Unit 12: Magnetic Fields and Electromagnetism

What the CED requires across Unit 12

Unit 12 of AP Physics C: Electricity and Magnetism is Magnetic Fields and Electromagnetism. 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 13 (Electromagnetic Induction). Units 8 and 11 carry the top band at 15 to 25%, and Unit 10 the lowest at 10 to 15%.

Four topics carry eight learning objectives, and they are distributed unevenly: Topic 12.1 alone holds three of the eight, while Ampere's law gets one.

Seven of those eight objectives open with the task verb "describe". The eighth does not. Objective 12.4.A reads "Use Ampere's law to describe the magnetic field created by a moving charge carrier", and it is the only objective in the unit that opens with "Use". The CED says "describe", used in nearly all learning objectives, "encompasses the range of possible graphical, mathematical, or verbal skill applications". Reading 12.4.A against that, the verb change is a hint about what the topic is: a tool you apply rather than a phenomenon you characterise.

The CED's framing is that "Unit 12 introduces students to magnetism and how magnetic fields are generated, behave, and relate to electricity", and that laboratory investigations or activities "should be provided for students to apply both the Biot-Savart Law (using calculations to determine the strength of a magnetic field) and Ampere's Law (deriving mathematical relationships which relate the magnitude of the magnetic field to current)." It closes by saying earlier units help students connect electric fields to magnetic fields and Gauss's Law to Ampere's Law, which is the structural idea to carry through the unit.

Its essential questions are why large-scale charged-particle accelerators such as those at CERN are shaped like a circle, how a guitar pickup works, why an electric current deflects a compass needle, and why the deflection of a pair of parallel conducting wires depends on the direction of current in the wires. Progress Check 12 lists about 18 multiple-choice questions and 4 free-response questions, one of each type.

What calculus changes: two integrals, and knowing which one to reach for

AP Physics C: Electricity and Magnetism is a calculus-based introductory college-level course, equivalent to the second course in a college sequence in calculus-based physics, and its prerequisites say students should have taken or be concurrently taking calculus. In Unit 12 the whole payoff is two integral laws, neither of which names a topic in the algebra-based course.

The [Biot-Savart law](/glossary/biot-savart-law) is a superposition integral. Statement 12.3.A.1 says it "defines the magnitude and direction of a magnetic field created by an electrical current" and prints

Every piece of that matters. The $1/r^2$ makes it a cousin of Coulomb's law, the cross product sets the direction, and the $d$ on both sides means you get the field of one infinitesimal segment and then add up. Statement 12.3.A.2 gives the geometric consequence: the magnetic field vectors around a small segment of a current-carrying wire are tangent to concentric circles centred on that wire, and the field has no component toward, away from, or parallel to the segment.

[Ampere's law](/glossary/amperes-law) is a closed line integral. Statement 12.4.A.1 relates the magnitude of the field to the current enclosed by a closed imaginary path called an Amperian loop:

That is the magnetic analogue of Gauss's law, and it works the same way: it is always true, but it only lets you solve for $B$ when the symmetry is good enough to pull $B$ outside the integral. Statement 12.4.A.2 defines the Amperian loop as a closed path around a current-carrying conductor, and 12.4.A.3 adds that the principle of superposition gives the net field from combinations of conductors, loops, segments or cylinders.

The two laws are not interchangeable, and choosing between them is the skill this unit tests. Biot-Savart handles a finite piece of wire or a loop, where symmetry is poor and you have to integrate contributions. Ampere handles a long wire, a solenoid or a current-carrying cylinder, where symmetry is high and the integral collapses. Each has its own boundary statement fencing off exactly which shapes are fair, and they are quoted whole in the next section.

One place calculus does not arrive is worth flagging. Objective 12.2.A, on the magnetic field produced by a moving charged object, prints no equation at all. Its three essential knowledge statements are purely qualitative: the field at a point depends on the object's velocity and the distance between the point and the object (12.2.A.1.i), its direction is perpendicular to both the velocity and the position vector and is found with the right-hand rule (12.2.A.1.ii), and its magnitude is maximum when those two vectors are perpendicular (12.2.A.1.iii). There is no point-charge Biot-Savart formula in this course.

