Unit 13: Electromagnetic Induction

What the CED requires across Unit 13

Unit 13 of AP Physics C: Electricity and Magnetism is Electromagnetic Induction. The course and exam description weights it at 10 to 20% of the multiple-choice section and suggests about 10 to 20 class periods. That is the same band as Unit 9 (Electric Potential) and Unit 12 (Magnetic Fields and Electromagnetism). Units 8 and 11 carry the top band at 15 to 25%, and Unit 10 the lowest at 10 to 15%.

Six topics, and exactly one learning objective each. That is unusual: most units in this course spread several objectives over fewer topics, and here the ratio is one to one all the way down.

Every one of those six objectives opens with the task verb "describe". The CED says that verb, used in nearly all learning objectives, "encompasses the range of possible graphical, mathematical, or verbal skill applications", and that students should be able to describe a concept graphically, mathematically, and verbally. So describing magnetic flux can mean drawing the area vector, writing the surface integral, or saying in words what gets through.

The CED's own framing of the unit is that it examines electromagnetism through the concept of electromagnetic induction and the application of Maxwell's equations, with students investigating the relationship between Faraday's law and Lenz's law. It says students are expected to call on earlier units, particularly charges, currents, and electric and magnetic fields, to be able to mathematically demonstrate as well as reason with how these fields are generated.

The essential questions printed on the unit opener are the everyday versions: how an electric motor works, how pushing a doorbell produces a sound inside the house, how an antenna works, how sound waves are generated by headphones or speakers from a digital recording of a song, and how a Wi-Fi internet connection works.

The unit has one boundary statement, and it is not where you would guess

Boundary statements are how the CED fences off a treatment, and in most units they land on the hardest calculation. Unit 13 prints exactly one across all six topics, and it sits under Topic 13.2 next to the speed of light. Quoted whole, second sentence included:

"AP Physics C: Electricity & Magnetism does not expect students to mathematically derive the speed of light in free space from Maxwell's equations. This relationship is included above solely as an indication of the further applications, implications, and connections to physical phenomena that students may study in more advanced physics courses."

That is the entire fencing in the unit. Topics 13.1, 13.3, 13.4, 13.5 and 13.6 print no boundary statement at all, which is worth knowing before you assume the differential equations in LR and LC circuits are bounded away. They are not bounded by a boundary statement. What bounds them is something else, and it is more precise.

The CED's front matter, on the Required Equations page, states that not all equations in the framework appear on the equation sheet, and that many are provided for reference and guidance "or to demonstrate the final results of derivations expected of students on the exam", labelled Derived Equations. Read the Unit 13 labels with that definition in hand and the treatment draws itself:

• 13.5.A.2 labels $\mathcal{E} = IR + L \frac{dI}{dt}$ a derived equation. So the differential equation itself is a derivation you are expected to produce, from Kirchhoff's loop rule applied to a series LR circuit with a battery. It is not printed on the sheet.
• 13.6.A.2 labels $\frac{d^2q}{dt^2} = -\frac{1}{LC}q$ a derived equation, and 13.6.A.3 does the same for the angular frequency, which it says "can be derived from the differential equation that describes an LC circuit".
• 13.2.A.4 labels $c = \frac{1}{\sqrt{\varepsilon_0 \mu_0}}$ a derived equation and then the boundary statement above cancels the expectation. It is the one derived equation in the unit you are told not to derive.

Nowhere in Unit 13 does the CED print a solved exponential such as a current-versus-time function for an LR circuit. What it prints instead is the behaviour: essential knowledge 13.5.A.4.ii states that the potential difference across an inductor, the current in the inductor, and the energy stored in the inductor "are exponential with respect to time and have asymptotes that are determined by the initial conditions of the circuit". The unit's "Building the Science Practices" page then names, as a valuable skill, "deriving a mathematical equation that shows the current in the solenoid as a function of time". So the exponential is expected knowledge and the derivation is a named skill, while the printed required content stops at the differential equation.

One more calibration comes from the CED's own sample free-response set. Its Question 4, on an RC circuit, asks students to "Derive, but do not solve, a differential equation" for a rate of change. That phrasing is the shape of the expectation across this part of the course: set up the equation from a fundamental principle, and be able to reason from it without necessarily integrating it.

