AP Physics 2 · Topic 12.1

Topic 12.1: Magnetic Fields

Unit 12: Magnetism and Electromagnetism12-15% of the multiple-choice section

Topic 12.1 defines the magnetic field as a vector field produced by dipoles and never by monopoles. Its field lines close on themselves, so breaking a bar magnet in half gives you two dipoles rather than a loose north pole. A material's magnetic behavior depends on how its own dipoles line up.

AP Physics: Unit 12 (topics 12.1 Magnetic Fields). Topic 12.1 carries three CED learning objectives: 12.1.A, describe the properties of a magnetic field; 12.1.B, describe the magnetic behavior of a material as a result of the configuration of magnetic dipoles in the material; and 12.1.C, describe the magnetic permeability of a material. The topic has no boundary statement and no relevant equation in the CED, and the vacuum permeability is its only entry on the AP Physics 2 equation sheet. The CED's suggested skills for this topic are 1.A, 2.C, 3.B and 3.C. Unit 12 is weighted at 12 to 15 percent of the multiple-choice section and estimated at about 10 to 14 class periods.

What Topic 12.1 requires

Topic 12.1 opens Unit 12, which the CED weights at 12 to 15 percent of the multiple-choice section and estimates at roughly 10 to 14 class periods. Three learning objectives sit under it:

12.1.A Describe the properties of a magnetic field.
12.1.B Describe the magnetic behavior of a material as a result of the configuration of magnetic dipoles in the material.
12.1.C Describe the magnetic permeability of a material.

Every verb is describe, and that is not a quirk of phrasing. The CED lists no relevant equation anywhere in Topic 12.1. Unit 12's first appears under Topic 12.2, and the first that gives you the field of a source appears under Topic 12.3. The only piece of 12.1 printed anywhere on the AP Physics 2 equation sheet is the vacuum permeability, and it is printed as a constant rather than as a relationship.

Topic 12.1 also carries no boundary statement. Unit 12 prints exactly one across all four of its topics, and it belongs to Topic 12.2. Nothing here is fenced off.

The suggested skills are 1.A, create diagrams, tables, charts, or schematics to represent physical situations; 2.C, compare physical quantities between two or more scenarios or at different times and locations in a single scenario; 3.B, apply an appropriate law, definition, theoretical relationship, or model to make a claim; and 3.C, justify or support a claim using evidence from experimental data, physical representations, or physical principles or laws. Two of the four are argumentation, one is drawing, one is comparison. This topic is examined in sentences and sketches.

A vector field, and the three things it acts on

EK 12.1.A.1 puts the definition in one sentence: a magnetic field is a vector field that can be used to determine the magnetic force exerted on moving electric charges, electric currents, or magnetic materials.

Read the list of three at the end. A magnetic field acts on moving electric charges, which is Topic 12.2; on electric currents, which is Topic 12.3; and on magnetic materials, which is the compass needle and the paperclip and the rest of this page. One field, three kinds of target, and the unit takes them in that order.

The word moving is load bearing. A charge sitting still in a magnetic field feels nothing from that field. That is the first structural difference from the electric field, which pushes a charge whether it moves or not; the electric field and potential guide covers that side.

EK 12.1.A.2 adds that a magnetic field is a vector quantity and can be represented using vector field maps. A vector field map is the picture with a small arrow at each point of a grid, its direction giving the field direction there and its length the relative strength. Skill 1.A is the drawing skill, and this is the drawing it wants.

Two statements pin down what those maps look like.

12.1.A.2.i Magnetic field lines form closed loops.
12.1.A.2.ii Magnetic fields in a bar magnet form closed loops, with the external magnetic field pointing away from one end (defined as the north pole) and returning to the other end (defined as the south pole).

The parenthesis in 12.1.A.2.ii is a definition of the poles, not a description of them. North is defined as the end the external field points away from. So when a question hands you a field map and asks which end is north, you need to know nothing about the magnet itself: follow the arrows outside it, and the end they leave from is north.