Unit 12's three boundary statements, quoted whole

Read these three before you practise anything, because they tell you which magnetostatics problems the exam can actually pose. Two sit under Topic 12.4 and one under Topic 12.3. Topics 12.1 and 12.2 print none.

Under 12.3.B: "AP Physics C: Electricity & Magnetism only expects students to perform quantitative analysis of certain cases of current-carrying conductors using the Biot-Savart law, such as at a location along the perpendicular bisector of a straight conductor, at a location along the central axis of a circular loop, or at the center of a segment of a circular loop."

Three named geometries, and the word "such as" in front of them, so the list illustrates rather than closes. What it rules out in practice is the off-axis field of a loop and any point at an arbitrary angle to a finite wire.

Under 12.4, first statement: "AP Physics C: Electricity & Magnetism only expects quantitative application of Ampere's law limited to situations involving symmetrical magnetic fields. Long straight wires, long solenoids carrying currents, as well as conductive slabs or cylindrical conductors carrying a current density, are the types of shapes to which Ampere's law will be applied on the AP Physics C: Electricity & Magnetism Exam."

Note the last item. A cylindrical conductor carrying a current density is explicitly on the list, which is where Unit 11's current density comes back: to get $I_{\text{enc}}$ inside a thick wire you integrate $\vec{J}$ over the part of the cross-section your loop encloses.

Under 12.4, second statement: "AP Physics C: Electricity & Magnetism does not expect students to use Maxwell's fourth equation with a changing electric field. However, students should understand that a changing electric field generates a magnetic field."

The second sentence is the half that gets dropped, and dropping it turns a limit on calculation into a licence to ignore the physics. Statement 12.4.A.4 is what the boundary is fencing: Maxwell's equations are the collection of equations that fully describe electromagnetism, Maxwell's fourth is Ampere's law with Maxwell's addition, and it states that magnetic fields can be generated by electric current (Ampere's law) and that a changing electric field creates a magnetic field, "similar to the way a moving charge creates a magnetic field" (Maxwell's addition). You are expected to know the displacement-current term exists and what it means; you are not expected to compute with it.

One essential knowledge statement behaves like a fourth boundary. Statement 12.4.A.1.ii reads: "Unless otherwise stated, all solenoids are assumed to be very long, with uniform magnetic fields inside the solenoids and negligible magnetic fields outside the solenoids." That single sentence is why solenoid problems are tractable at all, and it is also the sentence a magnetic flux question in Unit 13 leans on when it asks about a loop encircling a solenoid from outside.

The Unit 12 equations, and which ones are printed

Nine equations appear in Unit 12's required content. Six are printed on the AP Physics C: E&M formula sheet and three are not.

Three readings of that table are worth carrying into the exam.

The field of a long straight wire is not on the sheet. $B = \dfrac{\mu_0 I}{2\pi r}$ carries the Derived Equation label at 12.4.A.1.i, which the CED's Required Equations page defines as a final result of a derivation expected of students on the exam. So you are expected to produce it from Ampere's law with a circular Amperian loop, every time. The same is true of the field at the centre of a circular loop, $\dfrac{\mu_0 I}{2R}$, which 12.3.A.3 labels derived and offers as the worked example of what Biot-Savart is for.

Exactly one derived equation in this unit is printed anyway, and it is the solenoid. Three Unit 12 equations carry the derived label: the loop centre, the straight wire, and $B_{\text{sol}} = \mu_0 n I$. Only the third appears on the sheet, where $n$ is defined in the variable list as the number of loops per unit length. Do not read that as permission to skip its derivation, since the label still says the derivation is expected.

Both Maxwell equations in this unit are in the framework, but only one is printed. Statement 12.1.A.3 says magnetic field lines must form closed loops, as described by Gauss's law for magnetism, and 12.1.A.3.i identifies that as Maxwell's second equation with $\oint \vec{B} \cdot d\vec{A} = 0$, which is on the sheet. The fourth equation with the displacement-current term is not, and the second boundary statement above says why.

