Topic 13.2: Electromagnetic Induction

What Topic 13.2 requires

Topic 13.2 has one learning objective and eight essential-knowledge statements under it, plus the only boundary statement in Unit 13.

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

• 13.2.A.1 states that Faraday's law describes the relationship between changing magnetic flux and the resulting induced emf in a system, with the relevant equation $\mathcal{E} = -\dfrac{d\Phi_B}{dt} = -\dfrac{d(\vec{B} \cdot \vec{A})}{dt}$.
• 13.2.A.1.i states that when the area of the surface being considered is constant, the induced emf is equal to the area multiplied by the rate of change in the component of the magnetic field perpendicular to the surface.
• 13.2.A.1.ii states that when the magnetic field is constant, the induced emf is equal to the magnetic field multiplied by the rate of change in area perpendicular to the magnetic field.
• 13.2.A.1.iii states that when an emf is induced in a long solenoid, the total induced emf is equal to the induced emf in a single loop multiplied by the number of loops in the solenoid, with the relevant equation $\lvert \mathcal{E}_{\text{sol}} \rvert = N \left\lvert \dfrac{d\Phi_B}{dt} \right\rvert$.
• 13.2.A.2 states that Lenz's law is used to determine the direction of an induced emf resulting from a changing magnetic flux.
• 13.2.A.2.i states that an induced emf generates a current that creates a magnetic field that opposes the change in magnetic flux.
• 13.2.A.2.ii states that the right-hand rule is used to determine the relationships between current, emf, and magnetic flux.
• 13.2.A.3 states that Maxwell's equations are the collection of equations that fully describe electromagnetism, and that Maxwell's third equation is Faraday's law of induction, which describes the relationship between a changing magnetic flux and an induced electric field, with the relevant equation $\mathcal{E} = \oint \vec{E} \cdot d\vec{\ell} = -\dfrac{d\Phi_B}{dt}$.
• 13.2.A.4 states that Maxwell's equations can be used to show that electric and magnetic fields obey wave equations and that electromagnetic waves travel at a constant speed in free space, with the derived equation $c = \dfrac{1}{\sqrt{\varepsilon_0 \mu_0}}$.

The suggested skills are 1.B, create quantitative graphs with appropriate scales and units, including plotting data; 2.A, derive a symbolic expression from known quantities by selecting and following a logical mathematical pathway; 2.C, compare physical quantities between two or more scenarios or at different times and locations in a single scenario; 3.A, create experimental procedures that are appropriate for a given scientific question; and 3.C, justify or support a claim using evidence from experimental data, physical representations, or physical principles or laws. That is five suggested skills, the joint most in the unit, alongside Topic 13.6.

The unit's only boundary statement, and what it does not bound

Topic 13.2 carries the only boundary statement in Unit 13. Quoted whole, both sentences:

"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."

Read what it fences and what it leaves open.

It fences one derivation. You are not expected to get from Maxwell's equations to $c = 1/\sqrt{\varepsilon_0 \mu_0}$. That is the whole of it. Statement 13.2.A.4 remains required content, so knowing that Maxwell's equations imply electromagnetic waves travelling at a fixed speed in free space is fair game; producing the wave equation is not.

It leaves everything else in the topic open. Nothing here restricts the functional form of the field or the flux you can be asked to differentiate, the number of turns, the geometry, or the induced-electric-field form of Faraday's law at 13.2.A.3.

The boundary statement's second sentence is the part that gets dropped when this is paraphrased, and dropping it changes the meaning. It says the relationship is printed "solely as an indication of the further applications, implications, and connections to physical phenomena that students may study in more advanced physics courses". So the equation is signposting, not content to be examined by derivation. Without that sentence the first one reads as though the equation might still show up in some other guise.

One related bound comes from the previous unit rather than this one. Statement 12.4.A.4 introduces Maxwell's fourth equation, Ampere's law with Maxwell's addition, and its boundary statement says the course does not expect students to use Maxwell's fourth equation with a changing electric field, while adding that students should understand that a changing electric field generates a magnetic field. The two boundary statements have the same shape: know the symmetry, do not be asked to compute with it.

Faraday's law is a derivative, so differentiate

That is essential knowledge 13.2.A.1, and the second form is the instruction: the thing you differentiate is a dot product of two things that can both change.