Field lines close on themselves, so there is no monopole

EK 12.1.A.1.i is blunt: magnetic fields can be produced by magnetic dipoles or combinations of dipoles, but never by monopoles. EK 12.1.A.1.ii adds that magnetic dipoles have north and south polarity. Set those beside 12.1.A.2.i, that field lines form closed loops, and the argument is complete.

The cleanest way to hold it is to put the two fields side by side.

	Electric field	Magnetic field
Smallest possible source	a single charge, one sign on its own	a dipole, north and south together
Field lines	begin on positive charge, end on negative charge	form closed loops, no beginning and no end
Cut the source in half	you can end up holding one isolated charge	you end up holding two dipoles (EK 12.1.B.1.ii)

EK 12.1.B.1.ii states that last row outright: no magnetic north pole is ever found in isolation from a south pole, and if a bar magnet is broken in half, both halves are magnetic dipoles. There is no depth at which the halving finally frees a lone pole, because a dipole is not two separable pieces stuck together. It is charge going in circles, which the next section takes up.

One claim to state carefully. AP Physics C: Electricity and Magnetism writes the no-monopole rule as an equation, the closed-surface flux integral of the magnetic field being zero, printed alongside Gauss's law for the electric field on the C: E&M equation sheet. Neither integral is on the AP Physics 2 sheet. Count its Magnetism group and you find seven entries, all of them about force, the field of a straight wire, magnetic flux, or induced emf. In this course the absence of monopoles is a stated property you argue from, not a formula you can cite.

Magnetic dipoles are charge going in circles

EK 12.1.B.1 says magnetic dipoles result from the circular or rotational motion of electric charges, and that in magnetic materials this can be the motion of electrons.

That sentence ties the unit together. A dipole is circulating charge. A current loop in Topic 12.3 is circulating charge. A single charge with a velocity in Topic 12.2 is the smallest piece of the same idea. The unit's opening page says students will discover the natural symmetry between electricity and magnetism, and this is where it starts: magnetism is what charge in motion does.

Four statements sit under it.

12.1.B.1.i Permanent magnetism and induced magnetism are system properties that both result from the alignment of magnetic dipoles within a system. Both. The difference between a fridge magnet and a paperclip stuck to one is whether the alignment persists, not what causes it.
12.1.B.1.ii No magnetic north pole is ever found in isolation from a south pole.
12.1.B.1.iii Magnetic poles of the same polarity will repel; magnetic poles of opposite polarity will attract.
12.1.B.1.iv The magnitude of the magnetic field from a magnetic dipole decreases with increasing distance from the dipole.

Read 12.1.B.1.iv exactly as written. It says the field decreases with distance. It gives no power and no proportionality, and the CED prints no dipole field equation anywhere in AP Physics 2. So a question can ask you to say the field is weaker further out, or to rank two locations, which is skill 2.C. It cannot ask you for a factor. Do not import an inverse-cube law from a textbook: here you would be answering a question nobody asked, with a number you could not source.

EK 12.1.B.2 hands you the measuring instrument: a magnetic dipole, such as a magnetic compass, placed in a magnetic field will tend to align with the magnetic field. Scatter compasses around a bar magnet and every needle turns to lie along the local field, so the array of needles becomes the vector field map skill 1.A asks you to draw.

Ferromagnetic, paramagnetic, diamagnetic

EK 12.1.B.3 sets the classification up: a material's composition influences its magnetic behavior in the presence of an external magnetic field. Three sub-statements name the three behaviors, and each names its own examples.

Behavior	What the CED says it does	Materials the CED names
Ferromagnetic (12.1.B.3.i)	Can be permanently magnetized by an external field that causes the alignment of magnetic domains or atomic magnetic dipoles	iron, nickel, cobalt
Paramagnetic (12.1.B.3.ii)	Interacts weakly with an external magnetic field, in that the magnetic dipoles of the material do not remain aligned after the external field is removed	aluminum, titanium, magnesium
Diamagnetic (12.1.B.3.iii)	Electronic structure creates a usually weak alignment of the dipole moments of the material opposite the external magnetic field	all materials

Three details in that table decide questions, and each is a place where the plausible answer is the wrong one.

