Topic 12.1: Magnetic Fields

What Topic 12.1 requires

Topic 12.1 is the first topic of Unit 12, Magnetic Fields and Electromagnetism, and it carries three of the unit's eight learning objectives, more than any other topic in it. It prints no boundary statement.

12.1.A, describe the properties of a magnetic field.

• 12.1.A.1 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.
• 12.1.A.1.i magnetic fields can be produced by magnetic dipoles or combinations of dipoles, but never by monopoles.
• 12.1.A.1.ii magnetic dipoles have north and south polarity.
• 12.1.A.2 a magnetic field is a vector quantity and can be represented using vector field maps.
• 12.1.A.3 magnetic field lines must form closed loops, as described by Gauss's law for magnetism.
• 12.1.A.3.i Maxwell's equations are the collection of equations that fully describe electromagnetism. Gauss's law for magnetism is Maxwell's second equation. The relevant equation is $\oint \vec{B} \cdot d\vec{A} = 0$.
• 12.1.A.3.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).

12.1.B, describe the magnetic behavior of a material as a result of the configuration of magnetic dipoles in the material.

• 12.1.B.1 magnetic dipoles result from the circular or rotational motion of electric charges. In magnetic materials, this can be the motion of electrons.
• 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.
• 12.1.B.1.ii no magnetic north pole is ever found in isolation from a south pole. For example, if a bar magnet is broken in half, both halves are magnetic dipoles.
• 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.
• 12.1.B.2 a magnetic dipole, such as a magnetic compass, placed in a magnetic field will tend to align with the magnetic field.
• 12.1.B.3 a material's composition influences its magnetic behavior in the presence of an external magnetic field.
• 12.1.B.3.i ferromagnetic materials such as iron, nickel, and cobalt can be permanently magnetized by an external field that causes the alignment of magnetic domains or atomic magnetic dipoles.
• 12.1.B.3.ii paramagnetic materials such as aluminum, titanium, and magnesium interact weakly with an external magnetic field, in that the magnetic dipoles of the material do not remain aligned after the external field is removed.
• 12.1.B.3.iii all materials have the property of diamagnetism, in that their electronic structure creates a usually weak alignment of the dipole moments of the material opposite the external magnetic field.
• 12.1.B.4 Earth's magnetic field may be approximated as a magnetic dipole.

12.1.C, describe the magnetic permeability of a material.

• 12.1.C.1 magnetic permeability is a measurement of the amount of magnetization in a material in response to an external magnetic field.
• 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.
• 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.

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.

Notice what is not here. There is no force equation, no field of a wire, and no expression for how a dipole's field falls off. Those belong to 12.2, 12.3 and 12.4. Topic 12.1 prints exactly one equation, and it is the one that says the total is zero.

The one statement AP Physics 2 does not print

This is the honest headline for this page, and it is worth stating plainly rather than dressing up: AP Physics C Topic 12.1 and AP Physics 2 Topic 12.1 are almost the same required content. Objectives 12.1.B and 12.1.C, which together hold most of the words above, read the same in both course descriptions, statement for statement, down to the lists of named metals. Both courses list the same four suggested skills for the topic, 1.A, 2.C, 3.B and 3.C. Neither prints a boundary statement.

Set the two objectives 12.1.A side by side and the difference is three lines.

One statement reworded and promoted, one statement added, one statement renumbered. That is the whole difference, and the added statement is a Maxwell equation.

So do not spend Physics C study time re-reading the ferromagnetism list. Spend it on the surface integral, because that is the piece that is genuinely yours and it is the piece that gets examined at a different level. Everything else on this topic you can read on the algebra-based page.

The [AP Physics 2 Topic 12.1 page](/ap-physics-2/unit-12-magnetism-and-electromagnetism/12-1-magnetic-fields) is for AP Physics 2 students, and this page is for AP Physics C: Electricity and Magnetism students. If you are in the algebra-based course, that page covers your version of the topic and you do not need the surface integral. If you are in Physics C and you want the dipole and materials material explained at length, that page is the better read and this one is the shorter path to the part that differs.

Reading the closed surface integral

Statement 12.1.A.3.i prints this as a relevant equation, and it is on the AP Physics C: E&M equation sheet. Read it in pieces.

• The circle on the integral means the surface is closed: it has an inside and an outside, with no edge. A sphere qualifies, a cylinder with both ends capped qualifies, a flat disc does not.
• $d\vec{A}$ is an outward-pointing area element, so $\vec{B} \cdot d\vec{A}$ is positive where the field leaves the surface and negative where it enters.
• The integral totals those contributions over the whole surface. The magnetic flux through a closed surface is the net amount of field leaving it.
• It equals zero, always, for every closed surface you can draw, in every situation in the course.