The procedure is always the same three moves, and doing them in order is most of the marks on a free-response question.

1. Write the flux as a function of time, symbolically. Use Topic 13.1. Keep every constant as a symbol so you can see which factor carries the $t$.
2. Differentiate. Product rule if two factors change, chain rule if the changing quantity is nested.
3. Take the magnitude and argue the sign separately. The minus sign is Lenz's law, and it is easier to reason about a physical direction than to track a sign through a dot product with a chosen area vector.

Work one all the way through. A circular loop of radius $r$ sits perpendicular to a uniform field $B(t)$, with the area vector along the field.

The radius does not change, so $\pi r^2$ is a constant and comes straight out:

That is statement 13.2.A.1.i in symbols: with the area constant, the emf is the area times the rate of change of the perpendicular component of the field. Notice what this gives you that a finite difference does not. If $B(t) = \beta t^2$ then $dB/dt = 2\beta t$ and the emf grows linearly with time. An average over an interval would have hidden that.

Now the other case. A loop with a fixed shape sits in a constant field $B$ but its area changes:

which is statement 13.2.A.1.ii. If the loop is circular with a radius that changes, the chain rule is unavoidable:

so $\lvert \mathcal{E} \rvert = 2\pi B r \lvert dr/dt \rvert$. The emf depends on the current radius, so a loop shrinking at a steady rate produces an emf that falls steadily to zero. The third worked example does this one numerically.

And when both change, use the product rule. With the area vector along the field,

The two terms can cancel. A loop shrinking while the field grows can hold its flux momentarily constant, and at that instant the emf is exactly zero even though nothing in the situation is at rest. That is the strongest argument available for why the emf tracks the flux rather than the field.

There is no motional emf equation in this course

Search the AP Physics C: E&M equation sheet for a rod-on-rails result and it is not there. Search the Unit 13 framework and it is not there either. This is one of the sharper differences from the algebra-based course, whose sheet prints one.

What you get instead is statement 13.2.A.1.ii, in words: when the magnetic field is constant, the induced emf is equal to the magnetic field multiplied by the rate of change in area perpendicular to the magnetic field. Turning that sentence into the familiar product takes three lines, and you should be able to produce them cold.

Take a conducting rod of length $\ell$ sliding at speed $v$ along rails, perpendicular to a uniform field $B$, with the field perpendicular to the plane of the circuit.

1. The enclosed area at time $t$ is $A(t) = \ell x(t)$, where $x$ is the rod's position along the rails.
2. So $\Phi_B = B\ell x(t)$ and $\dfrac{d\Phi_B}{dt} = B\ell \dfrac{dx}{dt} = B\ell v$.
3. Therefore $\lvert \mathcal{E} \rvert = B\ell v$.

Three lines, and every one of them is a step a scoring guideline can award. The habit is worth more than the result, because the moment the geometry stops being a rectangle the memorised product fails and the derivative does not. A rod sliding along rails that are not parallel, a rotating rod pivoted at one end, a loop entering a field at an angle: all of these are still $d\Phi_B/dt$, and none of them is $B\ell v$.

The rotating-rod case is the standard demonstration of that. A rod of length $\ell$ pivoted at one end sweeps a circular sector; in time $dt$ it sweeps angle $d\theta$ and area $dA = \frac{1}{2}\ell^2 d\theta$, so $\lvert \mathcal{E} \rvert = B \cdot \frac{1}{2}\ell^2 \omega$. The factor of one half is entirely a consequence of taking the derivative properly, and no amount of remembering $B\ell v$ produces it.

Lenz's law is the sign, and it is an energy argument

Essential knowledge 13.2.A.2 says Lenz's law is used to determine the direction of an induced emf resulting from a changing magnetic flux, and 13.2.A.2.i says what the law is: 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. Not the flux, and not the external field. The two readings give opposite answers half the time.

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

The external field points the same way in all three rows. The current reverses. So "the induced field opposes the applied field" is false in the second row, where the induced field points the same way as the applied one, propping up a flux that is draining away.

A four-step routine that puts 13.2.A.2.ii to work:

1. State the direction of the flux through the loop. Into the page, out of the page, along the axis.
2. Decide whether it is growing, shrinking, or steady. This is the step people skip and it is the step that decides the answer.
3. Work out which way the induced field must point inside the loop to oppose that change. Against the existing flux if it is growing, along with it if it is shrinking. If it is steady, stop, there is no current.
4. Curl your right hand. Thumb along the required induced field, fingers give the current direction around the loop.