Diamagnetism is universal. EK 12.1.B.3.iii says all materials have the property of diamagnetism. Not that some materials are diamagnetic. Every material has it; in iron it is buried under a far stronger ferromagnetic response, and in aluminum under a paramagnetic one. An answer option offering "only certain materials are diamagnetic" is the trap.

The diamagnetic alignment is opposite the external field. The ferromagnetic and paramagnetic responses align with the external field; the diamagnetic one aligns against it. That direction is the substance of the statement, and it is why the CED hedges the size with "usually weak" instead of giving a number.

Paramagnetic is not weakly ferromagnetic. The CED draws the line at what happens after the field is taken away: paramagnetic dipoles do not remain aligned, ferromagnetic ones can. The word permanently in 12.1.B.3.i is doing real work.

EK 12.1.B.1.i already gave the mechanism behind the ferromagnetic case: permanent magnetism is a system property resulting from the alignment of dipoles within the system, so magnetizing a nail adds nothing to the nail. It turns dipoles that were already there until they point the same way.

Earth as a dipole, and the pole with the wrong name

The CED's statement about Earth is a single line. EK 12.1.B4 says Earth's magnetic field may be approximated as a magnetic dipole. (The CED prints that code without a second period; every sibling code in the unit uses the dotted form.) That is the whole of it: no field strength, no tilt angle, no pole location. Do not supply any of those from memory.

What you can do is combine the line with three statements you already have, and the result is a standing exam trap.

12.1.B.2 A compass needle, being a magnetic dipole, aligns with the local magnetic field.
12.1.B.1.iii Poles of opposite polarity attract.
12.1.A.2.ii Outside a magnet, the field points away from the north pole and returns to the south pole.

A compass needle's north end swings toward Earth's geographic north. By statement 2, the end of Earth's dipole up there has to be opposite in polarity to the needle's north end. So the magnetic pole near the geographic north pole behaves as the south pole of Earth's dipole. Statement 3 says the same thing from the field's side: outside Earth the field lines run into that northern pole, and field lines run into a south pole.

The naming is a historical accident rather than a physics error: the needle end that seeks north was called north-seeking long before anyone modelled Earth as a magnet. Keep two meanings apart and nothing conflicts. Geographic north is a place, magnetic north is a polarity, and at Earth's geographic north there is a magnetic south.

Magnetic permeability, and the one constant on the sheet

LO 12.1.C has three essential-knowledge statements and no equation of its own.

12.1.C.1 Magnetic permeability is a measurement of the amount of magnetization in a material in response to an external magnetic field. So permeability is a property of the material, not of the field. Two samples in the same external field magnetize by different amounts, and permeability is the number that says so.

12.1.C.2 Free space has a constant value of magnetic permeability, known as the vacuum permeability $\mu_0$ , that appears in equations representing physical relationships. On the AP Physics 2 sheet it is printed in the Constants and Conversion Factors group as

\mu_0 = 4\pi \times 10^{-7} \ (\mathrm{T} \cdot \mathrm{m})/\mathrm{A}

As a decimal that is about $1.26 \times 10^{-6}$ , but leave it in the $4\pi$ form when you substitute. The second worked example below shows why.

12.1.C.3 The permeability of matter has values different from that of free space and arises from the matter's composition and arrangement. It is not a constant for a material and varies based on many factors, including temperature, orientation, and strength of the external field.

Quote that one in full, exception clause included. Students file "permeability of a material" as a table lookup, like density. The CED says the opposite in as many words: it is not a constant for a material. Change the temperature, turn the sample, or turn up the external field, and the value moves.

Permeability is not permittivity. Both symbols are printed on the sheet, a few lines apart.

Symbol	Name	Value on the AP Physics 2 sheet	Shows up in
$\varepsilon_0$	vacuum permittivity	$8.85 \times 10^{-12} \ \mathrm{C}^2/(\mathrm{N} \cdot \mathrm{m}^2)$	electric field, capacitance
$\mu_0$	vacuum permeability	$4\pi \times 10^{-7} \ (\mathrm{T} \cdot \mathrm{m})/\mathrm{A}$	the magnetic field of a current

The sheet also prints the Coulomb constant as $k = 1/(4\pi\varepsilon_0) = 9.0 \times 10^9 \ \mathrm{N} \cdot \mathrm{m}^2/\mathrm{C}^2$ , so if you can recall where $\varepsilon_0$ lives in Coulomb's law, the other one is magnetic by elimination.

