Topic 12.4: Electromagnetic Induction and Faraday's Law

What Topic 12.4 requires

Topic 12.4 has a single learning objective, and five essential knowledge statements hang off it.

12.4.A, describe the induced electric potential difference resulting from a change in magnetic flux.

• 12.4.A.1 states that magnetic flux is a description of the amount of the component of a magnetic field that is perpendicular to a cross-sectional area.
• 12.4.A.2 states that magnetic flux through a surface is proportional to the magnitude of the component of the magnetic field perpendicular to the surface and to the cross-sectional area of the surface, with the relevant equation $\Phi_B = BA\cos\theta$.
• 12.4.A.2.i states that the area vector is defined to be perpendicular to the plane of the surface and directed outward from a closed surface.
• 12.4.A.2.ii states that the sign of the magnetic flux indicates whether the magnetic field is parallel to or antiparallel to the area vector.
• 12.4.A.3 states that Faraday's law describes the relationship between changing magnetic flux and the resulting induced emf in a system, with the relevant equation $\left| \mathcal{E} \right| = \left| \dfrac{\Delta \Phi_B}{\Delta t} \right|$.
• 12.4.A.4 states that Lenz's law is used to determine the direction of an induced emf resulting from a changing magnetic flux, with the relevant equation $\mathcal{E} = -\dfrac{\Delta \Phi_B}{\Delta t} = -\dfrac{\Delta (BA\cos\theta)}{\Delta t}$.
• 12.4.A.4.i states that an induced emf generates a current that creates a magnetic field that opposes the change in magnetic flux.
• 12.4.A.4.ii states that the right-hand rule is used to determine the relationships between current, emf, and magnetic flux.
• 12.4.A.5 states that a common example of electromagnetic induction is a conducting rod on conducting rails in a region with a uniform magnetic field, and gives the derived equation $\mathcal{E} = B\ell v$.

Topic 12.4 prints no boundary statement, and neither does Topic 12.3. The unit's only boundary statement sits under Topic 12.2 and restricts quantitative treatment of the force on a moving charge to angles of 0, 90, and 180 degrees. Nothing fences off the angle in $\Phi_B = BA\cos\theta$, so be ready to evaluate it.

What does bound this topic is the shape of the equations themselves. This is an algebra-based course, so Faraday's law arrives as a ratio of finite differences, $\Delta \Phi_B / \Delta t$, not as a derivative, and every emf you compute is an average over an interval $\Delta t$. Notice too what the CED's equations do not contain: there is no turn count. No $N$ appears in 12.4.A.3, in 12.4.A.4, or anywhere in the Magnetism group of the equation sheet.

The CED lists four suggested skills for this topic: 1.C, create qualitative sketches of graphs that represent features of a model or the behavior of a physical system; 2.B, calculate or estimate an unknown quantity with units from known quantities, by selecting and following a logical computational pathway; 2.D, predict new values or factors of change of physical quantities using functional dependence between variables; and 3.C, justify or support a claim using evidence from experimental data, physical representations, or physical principles or laws. Unit 12 is weighted at 12 to 15 percent of the multiple-choice section, with a suggested 10 to 14 class periods.

Magnetic flux, and the angle measured to the area vector (12.4.A.1 and 12.4.A.2)

Essential knowledge 12.4.A.1 defines flux in words before any symbol appears, and the words are the useful part: magnetic flux is a description of the amount of the component of a magnetic field that is perpendicular to a cross-sectional area. Read it as "how much of the field gets through the surface". Only the perpendicular part counts, which is the whole content of the $\cos\theta$.

Essential knowledge 12.4.A.2 names the two proportionalities: flux is proportional to the magnitude of the perpendicular component of the field, and to the cross-sectional area. The unit is the weber, one tesla times one square meter, worth remembering because $\mathrm{Wb/s}$ has to come out as volts. The AP Physics 2 equation sheet prints flux twice, as $\Phi_B = \vec{B} \cdot \vec{A}$ and as $\Phi_B = \left| \vec{B} \right| \cos\theta \left| \vec{A} \right|$; the second is what you compute with.

Now the part that goes wrong quietly, because the arithmetic still works and only the answer is off. Essential knowledge 12.4.A.2.i defines the area vector: it is perpendicular to the plane of the surface, and for a closed surface it is directed outward. A flat loop lying on your desk has an area vector pointing straight up out of the desk. And $\theta$ is the angle between the field and that vector, not between the field and the plane of the loop. The two differ by 90 degrees, which turns every cosine into a sine and every maximum into a zero.

The last row is essential knowledge 12.4.A.2.ii: the sign of the magnetic flux indicates whether the magnetic field is parallel to or antiparallel to the area vector. Parallel gives positive flux, antiparallel negative, and ninety degrees zero. Flux is a scalar, so it has a sign but no direction, and fluxes through a surface add as ordinary numbers.

Which way the area vector points for an open loop is your choice, and it is a choice you have to make out loud. Declare it before you calculate and keep it for the whole problem. Every worked example here takes the area vector into the page, so a field into the page gives positive flux. The reassuring part is that the physical answer does not depend on the choice: pick the other convention and both the flux and its change flip sign, the two cancel in Lenz's law, and the current still runs the same way round the loop. The convention is bookkeeping; the current is physics.