Two checks catch most errors. Flipping the flux from increasing to decreasing must flip the answer; if it does not, step 3 used the flux instead of its change. And reversing the field direction also flips the current for a fixed sense of change, so two reversals cancel.

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 produced it. More flux would drive a larger current, whose field would make more flux still, with no source 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 exactly that mechanical work. Topic 13.3 is that audit carried out with numbers.

One of the CED's optional sample activities for this topic 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. Another asks students to use Maxwell's equations to construct arguments for why there can be no magnetic monopoles, what eddy currents are and why they exist, why a surface entirely within conducting material must have zero net charge within, and why there must be electric currents within Earth's core. Both are Lenz's law worn as engineering.

The turn count, and why it multiplies

Statement 13.2.A.1.iii gives the reasoning before the equation: when an emf is induced in a long solenoid, the total induced emf is equal to the induced emf in a single loop multiplied by the number of loops in the solenoid. The turns are in series, each one sees the same changing flux, and emfs in series add.

The symbol is worth pinning down, because two different turn counts appear in this unit and they are not interchangeable.

The equation sheet's variable key for the Electricity and Magnetism table lists both, as "n = number of loops per unit length" and "N = number of loops". They are related by $n = N/\ell$ for a solenoid of length $\ell$, and mixing them is the fastest way to be out by a factor of the length.

$\Phi_B$ in that expression is the flux through one loop, not the total. Compute the flux through a single turn, differentiate it, then multiply by $N$. Doing the multiplication first is harmless arithmetically but it makes the flux graph wrong by a factor of $N$, and the CED asks for flux graphs.

The emf being $N$ times larger is also the whole design of a transformer and of the CED's first sample activity for this topic, which has students spin a magnet inside a coil, measure the spin rate, the area, the number of coils and the peak induced voltage, and use those to estimate the strength of the magnet's field. Read that activity as an equation solved backwards: with $\mathcal{E}_{\text{peak}} = NBA\omega$ for a coil rotating at angular frequency $\omega$, every quantity but $B$ is measured.

Maxwell's third equation, and the induced electric field

This is the form the sheet prints, and statement 13.2.A.3 identifies it: Maxwell's third equation is Faraday's law of induction, describing the relationship between a changing magnetic flux and an induced electric field. It is worth a paragraph on its own, because it says something the loop-and-emf picture hides.

A changing magnetic flux produces an electric field whether or not there is a wire there. The middle expression is a closed line integral of the electric field around a path, and that path does not have to be made of copper. Put no loop at all in the region and the electric field is still there, circulating. A wire simply gives the charges something to move through.

That electric field is not conservative. Its line integral around a closed path is non-zero whenever the enclosed flux is changing. Compare the electrostatic case, where $\Delta V = -\int_a^b \vec{E} \cdot d\vec{r}$ and going around a closed loop returns you to the same potential. An induced electric field has no potential function in the usual sense, which is why the quantity on the left is called an emf and not a potential difference between two points.

That is also why 13.2.A.3's picture and 13.2.A.1's picture are the same law: integrate the induced electric field around the loop and you get the work per unit charge that drives the current, which is the emf.

Statement 13.2.A.4 then closes the unit's conceptual arc: Maxwell's equations can be used to show that electric and magnetic fields obey wave equations and that electromagnetic waves travel at a constant speed in free space, $c = 1/\sqrt{\varepsilon_0 \mu_0}$. The Table of Information prints $\varepsilon_0 = 8.85 \times 10^{-12} \ \mathrm{C^2/(N \cdot m^2)}$ and $\mu_0 = 4\pi \times 10^{-7} \ (\mathrm{T \cdot m})/\mathrm{A}$ as constants, and $c = 3.00 \times 10^8$ m/s separately, so you can check the identity numerically in one line even though the boundary statement says you will not be asked to derive it.

The unit's essential questions land here. How does an antenna work, how does a Wi-Fi connection work, how are sound waves generated by headphones from a digital recording: all of them are 13.2.A.3 and 13.2.A.4, and the reason the CED prints an equation it then tells you not to derive is that the arc has to close somewhere.