Compare it with the electric case, which is on the same sheet in the same column:

Same shape, and the whole physics is in the right-hand side. Electric flux through a closed surface counts the charge inside, because you can isolate a positive charge and have field lines that start there and go off to infinity. Magnetic flux through a closed surface is zero because there is nothing to count. There is no magnetic charge to enclose.

That is statement 12.1.A.1.i expressed as mathematics: fields can be produced by dipoles or combinations of dipoles, but never by monopoles. And it is why 12.1.A.3 uses the word "must". Closed field lines are not an observed habit of magnets, they are forced. Every line that leaves a region has to come back into it, so the ledger balances on any surface you choose.

Three consequences worth having ready:

1. Enclosing one pole of a magnet does not enclose a source. Draw a box around only the north end of a bar magnet. Field leaves through the outer faces, and exactly as much re-enters through the face the magnet's body passes through, because the lines continue inside the magnet from south to north. Statement 12.1.A.3.ii is describing that loop: the external field points away from the north end and returns to the south end, and the path is completed inside the material.
2. Breaking a magnet cannot produce a monopole. Statement 12.1.B.1.ii says it outright: break a bar magnet in half and both halves are magnetic dipoles. Cut again and you get four dipoles. The equation does not permit any other outcome.
3. A closed surface is the wrong tool for finding $B$. Gauss's law for electricity is useful because the right-hand side is not zero, so a symmetric surface hands you the field. Gauss's law for magnetism has zero on the right, so it never solves for a field. The magnetic law that does solve for fields is a closed line integral, Ampere's law, and that difference between a surface and a loop is the structural point of this unit.

One more piece of framing the CED gives you free. Statement 12.1.A.3.i says Maxwell's equations are the collection of equations that fully describe electromagnetism, and names this one as the second. The fourth appears in Topic 12.4 at 12.4.A.4, and the third, Faraday's law of induction, appears in Unit 13 at 13.2.A.3. So the course does show you all of Maxwell's equations, spread across three topics, and this is where the set starts.

A vector field, and the three things it acts on

Statement 12.1.A.1 defines the magnetic field by what it does, and it names three targets:

1. moving electric charges, which is Topic 12.2 and the cross-product force law,
2. electric currents, which is Topic 12.3 and the force integral on a wire,
3. magnetic materials, which is 12.1.B on this page.

The definition is operational in exactly the way the electric field definition is. The electric field is defined on the sheet as $\vec{E} = \vec{F}_E / q$, force per unit charge, and it is a property of the space rather than of any test object. The magnetic field is the same kind of object, but there is no equally simple "force per unit something" definition, because the magnetic force depends on the velocity as well as the charge. That is why 12.1.A.1 defines $\vec{B}$ by what it lets you determine rather than by a quotient.

Statement 12.1.A.2 adds that it is a vector quantity representable by vector field maps. Skill 1.A, create diagrams, tables, charts, or schematics, is listed first for this topic, so drawing one is an examinable task. Four conventions that a scoring guideline can identify:

• A dot inside a circle means the field points out of the page toward you; a cross inside a circle means into the page. The CED's own sample free-response question uses exactly these two symbols and provides a key for them.
• Arrows in the plane of the page for a field lying in that plane.
• Density of lines stands for magnitude, so where lines crowd together the field is stronger.
• Direction at a point is the tangent to the field line there, and each line carries a single arrowhead direction.

The unit is the tesla, symbol T, which is on the sheet's unit-symbol list. Combining it with the vacuum permeability entry in the constants table, $\mu_0 = 4\pi \times 10^{-7}$ has units of tesla metre per ampere, which is worth noticing because it tells you before any derivation that magnetic fields will come out as (permeability) times (current) divided by (length).

A word about magnitudes, so you can sanity-check an answer. Earth's field at the surface is of order $10^{-5}$ T. A refrigerator magnet is of order $10^{-2}$ T. A laboratory electromagnet or an MRI machine reaches a few tesla. An answer of 200 T is wrong; an answer of $10^{-12}$ T is usually wrong too. The CED's fourth sample activity for this unit has students research the field strength of a typical MRI machine and then design a solenoid to match it, so this order-of-magnitude sense is something the course explicitly wants built.

Where a magnetic dipole comes from

Statement 12.1.B.1 is the sentence that makes magnetism and electricity one subject rather than two: magnetic dipoles result from the circular or rotational motion of electric charges, and in magnetic materials this can be the motion of electrons.

Read against 12.2.A.1, which says a single moving charged object produces a magnetic field, the logic closes. Charge in motion makes a field. Charge going round in a small closed circuit makes the field pattern of a dipole. Every magnet you have ever handled is that pattern, repeated over an enormous number of electrons that happen to be lined up.

