays constant induces nothing. A small flux that changes quickly can induce a large emf. Pushing a magnet into a coil in 0.1 s gives ten times the emf of pushing it in over 1 s, although the change of flux is the same.
One law, three situations\(\Phi = BA \cos \theta\) has three factors, and changing any of them changes the flux. The rest of this deck works through the three cards on the slide one by one. In every case use \(\Delta\Phi = \Phi_\mathrm{f} - \Phi_\mathrm{i}\), keeping the same positive area normal. The signed emf is \(-N\frac{\Delta\Phi}{\Delta t}\); its magnitude is \(N\frac{|\Delta\Phi|}{\Delta t}\).
Check the unit: \(\frac{\mathrm{Wb}}{\mathrm{s}} = \frac{\mathrm{T} \,\mathrm{m}^{2}}{\mathrm{s}} = \mathrm{V}\). One weber of flux change in one second induces one volt in one turn.


      Worked example
A coil in a changing field.
            200 turns · area \(4.0 \times 10^{-4} \,\mathrm{m}^{2}\), facing the field
            \(B\) rises from 0.10 T to 0.50 T in 0.020 s
            \(\Delta \Phi = A \Delta B\)
            \(\Delta \Phi = 4.0 \times 10^{-4} \times 0.40 = 1.6 \times 10^{-4} \,\mathrm{Wb}\)
            \(\varepsilon = -N \dfrac{\Delta \Phi}{\Delta t} = -\dfrac{200 \times 1.6 \times 10^{-4}}{0.020}\)
            \(\varepsilon = \class{res}{-1.6 \,\mathrm{V}}\)
          

            Only the change in \(B\) matters: 0.40 T
\(\Delta\Phi = \Phi_\mathrm{f} - \Phi_\mathrm{i}\); emf magnitude \(|\varepsilon| = 1.6 \,\mathrm{V}\)
If \(B\) then stays at 0.50 T, the emf is zero

          
D.4 INDUCTION · HL ONLY
What you could changeDouble the number of turns and the emf doubles. Make the same change of field in half the time and the emf doubles. Use a coil of twice the area and it doubles again. Each scales the magnitude \(N\frac{|\Delta\Phi|}{\Delta t}\). The negative emf here is relative to the loop direction associated with the chosen positive flux.
Where the field might come fromA common arrangement is a small “search” coil inside a solenoid. When the current in the solenoid changes, so does the field inside it, and an emf appears in the small coil even though nothing moves. This is the principle of the transformer: a changing current in one coil induces an emf in another.
Using the final field, 0.50 T, in place of the change. A coil sitting in a steady 0.50 T field has no emf at all.


      Reading graphs
Emf is the gradient of the flux graph.Emf = minus the gradient of a flux–time graph
Flux constant: no emf
Steeper: a larger emf
Flux falling: the emf changes sign
FLUX AND EMF AGAINST TIME
D.4 INDUCTION · HL ONLY
How to draw the lower graph from the upper oneWork along the time axis one section at a time. Where the flux graph is a straight sloping line, the gradient is constant, so the emf is a constant horizontal line. Where the flux graph is flat, the emf is zero. Where the flux falls twice as steeply as it rose, the emf is twice as big and on the other side of the axis.
Going the other wayThe area under an emf–time graph gives the size of the total change in flux linkage. On the slide the two blocks of the emf graph have equal areas, one below the axis and one above, because the flux returns to where it started.
The same relationship as velocity and displacement in A.1: one graph is the gradient of the other.