How 12.1 is examined, and what belongs to another topic

Unit 12's opening page describes how the unit is tested, and the advice lands squarely here. It says that when writing or identifying justifications for claims, simply referencing an equation, law, or physical principle is not sufficient, and gives an example: stating that "the force on a charged particle is to the right because of the 'right-hand rule'" is not a complete enough answer to earn points on the free-response section of the exam. Students should clearly and concisely explain the steps that lead from the equation, law or physical principle to the justification of their claim.

Topic 12.1 has no equation to hide behind, so that instruction is the whole exercise here: every answer is a chain of stated facts, joined up.

The unit lists science practices 2.B, 2.C, 2.D and 3.C as the ones it builds, and both 2.C and 2.D concern how one quantity changes when another does. In 12.1 the only such relationship the CED gives you is 12.1.B.1.iv, so ranking questions here rank by distance and by material rather than by formula.

Scope check. Three things that feel like Topic 12.1 are not.

The force a field exerts on a moving charge, and the right-hand rule for its direction, belong to Topic 12.2.
The field produced by a current-carrying wire, including $B = \mu_0 I/(2\pi r)$ , belongs to Topic 12.3.
Magnetic flux, Faraday's law and Lenz's law belong to Topic 12.4.

Errors that cost marks on 12.1.

Drawing field lines that stop at the magnet's surface. They continue through the magnet and close the loop. A map where the lines end is a map of an electric field.
Calling the two pieces of a broken magnet a north and a south. EK 12.1.B.1.ii: both halves are dipoles.
Treating diamagnetism as a category some materials fall into. Every material has it.
Quoting a strength for Earth's field, or a power law for a dipole field. The CED gives neither, so neither has a source.
Saying a magnet attracts a charge. EK 12.1.A.1 lists the three targets and a stationary charge is not among them. Reading that statement carefully is the door into Topic 12.2.

Break the magnet, then break it again

A bar magnet is labelled N at its left end and S at its right end. A student cuts it in half at the midpoint and asks which piece is the north pole and which is the south pole. Answer the student, and say what the external field of each piece looks like, citing CED statements as evidence.

Name the wrong assumption first. The question assumes a magnet is a north pole joined to a south pole, so that cutting between them separates the two. EK 12.1.A.1.i rules that out: magnetic fields can be produced by magnetic dipoles or combinations of dipoles, but never by monopoles.
Apply the statement that covers this exact case. EK 12.1.B.1.ii says no magnetic north pole is ever found in isolation from a south pole, and that if a bar magnet is broken in half, both halves are magnetic dipoles. So the student's question has no answer as asked: neither piece is a single pole.
Get each piece's orientation from the field map rather than from memory. EK 12.1.A.2.i says the lines form closed loops, so the lines leaving the original north end on the left and returning to the south end on the right must run back through the inside of the magnet, right to left. Cutting does not redirect them.
Read the left-hand piece. Inside it the lines still run toward its left end, so the left end is still the end they leave from, which 12.1.A.2.ii defines as north, and the fresh cut face is the end they return into, which is south. The same reading of the right-hand piece puts its north at the cut face. The two cut faces are therefore a south and a north facing each other, and 12.1.B.1.iii says opposite polarities attract, so the halves pull back together the way they came apart.
Extend it. Cut each piece again and you have four dipoles. The result does not depend on how finely you cut, because there is no pair of separable pole objects inside to pull apart: EK 12.1.B.1 says the dipole comes from the circular or rotational motion of electric charges.

Neither piece is a single pole. Both halves are complete magnetic dipoles, each with its own north end and south end, which is EK 12.1.B.1.ii stated directly. Both keep the original orientation, north on the left and south on the right, so the two freshly cut faces are a south and a north facing each other and the pieces attract. No number of cuts produces an isolated pole, because EK 12.1.A.1.i states that a magnetic field is never produced by a monopole.