The Coulomb constant, $\varepsilon_0 = 8.85 \times 10^{-12}$ and $\mu_0 = 4\pi \times 10^{-7}$ are all in the sheet's constants table, so $\mu_0/4\pi$ is exactly $1.0 \times 10^{-7}$ and $\mu_0/2\pi$ is exactly $2.0 \times 10^{-7}$. Those two shortcuts save real time. Because the C: E&M sheet reprints the whole C: Mechanics table, $a_c = v^2/r = r\omega^2$, $\vec{F}_{\text{net}} = d\vec{p}/dt$ and $T = 2\pi/\omega = 1/f$ are on the same page, which is what a charged particle circling in a magnetic field needs. Note what is not printed anywhere: the radius of that circular path. $r = mv/(qB)$ has to be assembled from $\vec{F}_B = q(\vec{v} \times \vec{B})$ and the centripetal condition.

Traps that span more than one topic

"Because of the right-hand rule" is not a justification, and the CED says so about Lenz's law by name. The unit's exam-preparation page warns that "simply referencing an equation, law or physical principle is not sufficient", and gives its own example: "stating that the induced current in a loop is clockwise because of 'Lenz's Law' is not complete enough of an answer to earn credit on the free-response section of the exam." Instead students "should cultivate the habit of including references to the right hand rule and the resistance to the change in magnetic flux in their responses." Name the rule and then show it being applied.

Ampere's law is always true and only sometimes useful. The equation holds for any closed path. It solves for $B$ only when symmetry makes $\vec{B}$ constant in magnitude and parallel to $d\vec{\ell}$ along the whole loop. That is the same limitation Gauss's law has, and the first Topic 12.4 boundary statement names the four shapes where it holds on this exam.

Biot-Savart and Ampere are not alternatives for the same problem. A finite straight segment or a single loop is a Biot-Savart problem, because no Amperian loop has the right symmetry. An infinite wire, a long solenoid or a thick cylinder is an Ampere problem, because integrating Biot-Savart there is unnecessary work. Picking the wrong tool is how a 12-point question turns into a 3-point one.

Inside a thick conductor, only the enclosed current counts. For a cylinder carrying uniform current density, an Amperian loop of radius $r < a$ encloses $I(r^2/a^2)$, so $B$ grows linearly with $r$ inside and falls as $1/r$ outside, peaking at the surface. Two different radii can give the same field, which makes this a favourite for skill 2.C comparisons.

A magnetic force never changes a particle's speed. Because $\vec{F}_B = q(\vec{v} \times \vec{B})$ is always perpendicular to $\vec{v}$, it does no work. It bends the path without changing the kinetic energy, which is the answer to the CED's own essential question about why particle accelerators are circular. Statement 12.2.B.2 adds the companion point: in a region containing both an electric and a magnetic field, a moving charged object experiences independent forces from each field, so you superpose them rather than combining them into one rule.

A monopole is never the answer. Statement 12.1.A.1.i says magnetic fields can be produced by magnetic dipoles or combinations of dipoles, but never by monopoles, and 12.1.B.1.ii adds that no magnetic north pole is ever found in isolation from a south pole: break a bar magnet in half and both halves are dipoles. That is the physical content of $\oint \vec{B} \cdot d\vec{A} = 0$.

Permeability is not a constant of a material. Statement 12.1.C.2 says free space has a constant value, the vacuum permeability $\mu_0$. Statement 12.1.C.3 then says the permeability of matter differs from that of free space, arises from the matter's composition and arrangement, and "is not a constant for a material and varies based on many factors, including temperature, orientation, and strength of the external field."

All three magnetic behaviours are in the framework, and the third is universal. Ferromagnetic materials such as iron, nickel and cobalt can be permanently magnetized (12.1.B.3.i); paramagnetic materials such as aluminum, titanium and magnesium interact weakly and do not stay aligned once the external field is removed (12.1.B.3.ii); and 12.1.B.3.iii says all materials have the property of diamagnetism, aligning weakly opposite the external field. Diamagnetism is not a third category of substance, it is a property everything has.