Faraday's law, and the three ways to change a flux (12.4.A.3)

That is what the equation sheet prints, and it is what essential knowledge 12.4.A.3 gives as the relevant equation. Two things about it are worth stating plainly.

First, the emf depends on the rate of change of flux, not on the flux. A loop in the strongest steady field you can build has an enormous flux through it and zero induced emf. A loop in a weak field changing fast has a small flux and a real emf.

Second, the printed form is an absolute value on both sides. The sheet gives you the size of the emf and nothing about its direction. Direction is a separate statement, 12.4.A.4, and it is the next section.

Because $\Phi_B = BA\cos\theta$ is a product of three things, there are exactly three ways to change it, and question writers use all three.

1. Change $B$. Move a magnet toward or away from the loop, ramp the current in a nearby electromagnet, or switch a neighboring circuit on or off. The unit's essential question about the induction stovetop is this one.
2. Change $A$. Slide, stretch, or squash the loop so the area inside it grows or shrinks. The conducting rod on rails in 12.4.A.5 is exactly this, and so is a loop pulled out of a field region.
3. Change $\theta$. Rotate the loop in a steady field. This is the generator, and it is how the second worked example produces an emf with $B$ and $A$ both fixed.

Combinations are worth watching for. A loop can rotate while the field ramps, and the flux may then hold steady while both factors move. Since $\mathcal{E}$ tracks the change, a momentarily flat flux produces no emf even while $B$ and $\theta$ are both changing.

Lenz's law, and what the minus sign actually says (12.4.A.4)

That signed form is printed in the CED at essential knowledge 12.4.A.4, under the heading of Lenz's law. It is not printed on the equation sheet: the sheet's Magnetism group gives Faraday's law only as $\left| \mathcal{E} \right| = \left| \Delta \Phi_B / \Delta t \right|$, in absolute-value bars, with no minus sign anywhere in the group. That division of labor is the design of the topic. The sheet hands you the size. Lenz's law is the physics you have to supply, and 12.4.A.4 says exactly what it is for: determining the direction of an induced emf resulting from a changing magnetic flux.

Now the sentence that decides whether the rest of this topic works. Essential knowledge 12.4.A.4.i reads: an induced emf generates a current that creates a magnetic field that opposes the change in magnetic flux.

Read the object of "opposes". It is the change in magnetic flux. It is not the flux, and it is not the external magnetic field. This is the single most mangled idea in the topic, and the difference is not a technicality, because the two readings give opposite answers half the time.

Hold the external field fixed, pointing into the page through a loop, and compare the two cases.

The external field points the same way in all three rows. The induced current reverses. So "the induced field opposes the applied field" is simply false in the second row: there, the induced field points the same way as the applied one, propping up a flux that is draining away. What stays true in every row is the CED's wording, that the induced field opposes the change.

A reliable way to hold it: the loop is conservative about its own flux. Push more through and it pushes back; take flux away and it tries to keep it; leave it alone and it does nothing.

The third row deserves saying on its own, because it is the fastest available multiple-choice elimination. A constant magnetic field, however strong, induces no current in a stationary loop. A magnet resting inside a coil generates nothing; the same magnet moving generates an emf while it moves and nothing once it stops.

The minus sign is also where energy conservation enters, and it is why the sign could not have come out the other way. Suppose the induced current helped the change. More flux would drive a current whose field made still more flux, driving a larger current, without limit and with nothing supplying the energy. The minus sign is what makes induction cost work: pushing a magnet into a coil takes a real force through a real distance, and the electrical energy appearing in the circuit is that mechanical work, exactly as conservation of energy requires. The third worked example checks that the two powers match to the digit.

A four-step routine for the direction of the induced current (12.4.A.4.ii)

Essential knowledge 12.4.A.4.ii says the right-hand rule is used to determine the relationships between current, emf, and magnetic flux. Here is the routine that puts it to work. Do the four steps in order and the direction comes out right without hand-waving.

1. Which way is the flux through the loop? State the field's direction and confirm it actually passes through the loop rather than skimming across it. "Into the page" or "out of the page" is usually enough.
2. Is that flux growing or shrinking? This is the step people skip, and it is the step that decides the answer. Increasing, decreasing, or steady.
3. Which way must the induced field point inside the loop? By 12.4.A.4.i it opposes the change: against the existing flux if the flux is growing, along with it if the flux is shrinking. If the flux is steady, stop here, because there is no current.
4. Curl your right hand to produce that field. Point your right thumb along the direction you need the induced field to have inside the loop, and your fingers curl the way the induced current flows. This is the loop rule from Topic 12.3 used backwards: there you knew the current and wanted the field, here you know the field and want the current.

Worked once, with the field into the page and increasing. Step 1: flux into the page. Step 2: increasing. Step 3: the induced field inside the loop must point out of the page. Step 4: thumb out of the page, fingers curl counterclockwise, so the induced current runs counterclockwise. That is the first row of the table above, derived rather than recalled.