Putting a number on the vacuum permeability

The AP Physics 2 equation sheet prints the vacuum permeability as $\mu_0 = 4\pi \times 10^{-7} \ (\mathrm{T} \cdot \mathrm{m})/\mathrm{A}$ . Write it as a decimal, then use the Topic 12.3 field equation to find the magnetic field 1.00 m from a long straight wire carrying 1.00 A.

Decimal form first. $4\pi = 12.566...$ , so $\mu_0 = 12.566 \times 10^{-7} = 1.2566 \times 10^{-6}$ , which is $1.26 \times 10^{-6} \ (\mathrm{T} \cdot \mathrm{m})/\mathrm{A}$ to three significant figures.
Take the relationship from the Magnetism group of the sheet. The field of a long straight current-carrying wire, which Topic 12.3 owns, is $B = \frac{\mu_0 I}{2\pi r}$ .
Substitute with $\mu_0$ still in its $4\pi$ form, because the $\pi$ cancels: $B = \frac{(4\pi \times 10^{-7})(1.00)}{2\pi(1.00)} = \frac{4\pi}{2\pi} \times 10^{-7} = 2 \times 10^{-7}$ .
Carry the units through: $[(\mathrm{T} \cdot \mathrm{m})/\mathrm{A}] \times \mathrm{A} \div \mathrm{m} = \mathrm{T}$ , so $B = 2.00 \times 10^{-7} \ \mathrm{T}$ .
Notice why the sheet prints $4\pi \times 10^{-7}$ rather than the decimal. On the AP Physics 2 sheet $\mu_0$ appears inside exactly one equation, and there it arrives divided by $2\pi$ . Substituting the decimal first buys you an extra multiplication and an extra rounding step for nothing.
Sanity check the size. Two ten-millionths of a tesla, a metre out from a one-amp wire. Fields from ordinary currents are small, which is why the distance $r$ matters so much in Topic 12.3.

As a decimal, $\mu_0 = 1.26 \times 10^{-6} \ (\mathrm{T} \cdot \mathrm{m})/\mathrm{A}$ . The field is $2.00 \times 10^{-7} \ \mathrm{T}$ , which is 0.200 microtesla. Keep the constant in its $4\pi \times 10^{-7}$ form when you substitute: the $2\pi$ underneath cancels it down to a factor of 2, and the calculator never comes out.

Sorting three rods with EK 12.1.B.3

A student is handed three unlabelled rods of identical size and told they are iron, aluminum and titanium in some order. Each is brought close to a strong permanent magnet, held there, then taken away. Using only EK 12.1.B.3, say how the student can identify the iron rod, and say what is true of all three.

Read the three sub-statements in the table above as a test procedure rather than as definitions. 12.1.B.3.i names iron among the ferromagnetic materials, 12.1.B.3.ii names both aluminum and titanium among the paramagnetic ones, and 12.1.B.3.iii applies to every material there is.
Pick the observation that actually separates the cases. All three rods respond in some way while the magnet is present, so a test run with the magnet in place does not sort them. The CED's distinction is about what is left afterwards, so the test has to happen after the magnet is taken away.
Run it. Remove the magnet, then hold each rod against a fresh, unmagnetized paperclip. Iron is the rod that still attracts it: of the three statements, only 12.1.B.3.i permits a result that outlasts the external field, and 12.1.B.3.ii says outright that paramagnetic dipoles do not remain aligned once that field is removed.
Do not try to separate the other two. Aluminum and titanium sit in the same category in 12.1.B.3.ii, alongside magnesium, and the CED asks for the category rather than a ranking within it.
State what holds for all three. Every one has a diamagnetic response, per 12.1.B.3.iii, a usually weak alignment of the material's dipole moments opposite the external field. In the iron rod it sits under the ferromagnetic response and in the other two under the paramagnetic one. Diamagnetism is not a fourth kind of material.
One caution on the word permanently. EK 12.1.C.3 says the permeability of matter is not a constant for a material and varies with factors including temperature, so the magnetized rod is in a state rather than stamped with a fixed property.