How the six topics build

[13.1 Magnetic Flux](/ap-physics-c-electricity-magnetism/unit-13-electromagnetic-induction/13-1-magnetic-flux) is the definition the rest of the unit differentiates. Objective 13.1.A asks you to describe the magnetic flux through an arbitrary area or geometric shape. For a field constant across an area it is a dot product, and 13.1.A.2 gives the general case as a surface integral:

The words "arbitrary area or geometric shape" in the objective are the reason the integral is there.

[13.2 Electromagnetic Induction](/ap-physics-c-electricity-magnetism/unit-13-electromagnetic-induction/13-2-electromagnetic-induction) takes the derivative. Faraday's law arrives at 13.2.A.1 as $\mathcal{E} = -d\Phi_B/dt = -d(\vec{B} \cdot \vec{A})/dt$, Lenz's law arrives at 13.2.A.2 as the rule that fixes the sign, and 13.2.A.3 identifies Faraday's law as Maxwell's third equation, in the form with a closed line integral of the electric field. This is the only topic in the unit with a boundary statement.

[13.3 Induced Currents and Magnetic Forces](/ap-physics-c-electricity-magnetism/unit-13-electromagnetic-induction/13-3-induced-currents-and-magnetic-forces) closes the loop back onto mechanics. The induced current sits in the field that induced it, so the field pushes on it, with $\vec{F}_B = \int I (d\vec{\ell} \times \vec{B})$. Statement 13.3.A.4 then says Newton's second law can be applied to a conducting loop moving in a magnetic field as it experiences an induced emf, which is where the free-response questions live.

[13.4 Inductance](/ap-physics-c-electricity-magnetism/unit-13-electromagnetic-induction/13-4-inductance) turns induction into a circuit element. Statement 13.4.A.1 defines inductance as the tendency of a conductor to oppose a change in electrical current, 13.4.A.1.iii gives the solenoid's inductance from its geometry and core, 13.4.A.2 gives the stored energy $U_L = \frac{1}{2}LI^2$, and 13.4.A.3 gives the self-induced emf $\mathcal{E}_i = -L\,dI/dt$.

[13.5 LR Circuits](/ap-physics-c-electricity-magnetism/unit-13-electromagnetic-induction/13-5-circuits-with-resistors-and-inductors-lr-circuits) puts that element in a circuit with resistors. Kirchhoff's loop rule produces a differential equation, the time constant is $\tau = L/R_{\text{eq}}$, and four separate essential-knowledge statements describe what that constant means and how the circuit behaves at the two ends of time.

13.6 LC Circuits removes the resistor and the circuit oscillates. Statement 13.6.A.2 says outright that the time dependence of the charge on the capacitor "can be modeled as simple harmonic motion", which makes this the one topic in the whole E&M course that hands you a mechanics result to reuse rather than a new one to learn.

Read in order, the unit is one sentence with six clauses: define the flux, differentiate it, feel the force that comes back, package the effect as an inductor, put the inductor with a resistor, then put it with a capacitor.

The Unit 13 equations, and which ones are printed

Fourteen equations appear in Unit 13's required content or on the equation sheet. Nine of the fourteen are printed on the AP Physics C: E&M formula sheet, and knowing which five are not is worth more than knowing the nine.

Three observations from that table are worth carrying into the exam.

The plain dot-product flux is not printed. The sheet gives you the surface integral and nothing simpler. Collapsing $\int \vec{B} \cdot d\vec{A}$ to $BA\cos\theta$ for a uniform field is a step you take, and on a free-response question it is a step you justify.

There is no motional-emf equation anywhere in this course. Search the sheet and the framework for a rod-on-rails result and it is not there. The algebra-based course prints one on its sheet; this one does not, so a sliding-rod problem starts from $d\Phi_B/dt$ every time. Statement 13.2.A.1.ii is the closest the CED comes, and it is a sentence rather than a formula: when the magnetic field is constant, the induced emf equals the magnetic field multiplied by the rate of change in area perpendicular to the field.

The two differential equations are yours to write. Neither the LR loop equation nor the LC oscillator equation is on the sheet, and both carry the derived-equation label. What is on the sheet is what you need in order to write them: $\mathcal{E} = -L\,dI/dt$, $I = dq/dt$, $C = Q/\Delta V$, and $I = \Delta V / R$.

Because the C: E&M sheet reprints the whole C: Mechanics table, the mechanics results are available on the same page. That matters most in Topic 13.6, where $T = 2\pi/\omega = 1/f$ and $x = x_{\max}\cos(\omega t + \phi)$ are already printed, and in Topic 13.3, where $\vec{F}_{\text{net}} = d\vec{p}/dt$ and $P_{\text{avg}} = W/\Delta t = \Delta E/\Delta t$ do the work.