Two checks that catch most errors.

• Flip the flux from increasing to decreasing and the answer must flip. If it does not, you used the flux rather than its change in step 3.
• Reversing the field direction also flips the current, for a fixed sense of change, so two reversals cancel: a field out of the page and decreasing gives the same current as a field into the page and increasing.

Skill 3.C is listed for this topic, and the unit's "Preparing for the AP Exam" page warns that simply naming the right-hand rule is not a complete enough answer to earn free-response points, a passage quoted in full on Topic 12.3. The four steps above are the fix: write them as sentences and the reasoning is on the page rather than in your hand.

The rod on rails, and emf = B l v (12.4.A.5)

Essential knowledge 12.4.A.5 singles out one setup as the common example of electromagnetic induction: a conducting rod on conducting rails in a region with a uniform magnetic field. Slide the rod and the circuit's enclosed area changes, so the flux changes, so an emf appears. This is case 2 of the three, with $B$ and $\theta$ both fixed.

The CED labels $\mathcal{E} = B\ell v$ a derived equation rather than a relevant equation, and the framework's front matter defines the difference: derived equations are provided to demonstrate the final results of derivations expected of students on the exam. The expectation is that you can produce it, not merely quote it. It is also printed on the equation sheet, but the label tells you what is being asked for.

The derivation is three lines. Take a rod of length $\ell$ moving at speed $v$ perpendicular to a uniform field $B$.

1. In a time $\Delta t$ the rod sweeps out extra area $\Delta A = \ell v \Delta t$.
2. With $B$ and $\cos\theta$ constant, $\Delta \Phi_B = B \, \Delta A = B \ell v \Delta t$.
3. So $\left| \mathcal{E} \right| = \left| \Delta \Phi_B / \Delta t \right| = B\ell v$, and $\Delta t$ cancels.

The unit's "Building the Science Practices" page describes students being asked to justify why they can simplify $\mathcal{E} = -\Delta \Phi_B/\Delta t = -\Delta (BA\cos\theta)/\Delta t$ to $\mathcal{E} = -B\ell v$ in some cases. The words "in some cases" are the assessable part, and the cases are the conditions used above: the field is uniform and constant in time, the rod stays perpendicular to both the field and its velocity, and only the area changes. Change any of those and you are back to the full expression.

Once the emf exists the circuit behaves like any other: with total resistance $R$, the induced current follows from $I = \Delta V / R$, the form Ohm's law takes on the AP Physics 2 sheet.

The rod then feels a force, because it is now a current-carrying wire in a magnetic field: $F_B = I \ell B \sin\theta$, which is Topic 12.3 arriving at the end of Topic 12.4. Lenz's law gives the direction before any arithmetic. The induced effects oppose the change, the change is the rod's motion, so the force opposes the motion and never speeds the rod up. Holding the rod at constant speed therefore takes a steady applied force, that force does work at a rate $Fv$, and that rate equals the electrical power $\mathcal{E}I$ delivered to the circuit. The third worked example carries both out and they agree.

Sketching the graphs, and how Topic 12.4 is tested (skills 1.C, 2.D, 2.B)

Skill 1.C, creating qualitative sketches of graphs that represent features of a model or the behavior of a physical system, is listed first among this topic's suggested skills, so expect to draw as well as to calculate.

The pairing to internalize is that emf is the slope of the flux-against-time graph, up to the minus sign. Everything about reading these graphs follows from that.

The second row is the one to have ready: a flux ramping steadily gives a constant emf, so the emf-against-time graph is a horizontal step, not a ramp. Students often draw a sloping emf because the flux is sloping. The fourth row is the other favorite: at a maximum of flux the slope is zero, so the emf is zero exactly where the flux is largest. Flux at its peak, emf at zero, which is the graph version of the spine of this topic.

Skill 2.D, functional dependence, reads off the same equation.

• Double the rate the field changes at, or the loop's area, and the emf doubles.
• Double the radius of a circular loop and the area quadruples, so the emf quadruples.
• Halve the time for the same flux change, and the emf doubles.
• Double the field but hold it steady, and the emf stays zero.

The question patterns are narrow, and rehearsing them is most of the preparation.

1. Compute a flux from $B$, $A$, and an angle measured to the area vector, then an average emf from a stated flux change, then a current from $I = \Delta V/R$ (skill 2.B).
2. Give the direction of an induced current and justify it in steps (skill 3.C), using the four-step routine.
3. Sketch flux, emf, or current against time for a described motion, or read one that is given (skill 1.C).
4. Predict how the emf changes when the area, the field, the rate, or the angle changes (skill 2.D).
5. Explain why a force is needed to keep an induced-current situation going, using energy conservation.

Topic 12.1 gives the field, Topic 12.2 the force it exerts on a moving charge, Topic 12.3 the two-way relationship between currents and fields, and Topic 12.4 closes it: a changing magnetic field produces an electric effect, the symmetry the unit opener has in mind when it says students will discover the natural symmetry between electricity and magnetism.