The iron rod is the one that still picks up a paperclip after the magnet has been taken away. It is ferromagnetic (12.1.B.3.i), so an external field can align its domains permanently. Aluminum and titanium are both paramagnetic (12.1.B.3.ii), their dipoles do not remain aligned once the field is removed, so neither holds a magnetization and this test cannot tell the two apart. All three rods, and every other material, also carry a diamagnetic response that aligns opposite the external field (12.1.B.3.iii).

Frequently asked questions

What does AP Physics 2 Topic 12.1 cover?

Three learning objectives. 12.1.A asks you to describe the properties of a magnetic field: that it is a vector field determining the force on moving charges, currents and magnetic materials, that it comes from dipoles and never from monopoles, and that its field lines form closed loops. 12.1.B asks you to describe a material's magnetic behavior in terms of how its dipoles are configured, covering ferromagnetic, paramagnetic and diamagnetic responses, compasses, and Earth's field as a dipole. 12.1.C asks you to describe magnetic permeability. The topic has no boundary statement and no relevant equation.

Why are there no magnetic monopoles?

AP Physics 2 treats it as a stated property of the magnetic field rather than as something you derive. EK 12.1.A.1.i says magnetic fields can be produced by magnetic dipoles or combinations of dipoles, but never by monopoles, and EK 12.1.A.2.i says magnetic field lines form closed loops. Those two fit together: a line that closes on itself has no start and no end, so there is nowhere for a lone source to sit. An electric field line, by contrast, begins on a positive charge and ends on a negative one, so a single electric charge can exist on its own.

What happens if you cut a bar magnet in half?

You get two smaller bar magnets, not a separate north pole and south pole. EK 12.1.B.1.ii states it directly: no magnetic north pole is ever found in isolation from a south pole, and if a bar magnet is broken in half, both halves are magnetic dipoles. Cut those halves and you have four dipoles. The reason is EK 12.1.B.1, which says a magnetic dipole results from the circular or rotational motion of electric charges. There is no pair of separable pole objects inside to pull apart.

What is the difference between ferromagnetic, paramagnetic and diamagnetic materials?

The AP Physics 2 CED separates them by what survives after the external field is removed. Ferromagnetic materials, which it names as iron, nickel and cobalt, can be permanently magnetized by an external field that causes the alignment of magnetic domains or atomic magnetic dipoles. Paramagnetic materials, named as aluminum, titanium and magnesium, interact weakly with an external field, and their dipoles do not remain aligned once that field is gone. Diamagnetism is different in kind: the CED says all materials have it, and describes it as a usually weak alignment of the material's dipole moments opposite the external magnetic field.

Is Earth's geographic north pole a magnetic north or a magnetic south?

A magnetic south. The AP Physics 2 CED states only that Earth's magnetic field may be approximated as a magnetic dipole, but the polarity follows from two of its other statements. A compass needle is itself a magnetic dipole and aligns with the local field, and poles of opposite polarity attract, so the needle's north-seeking end is drawn toward a south polarity. The pole sitting near the geographic north pole is therefore the south pole of Earth's dipole. The naming is historical: the needle end was called north because it seeks north, not because of what it faces.

What is magnetic permeability, and what is the value of mu naught?

Magnetic permeability is a measurement of the amount of magnetization in a material in response to an external magnetic field, so it is a property of the material rather than of the field. Free space has a constant value, the vacuum permeability, printed on the AP Physics 2 equation sheet as mu naught equals 4 pi times ten to the power minus seven, in units of tesla metres per ampere, which is about 1.26 times ten to the power minus six. The permeability of matter differs from that of free space and, as the CED stresses, is not a constant for a material: it varies with factors including temperature, orientation and the strength of the external field.

Are any equations for Topic 12.1 on the AP Physics 2 formula sheet?

No equation belongs to Topic 12.1. The only item on the sheet the topic owns is the vacuum permeability, printed in the Constants and Conversion Factors group. The Magnetism group of the AP Physics 2 sheet holds seven entries, and every one belongs to a later topic in the unit: the force on a moving charge, the field of a long straight wire, the force on a current-carrying wire, magnetic flux in two forms, and two expressions for induced emf. The closed-surface flux integral that states the no-monopole rule as an equation is on the AP Physics C: Electricity and Magnetism sheet, not on this one.