How Unit 12 is assessed

The exam is 3 hours long: 42 multiple-choice questions in 85 minutes for half the score, then 4 free-response questions in 95 minutes for the other half, one of each type in a fixed order. A four-function, scientific, or graphing calculator is allowed on both sections.

Skill 2.A, deriving a symbolic expression by following a logical mathematical pathway, is listed for three of this unit's four topics, and on the multiple-choice section it carries 25 to 30% of the weighting, the largest share of any single skill. On the free-response section Science Practice 2 as a whole carries 40 to 45%, more than either other practice. For a unit whose two central equations are both integrals you are expected to evaluate rather than quote, that alignment is the whole story.

The unit's Building the Science Practices page names skills 2.C, 2.D, 3.B and 3.C, and its worked illustration is a functional-dependence question: describe what happens to the net force on an electron moving through a magnetic field if its speed increases, then justify what that does to the radius of its path. The CED says students should be comfortable "making claims about the reasonableness of their claims and justifications made with functional dependence (2.D, 3.C), starting with the first principles of physics."

Unit 12 appears twice in the CED's own sample free-response set, out of four questions.

• Sample Question 2, a Translation Between Representations question worth 12 points, aligns to 12.4.A, 13.1.A and 13.2.A. It puts a small conducting loop inside a long solenoid, asks students to indicate the field direction inside the solenoid on and off its axis and outside it, then to derive an induced emf using $B_{\text{sol}} = \mu_0 n I$. The three-region field question is 12.4.A.1.ii doing its work.
• Sample Question 3, an Experimental Design and Analysis question worth 10 points, aligns to 11.3.B, 12.3.B and 12.4.A. Two parallel wires carry currents in opposite directions and a force sensor measures the force between them. Students plot force against current squared, then extract the vacuum permeability from the slope. The CED's own worked solution gets $\mu_0 = 1.48 \times 10^{-6}\ \mathrm{N/A^2}$ from the drawn line, which is close to but not equal to $4\pi \times 10^{-7}$, and that gap is the point of a lab question.

Two of the fifteen sample multiple-choice questions align to Unit 12, one to 12.4.A and one to 12.3.B. Of the unit's five optional sample instructional activities, three are on Topic 12.4: one on how far from a high-tension power line you must stand before its field equals Earth's, one on designing a solenoid to match an MRI machine's field, and one desktop task using a compass deflection and Ampere's law to measure Earth's magnetic field. The course requires that 25 percent of instructional time be spent in hands-on laboratory work.

Where AP Physics 2 stops, and who each page is for

AP Physics 2 has a Unit 12 as well, called Magnetism and Electromagnetism, and it also has four topics. Two of the four titles match this unit exactly: 12.1 Magnetic Fields and 12.2 Magnetism and Moving Charges. The other two diverge, and the divergence tells you how the courses split the material. Physics 2's 12.3 is Magnetism and Current-Carrying Wires and its 12.4 is Electromagnetic Induction and Faraday's Law, so the algebra-based course finishes magnetism and induction inside one unit. Physics C spends this whole unit on fields and their sources, then gives induction a unit of its own.

The calculus difference in the two topics that diverge is not decoration.

• Physics C's 12.3 adds the Biot-Savart law by name, as a superposition integral over current elements, plus the force on a wire as $\int I (d\vec{\ell} \times \vec{B})$ rather than a product.
• Physics C's 12.4 is Ampere's law, which is not a topic in the algebra-based course at all, along with Maxwell's second and fourth equations.

The [AP Physics 2 Unit 12 hub](/ap-physics-2/unit-12-magnetism-and-electromagnetism) is for AP Physics 2 students; this page is for AP Physics C students. They are not the same material at two reading levels. If you are in the algebra-based course you want the field of a wire as a formula to use, and induction in the same unit. If you are in Physics C you want the two integral laws, the symmetry argument that makes one of them solvable, and the derivations the sheet withholds on purpose.

Unit 12 is also the setup for what follows. Everything in Unit 13, Electromagnetic Induction starts from a magnetic flux, and a flux needs a field: the solenoid field from 12.4.A.1.iii and the wire field from 12.4.A.1.i are what get differentiated there. The full course map is on the AP Physics C: Electricity and Magnetism hub.