      Quick check
Flux and Faraday’s law.01What is the unit of magnetic flux?
The weber, Wb (T m²)
02The plane of a coil lies along the field. Flux through it?
Zero
03A magnet rests inside a coil. Induced emf?
Zero: the flux is not changing
04Flux through one turn changes by 0.020 Wb in 0.10 s. Emf in 50 turns?
\(\frac{50 \times 0.020}{0.10} = 10 \,\mathrm{V}\)
05The same change happens in half the time. The emf?
It doubles
06What does the gradient of a flux–time graph give?
The size of the induced emf, per turn
PRESS → TO REVEAL EACH ANSWER


      
      Part 02 · HL only
Lenz’s law and moving conductors.Which way the induced current flows, and why it could not be otherwise
D.4 INDUCTION · HL ONLY


      The direction of the induced emf
Lenz’s law.The induced current flows in the direction that opposes the change that causes it
Flux increasing: its field is against the applied field
Flux decreasing: its field is with it
It opposes the change in flux, not the flux.
A MAGNET AND A COIL
D.4 INDUCTION · HL ONLY
A method that always worksOne: which way does the applied field point through the coil? Two: is the flux increasing or decreasing? Three: the induced current must make a field through the coil that points against the applied field if the flux is increasing, and with it if the flux is decreasing. Four: use the right-hand grip rule from D.2 to find the current that makes that field.
The poles pictureFor a magnet and a coil there is a shortcut. The coil becomes an electromagnet whose near end repels an approaching pole and attracts a departing one. Either way, you have to do work to move the magnet.
Heinrich Lenz, working in St Petersburg, stated the rule in 1834, three years after Faraday’s discovery. It is the reason for the minus sign in Faraday’s law.


      Why it must oppose
Lenz’s law is conservation of energy.ONE MAGNET, TWO TUBESPRESS → TO DROP
If the current helped the change, the magnet would speed up by itself: energy from nothing
Because it opposes, work must be done to move the magnet
That work becomes electrical energy in the circuit
D.4 INDUCTION · HL ONLY
The copper tubeCopper is not magnetic: a magnet does not stick to it. But as the magnet falls, the flux through each ring of the tube wall changes, so currents are induced round the tube. By Lenz’s law their fields oppose the motion: the copper below repels the falling magnet and the copper above attracts it. The magnet soon reaches a steady speed at which this upward force equals its weight.
Where the energy goesAt a steady speed the magnet is losing gravitational potential energy without gaining kinetic energy. All of it is transferred, through the induced currents, to thermal energy in the copper. In the plastic tube no current can flow, so nothing opposes the fall.
The same effect brakes trains and roller coasters without any contact or wear: magnets move past a metal fin, and the induced currents slow them.


      Changing the area
A straight conductor moving across a field.
              In a time \(\Delta t\) the rod moves \(v \Delta t\)
              \(\Delta \Phi = B \Delta A = B L v \Delta t\)
              \(|\varepsilon| = \dfrac{|\Delta \Phi|}{\Delta t}\)
              \(|\varepsilon| = BvL\)
            
Field, velocity and rod all at right angles.
A ROD ON TWO RAILS
D.4 INDUCTION · HL ONLY
The same result from the force on a chargeThe free electrons in the rod are moving to the right with the rod, across the field. Each feels a magnetic force \(qvB\) along the rod, from D.3. Charge is pushed towards one end until the electric field set up in the rod balances the magnetic force: \(qE = qvB\), so \(E = vB\). The potential difference across a rod of length \(L\) is \(EL = BvL\). Faraday’s law and the force on a moving charge give the same emf.
The directionThe flux through the circuit is increasing, so by Lenz’s law the induced current must make a field inside the loop opposite to \(B\). Seen from above, looking along \(B\), the grip rule gives an counterclockwise current: along the rod towards the far rail, as shown.
A rod moving alone, with no rails, still has the emf \(BvL\) between its ends, but no current flows. The wings of an aircraft crossing the Earth’s field have a small emf between their tips.