What calculus changes, and where AP Physics 2 stops

AP Physics C: Electricity and Magnetism is a calculus-based introductory college-level course, equivalent to the second course in an introductory college sequence in calculus-based physics. Its prerequisites say students should have taken or be concurrently taking calculus, and should have taken AP Physics C: Mechanics, AP Physics 1, or another mechanics-based physics course first.

Unit 13 is where that prerequisite earns its place, and the difference from the algebra-based treatment is not decoration.

• Flux is a surface integral, not a product. A field that varies across the surface, such as the field near a long straight wire, gives a flux you have to integrate for.
• Faraday's law is a derivative, not a ratio of differences. You get the emf at an instant rather than an average over an interval, so a field that changes non-linearly in time is fair game, and the answer is a function of time rather than a number.
• Inductance, LR circuits and LC circuits do not exist in AP Physics 2 at all. The words inductance and inductor appear nowhere in the AP Physics 2 course and exam description. Half of Unit 13, Topics 13.4 through 13.6, has no algebra-based counterpart in that course.
• The LC circuit is a second-order differential equation, which is why the CED can say its solution is simple harmonic motion instead of asking you to find it from scratch.

The nearest algebra-based page on this site is AP Physics 2 Topic 12.4, Electromagnetic Induction and Faraday's Law, which compresses flux, Faraday, Lenz and the sliding rod into a single topic with $\Phi_B = BA\cos\theta$ and $\lvert \mathcal{E} \rvert = \lvert \Delta\Phi_B/\Delta t \rvert$. That page is for AP Physics 2 students; this unit is for AP Physics C students, and the two are not the same material at different reading levels. If you are in Physics 2 you want the finite differences and you can stop before inductance. If you are in Physics C you want the derivative, the integral, and three topics the other course never reaches.

The corresponding AP Physics 2 Unit 12 hub covers magnetic fields, moving charges, wires and induction in four topics. Physics C splits the same ground across two units: Unit 12 for fields, the Biot-Savart law and Ampere's law, and this unit for everything that follows from a flux that changes.

Lenz's law is an energy argument, and it runs through the whole unit

Essential knowledge 13.2.A.2.i states it: an induced emf generates a current that creates a magnetic field that opposes the change in magnetic flux. Read the object of that verb carefully. The induced field opposes the change, not the flux and not the external field. With a flux into the page and growing, the induced field inside the loop points out of the page. With that same flux into the page but shrinking, the induced field points into the page, the same way as the external field, propping up what is draining away. Learning the rule as "opposes the field" gets every decreasing case backwards.

The minus sign could not have gone the other way, and the reason is conservation of energy. Suppose an induced current reinforced the change that made it. More flux would drive a current whose field made more flux still, with no source supplying the energy. The minus sign is what makes induction cost work.

That argument reappears in every later topic, which is why it is worth learning once properly.

• In 13.3, the force on the induced current opposes the motion that produced it, so keeping a loop moving through a field boundary takes a sustained applied force, and the mechanical power that force delivers equals the electrical power dissipated.
• In 13.4, an inductor's self-induced emf opposes the change in its own current, which is what statement 13.4.A.1 means by "the tendency of a conductor to oppose a change in electrical current".
• In 13.5, that opposition is why the current in an LR circuit rises smoothly instead of jumping, and why 13.5.A.1 says a resistor will dissipate energy that was stored in an inductor as the current changes.
• In 13.6, with no resistor to dissipate anything, the same opposition returns the energy instead of spending it, and the circuit oscillates. Statement 13.6.A.1 names conservation of energy as the route to the maximum current.

One of the CED's optional sample activities for Topic 13.2 asks students to describe qualitatively how electromagnetic braking works, including how electromagnetic brakes are structured, how they can recharge a battery, and how they can double as electric motors. That single activity is Lenz's law, energy conservation and Topic 13.3 in one object.

Traps that span more than one topic

A large flux is not an emf. A loop sitting in the strongest steady field you can build has an enormous flux and exactly zero induced emf. Faraday's law contains a derivative, so the quantity that has to be non-zero is $d\Phi_B/dt$.

A zero flux is not a zero emf either. A rotating coil passes through the orientation where the flux is zero at the moment its emf is largest, because that is where the cosine is steepest. Flux at its maximum, emf at zero; flux at zero, emf at its maximum. The two peak a quarter cycle apart.