Graphs: emf is the slope of the flux graph

Skill 1.B, creating quantitative graphs with appropriate scales and units, is listed first for this topic, and skill 1.C, qualitative sketches, sits next door in Topics 13.4 and 13.5. The one relationship to internalise is that the emf is minus the slope of the flux-against-time graph. Everything else follows.

Row two is the one that gets drawn wrong: a flux ramping steadily gives a constant emf, so the emf graph is flat, not sloping. Row five is the other: at a maximum of flux the slope is zero, so the emf is zero exactly where the flux is largest.

Row four is the CED's own sample question. Its Question 2 puts a small loop inside a long solenoid whose current runs as $I(t) = I_0(1 - bt^2)$, and asks students to sketch the flux against time from $t = 0$ until the flux reaches zero and to label the time at which it vanishes. The scoring guidelines look for a curve that is concave downward everywhere, a positive maximum on the vertical axis with zero slope there, and the horizontal intercept at $1/\sqrt{b}$ clearly indicated. Zero slope at $t = 0$ is the graphical statement that the emf starts at zero, because $dI/dt = -2I_0bt$ vanishes there.

When you sketch, get three things right and the rest is detail: the value at $t = 0$, whether the curve is concave up or down, and where it crosses zero. Label the axes with units. The CED distinguishes skill 1.B, quantitative graphs with scales and units, from skill 1.C, qualitative sketches, and only one of those lets you leave the axes bare.

If you landed here from AP Physics 2

The algebra-based course covers induction in one topic, AP Physics 2 Topic 12.4, Electromagnetic Induction and Faraday's Law. That page is for students in AP Physics 2. This page is for students in AP Physics C: Electricity and Magnetism.

The physical law is the same. Four things about the treatment are not.

The practical consequence is that a Physics 2 induction question hands you two flux values and a time interval, while a Physics C question hands you a field or a current as a function of time and expects a function back. If you are in the algebra-based course, use the finite differences, and $B\ell v$ is yours to quote. If you are in Physics C, the derivative is the whole point, and the algebra-based shortcut is a result you re-derive rather than recall.

Lenz's law is identical in both, word for word in substance, so any Lenz's law reasoning transfers intact.

How Topic 13.2 is tested

This topic is the hinge of the unit, so it appears in combination more often than alone. The CED's own sample free-response Question 2 aligns to learning objectives 12.4.A, 13.1.A and 13.2.A together: the field of a solenoid from the previous unit, the flux from Topic 13.1, the derivative from here.

That question is worth reading as a template. It is a 12-point Translation Between Representations question. Part A asks students to indicate the direction of the solenoid's field in three regions, including writing "zero" outside. Part B asks students to derive an expression for the absolute value of the induced emf in a small internal loop, beginning the derivation by writing a fundamental physics principle or an equation from the reference information. Part C asks for a sketch of flux against time with the zero labelled. Part D asks for the factor by which the graph's vertical intercept changes if the loop's radius doubles, with a brief justification.

The scoring guidelines for part B award four points, and reading them tells you what a derivation is worth:

• one point for a multistep derivation that starts with $\mathcal{E} = -d\Phi/dt$,
• one for substituting the small loop's area into an expression for the magnetic flux,
• one for substituting a correct expression for the magnitude of the magnetic field inside a solenoid,
• one for correctly taking the time derivative of an expression for either the flux, the field, or the current, with a scoring note that the expression need not be correct to earn this point.

Three of the four points are for pathway rather than answer. That is the recurring lesson of the C free-response section, and the CED's own science-practices page for this unit says it plainly: simply being able to solve for a final answer is insufficient, and students may benefit from practice crafting clear, concise arguments, derivations, and calculations that follow a logical pathway.

The question patterns worth rehearsing:

1. Write a flux as a function of time, differentiate it, and report a symbolic emf (skill 2.A).
2. Evaluate that emf at an instant, then get a current from $I = \Delta V/R$ (skill 2.C).
3. Argue the direction of the induced current in steps (skill 3.C).
4. Sketch or plot flux, emf or current against time with labelled axes (skill 1.B).
5. Design a measurement, such as the spinning-magnet activity, and say what you would graph (skill 3.A).
6. Compare two instants or two scenarios and say which emf is larger and why (skill 2.C).