      Worked example
A rod pulled along rails.
            \(B = 0.40 \,\mathrm{T}\) · \(L = 0.25 \,\mathrm{m}\) · \(v = 3.0 \,\mathrm{m} \,\mathrm{s}^{-1}\) · \(R = 1.5 \,\Omega\)
            \(\varepsilon = BvL = 0.40 \times 3.0 \times 0.25 = \class{res}{0.30 \,\mathrm{V}}\)
            \(I = \dfrac{\varepsilon}{R} = \dfrac{0.30}{1.5} = \class{res}{0.20 \,\mathrm{A}}\)
            \(F = BIL = 0.40 \times 0.20 \times 0.25 = \class{res}{0.020 \,\mathrm{N}}\)
          

            Power to keep the rod moving
            \(P = Fv = 0.020 \times 3.0 = \class{res}{0.060 \,\mathrm{W}}\)
            Power dissipated in the resistor
            \(P = I^{2} R = 0.20^{2} \times 1.5 = \class{res}{0.060 \,\mathrm{W}}\)
          
D.4 INDUCTION · HL ONLY
What the two powers showThe induced current flows through the rod, and a current in a field feels a force \(BIL\). By Lenz’s law this force is against the motion. To keep the rod moving at a steady speed, someone must pull forwards with 0.020 N, doing 0.060 J of work every second. Exactly that much energy appears as heating in the resistor. Energy is conserved, and Lenz’s law is what guarantees it.
If nobody pullsThe opposing force slows the rod. A slower rod gives a smaller emf, a smaller current and a smaller force, so the rod slows more and more gently and never reverses.
Three equations from three topics in one question: \(\varepsilon = BvL\) (D.4), \(V = IR\) (B.5) and \(F = BIL\) (D.3). This combination is a favorite in exams.


      In and out of a field
A rectangular coil moving through a field.Entering: flux rising, an emf
Fully inside: flux constant, no emf
Leaving: flux falling, an emf of opposite sign
While entering or leaving: \(\varepsilon = NBvL\).
CONSTANT SPEED
D.4 INDUCTION · HL ONLY
Why nothing happens in the middleWhen the whole coil is inside the field, the front and back edges are both cutting field lines. Each has an emf \(BvL\), but they drive current in opposite directions round the coil, so they cancel. In terms of flux: the coil has as much field through it at the end of each instant as at the start.
While enteringOnly the front edge is in the field. It acts as the moving rod of the last two slides, with \(L\) the length of that edge. For \(N\) turns the emf is \(NBvL\). It lasts for the time the coil takes to travel its own width.
The directionEntering, the flux is increasing and the induced current opposes the increase. Leaving, the flux is decreasing and the current reverses to oppose the decrease. In both cases the force on the coil is backwards, against its motion.
A faster coil gives taller, narrower pulses on the graph. The area of each pulse, emf × time, stays the same: it equals the change in flux linkage.


      Induction and oscillations
A magnet oscillating above a coil.The flux through the coil rises and falls as the magnet moves
The emf alternates, at the frequency of the oscillation
Largest when the magnet moves fastest
If a current flows, the oscillation is damped
A MAGNET ON A SPRING
D.4 INDUCTION · HL ONLY
Linking the emf to the motionThe magnet moves fastest as it passes through its equilibrium position, so the flux changes fastest there and the emf is at its peak. At the top and bottom of the motion the magnet is momentarily at rest and the emf is zero. The emf therefore follows the velocity of the magnet, not its displacement: for a small swing, near enough in step with it.
Why the oscillation dies awayWhen the coil is part of a complete circuit, a current flows and energy is dissipated in its resistance. That energy comes from the oscillation, by way of the Lenz’s law force on the magnet, which always acts against its motion. With the circuit open no current flows, there is no opposing force, and the magnet oscillates for much longer.
C.1 and C.4 treat oscillations and damping. This is electromagnetic damping.