The area in a flux is the area the field actually crosses. A small loop inside a long solenoid has flux $B \cdot \pi r_{\text{loop}}^2$. A large loop encircling that same solenoid from outside has flux $B \cdot \pi r_{\text{sol}}^2$, because statement 12.4.A.1.ii of the previous unit says all solenoids are assumed to be very long, with uniform fields inside and negligible fields outside. Using the wrong radius is the most mechanical error available in Topic 13.1.

Only the segments inside the field feel a force. Statement 13.3.A.2 says so directly, and adds that these forces may cause translational or rotational acceleration. A loop entirely inside a uniform field has no flux change, no induced current, and no net force. A loop straddling the boundary has all three.

An inductor is not a resistor at any instant, and it is not an inductor after a long time. At the moment a switch changes, statement 13.5.A.4.i says the induced emf is equal in magnitude and opposite in direction to the applied potential difference across the branch containing the inductor, so the current through it does not jump. After a time much greater than the time constant, 13.5.A.4.iii says the inductor behaves as a conducting wire with zero resistance. The two limits are the two questions that need no calculus, and they are asked constantly.

The time constant carries the equivalent resistance, not the nearest resistor. The CED writes $\tau = L/R_{\text{eq}}$ with the subscript in place. In any circuit with more than one resistor, the resistance in that expression is the one the inductor actually sees, and it is often neither of the printed values.

An LC circuit has no time constant. It has an angular frequency. Reaching for $\tau$ in Topic 13.6, or for $\omega$ in Topic 13.5, is the single cleanest way to tell which circuit you are looking at: exponential approach for LR, oscillation for LC.

How Unit 13 is assessed

The AP Physics C: Electricity and Magnetism exam is 3 hours long. Section I is 42 multiple-choice questions in 85 minutes for 50% of the score. Section II is 4 free-response questions in 95 minutes for the other 50%, always one of each type in a fixed order: Mathematical Routines, Translation Between Representations, Experimental Design and Analysis, and Qualitative/Quantitative Translation. A four-function, scientific, or graphing calculator is allowed on both sections.

With four free-response questions and six units, a Unit 13 free-response question is not guaranteed in a given year. The reliable figure is the multiple-choice weighting, 10 to 20%. The unit's AP Classroom Progress Check runs about 18 multiple-choice questions and 4 free-response questions, one of each type.

The suggested skills the CED lists per topic are teaching suggestions rather than a promise about the exam, but they read as a map of the question styles:

Skill 2.C, comparing physical quantities between two or more scenarios or at different times in a single scenario, is the only skill listed for all six topics. Skill 2.A, deriving a symbolic expression by following a logical mathematical pathway, is listed for four of the six. Science Practice 2 as a whole carries a 40 to 45% weighting on the free-response section; the CED merges that column across skills 2.A through 2.D and publishes no per-skill figure, so treat it as a statement about mathematical routines generally rather than about 2.A alone. Symbolic derivation is the centre of gravity of this unit.

Topics 13.2 and 13.6 are the two that list 3.A, creating experimental procedures, which fits a unit whose lab work is spinning magnets in coils and measuring circuit behaviour. The CED's exam-preparation note for Unit 13 says laboratory investigations about the behavior of circuits when a solenoid is in the circuit are valuable exercises that concretely demonstrate principles and ideas that are often abstract. Its five optional sample activities cluster on exactly two topics: three on 13.2 and two on 13.5, including building a solenoid and measuring its inductance by getting the time constant and the resistance of an LR circuit, then repeating with an iron or steel core to get the increased inductance.

One worked example of what a Unit 13 free-response question looks like is printed in the CED itself. Its sample Question 2, a Translation Between Representations question worth 12 points, aligns to learning objectives 12.4.A, 13.1.A and 13.2.A. It puts a small loop of radius $r_0$ inside a long solenoid of radius $3r_0$ with $n_0$ turns per unit length and current $I(t) = I_0(1 - bt^2)$, then asks students to indicate field directions in three regions, derive the magnitude of the induced emf in the loop, sketch the flux against time and label where it reaches zero, and state by what factor the graph's vertical intercept changes if the loop's radius doubles. The scoring guidelines award a point for a multistep derivation that starts from $\mathcal{E} = -d\Phi/dt$, and a further point for correctly taking the time derivative, explicitly noting that the expression being differentiated need not be correct to earn it. Method is worth points on its own.