      A coil acting on itself · qualitative only
Self-induction.A coil carrying a current sits in its own magnetic field
When the current changes, the flux through the coil changes, so an emf is induced in the coil itself
By Lenz’s law that emf opposes the change of current
Switching on
The current rises graduallyThe induced emf acts against the supply while the current is growing
Switching off
The current is hard to stopThe induced emf tries to keep it flowing, and can make a spark at the switch
D.4 INDUCTION · HL ONLY
An analogySelf-induction gives a current something like inertia. A heavy flywheel cannot be set spinning or stopped in an instant; a coil does not let its current start or stop in an instant. In both cases a steady state is not opposed at all: a constant current in a coil feels no induced emf.
Why the sparkOpening a switch tries to bring the current to zero in a very short time. The flux collapses quickly, so \(\frac{\Delta \Phi}{\Delta t}\) is very large and the induced emf can be far greater than the supply voltage, enough to drive a spark across the opening contacts. The ignition coil of a petrol engine uses this to produce thousands of volts from a 12 V battery.
Only this description in words is required. Inductance and circuits with resistors and inductors are not part of the course.


      Quick check
Lenz’s law and moving conductors.01State Lenz’s law.
The induced current opposes the change that causes it
02Which conservation law does it follow from?
Conservation of energy
03A 0.50 m rod moves at 4.0 m s⁻¹ across a 0.20 T field. Emf?
\(0.20 \times 4.0 \times 0.50 = 0.40 \,\mathrm{V}\)
04A coil is completely inside a uniform field and moving. Emf?
Zero
05A north pole approaches a coil. Polarity of the near end?
North
06Why does a magnet fall slowly through a copper tube?
Induced currents oppose its motion
PRESS → TO REVEAL EACH ANSWER


      
      Part 03 · HL only
The rotating coil.Changing the angle: how a generator makes an alternating emf
D.4 INDUCTION · HL ONLY


      Changing the angle
A coil rotating in a uniform field.ONE ROTATIONCOIL AND EMF IN STEP
The flux linkage is \(NBA \cos \theta\). As the coil turns steadily, \(\theta\) grows steadily and the induced emf varies as a sine curve.
D.4 INDUCTION · HL ONLY
Follow one turnWhen the normal to the coil points along the field, the flux through the coil is at its greatest, but for an instant it is not changing: the sides of the coil are moving parallel to the field lines. The emf is zero. A quarter of a turn later the plane of the coil lies along the field. The flux is zero, but it is changing at its fastest: the sides are cutting straight across the field lines. The emf is at its peak.
Why it alternatesIn the second half of the turn each side of the coil moves through the field in the opposite direction, so the emf reverses. One complete rotation gives one complete cycle of alternating emf.
The emf is greatest when the flux is zero, and zero when the flux is greatest. Examiners often ask for this.


      Two sine curves
Flux and emf, a quarter of a cycle apart.Flux linkage: \(N \Phi = NBA \cos \omega t\)
Emf: minus its gradient, \(\varepsilon = NBA \omega \sin \omega t\)
Peak emf: \(\varepsilon_{0} = NBA \omega\), with \(\omega = 2 \pi f\)
More turns, a stronger field, a larger coil or faster rotation: a larger emf.
ONE AND A HALF ROTATIONS
D.4 INDUCTION · HL ONLY
Seeing the gradient on the graphLook where the flux curve crosses the axis: it is at its steepest there, and the emf curve is at a peak. Look at the top of the flux curve: for an instant it is flat, and the emf curve crosses zero. Where the flux is falling, the emf is positive; where the flux is rising, it is negative. That is the minus sign in Faraday’s law.
Where \(\omega\) comes fromThe coil turns through the angle \(\theta = \omega t\). The faster it turns, the more quickly the flux swings from positive to negative, so the gradient of the flux curve at each point is proportional to \(\omega\). That is why \(\omega\) appears in the peak emf.
\(\varepsilon_{0} = NBA \omega\) is not in the data booklet. Be ready to reason it from the graphs and from Faraday’s law.


      Turning the coil faster
Changing the frequency of rotation.Double the frequency of rotation:
The period halves: twice as many cycles each second
The peak emf doubles: the flux changes twice as fast
Both change together.
THE SAME COIL AT TWO SPEEDS
D.4 INDUCTION · HL ONLY
Why both changeThe maximum flux through the coil, \(BA\), does not depend on how fast it turns. But at twice the frequency the flux goes from its greatest value to zero in half the time, so its rate of change, and with it the emf, is doubled at every point of the cycle. And since each rotation still gives one cycle, twice the rotations each second means twice the frequency of the emf.
Sketching the new curveStart from the original. Squash it to half its width and stretch it to twice its height. A common exam question gives one curve and asks for the other on the same axes.
Doubling only the frequency of the curve and leaving its height unchanged. The peak emf is proportional to the frequency.
Power stations hold their generators at a fixed frequency, 50 Hz or 60 Hz, and control the output in other ways.


      Worked example
A small generator.
            100 turns · \(A = 0.020 \,\mathrm{m}^{2}\) · \(B = 0.30 \,\mathrm{T}\)
            Rotating at \(f = 50 \,\mathrm{Hz}\)
            \(\varepsilon_{0} = NBA \omega\)  with  \(\omega = 2 \pi f\)
            \(\omega = 2 \pi \times 50 = 314 \,\mathrm{rad} \,\mathrm{s}^{-1}\)
            \(\varepsilon_{0} = 100 \times 0.30 \times 0.020 \times 314\)
            \(\varepsilon_{0} = \class{res}{190 \,\mathrm{V}}\)
          

            Greatest flux linkage: \(NBA = 0.60 \,\mathrm{Wb}\)
At 25 Hz the peak emf would be 94 V
The emf passes through zero 100 times a second

          
D.4 INDUCTION · HL ONLY
Checking it with Faraday’s lawIn a quarter of a turn, 0.0050 s, the flux linkage falls from 0.60 Wb to zero. The average emf over that time is \(\frac{0.60}{0.0050} = 120 \,\mathrm{V}\). The peak is larger than the average, as it must be for a curve that starts from zero: 190 V.
How generators get moreReal generators wind the coil on an iron core, which increases the field through it many times, and use many more turns. Large ones rotate the magnet and keep the coils still, so that the large output current does not have to pass through sliding contacts.
Using the frequency \(f\) where the angular speed \(\omega\) is needed. \(\omega = 2 \pi f\): its numerical value in rad s⁻¹ is about six times the numerical value of \(f\) in Hz.


      Quick check
The rotating coil.01What shape is the graph of emf against time for a steadily rotating coil?
A sine curve
02In which position of the coil is the emf greatest?
Plane of the coil along the field
03What is the flux through the coil at that moment?
Zero
04The frequency of rotation is tripled. What happens to the peak emf?
It triples
05And to the period of the emf?
It falls to one third
06Give two other ways to increase the peak emf.
More turns, a stronger field or a larger area
PRESS → TO REVEAL EACH ANSWER


      Summary
Induction on one slide.Flux
\(\Phi = BA \cos \theta\)In webers. \(\theta\) is measured from the normal
Flux linkage \(N \Phi\)
Faraday’s law
\(\varepsilon = -N \frac{\Delta \Phi}{\Delta t}\)The rate of change of flux linkage
Gradient of a flux–time graph
Lenz’s law
Opposes the changeA consequence of conservation of energy
Moving conductor
\(\varepsilon = BvL\)A coil entering or leaving a field
No emf while fully inside
Self-induction
A coil opposes changes in its own currentQualitative only
Rotating coil
A sinusoidal emfPeak when the flux is zero
Double the frequency: double the peak, half the period
D.4 INDUCTION · HL ONLY


      Summary
Key words.Magnetic fluxField × area facing the field
WeberThe unit of flux: 1 T m²
Flux linkageFlux × number of turns
Induced emfCaused by a changing flux
Faraday’s lawEmf = rate of change of flux linkage
Lenz’s lawThe emf opposes the change
Self-inductionA coil’s emf from its own changing current
Alternating emfReverses direction every half cycle
Peak emfThe greatest value in a cycle
D.4 INDUCTION · HL ONLY


    