Which is what 12.1.B.1.i means by calling permanent magnetism and induced magnetism system properties: neither is a property of one electron, both come from the alignment of many dipoles within a system. Change the alignment and you change the magnet. This is why heating or hammering a magnet weakens it, and it is the physical content behind the temperature dependence that 12.1.C.3 mentions.

The four sub-statements under 12.1.B.1 are each examinable on their own:

• 12.1.B.1.ii, no isolated north pole. Break a bar magnet and you get two dipoles, not a north and a south. This is 12.1.A.1.i and $\oint \vec{B} \cdot d\vec{A} = 0$ again, said in terms of objects instead of flux.
• 12.1.B.1.iii, like poles repel and opposite poles attract. The same sign rule as electric charge, applied to poles rather than charges.
• 12.1.B.1.iv, the field of a dipole decreases with increasing distance. Read that statement carefully, because it is deliberately weak. It says the magnitude decreases. It does not give a functional form, and neither the CED nor the equation sheet prints one. Do not assume the dipole field falls as $1/r^2$ by analogy with a point charge; a point charge is a monopole and a magnet is not, so the analogy has no force here. If a question needs a functional dependence for a dipole field, it will supply one or give you data to extract it from. What you are responsible for is the direction of the trend.
• 12.1.B.2, a dipole placed in a field tends to align with it. The compass is the CED's own example. Note that the statement says "tend to align", a direction of behaviour rather than an equation: no torque equation for a magnetic dipole appears anywhere in this course description or on the equation sheet. A question can ask which way a compass needle points, or which way it rotates, but not for a numerical torque on it.

That last pair is a good illustration of how to read a CED. Two statements about the same object, both qualitative on purpose, and the useful move is to notice the absence of an equation rather than to import one from a textbook.

Ferromagnetic, paramagnetic, diamagnetic, and Earth

Statement 12.1.B.3 says a material's composition influences its magnetic behavior in an external field, and the three sub-statements give the three behaviours. The trap here is a categorisation error, and it is built into how the list reads.

Diamagnetism is not a third category of substance. The statement says all materials have the property of diamagnetism, in that their electronic structure creates a usually weak alignment of the dipole moments of the material opposite the external magnetic field. So iron is diamagnetic too; its ferromagnetism simply swamps it. Calling a material "a diamagnet" in casual speech means the diamagnetic response is the only one it has that is worth noticing, but the CED's own wording is universal, and a question that asks which materials show diamagnetism has the answer "all of them".

The word to attach to each row is the direction of alignment. Ferromagnetic and paramagnetic materials align with the field and are attracted; diamagnetic response is opposite the field and is repulsive. Getting the sign right on that one word is most of what a 3.B question on this statement is checking.

Earth as a dipole (12.1.B.4). The statement is a modelling claim, and it is the modelling that matters: Earth's magnetic field may be approximated as a magnetic dipole. It is an approximation, and the CED says so in the word "approximated".

The naming confusion is worth having straight, because it is a favourite. A compass needle is itself a magnetic dipole, and 12.1.B.2 says it tends to align with the field it sits in. The end of the needle we call its north pole points toward Earth's geographic north. But 12.1.B.1.iii says opposite poles attract. So the magnetic pole of Earth that lies near the geographic north pole behaves as a south magnetic pole. Nothing physical is odd here; the labels were fixed by which way compasses point, centuries before anyone knew why.

The CED's own Unit 12 activity list has students measure Earth's field with a compass and a current-carrying strip of aluminium foil, using Ampere's law to convert a deflection angle between about 30 and 60 degrees into a field strength. That is 12.1.B.2 and 12.1.B.4 doing measurement work rather than sitting as facts.

Permeability is a property of conditions, not a constant of a material

Objective 12.1.C exists to prevent one specific mistake, and its three statements build to it.

12.1.C.1 defines magnetic permeability as a measurement of the amount of magnetization in a material in response to an external magnetic field. So it is a responsiveness: how much magnetization you get out for the field you put in.

12.1.C.2 says free space has a constant value, the vacuum permeability $\mu_0$, and that it appears in equations representing physical relationships. On the C: E&M sheet's constants table it is printed as

exactly, not as a decimal, which is a small gift: $\mu_0 / 4\pi$ is exactly $1.0 \times 10^{-7}$ and $\mu_0 / 2\pi$ is exactly $2.0 \times 10^{-7}$. Both combinations turn up constantly in Topics 12.3 and 12.4, and using them saves a keystroke and a rounding error every time.

12.1.C.3 is the one that carries the weight: the permeability of matter has values different 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."

Read the two halves of that. Composition and arrangement, so two samples of the same element can differ. And then the flat statement that it is not a constant, with three named factors. That rules out the mental model most students arrive with, which is that permeability is a lookup value like a density. It is not. The AP equation sheet prints no table of permeabilities and no relative-permeability values, and the reason is 12.1.C.3.

Where does $\mu$ for matter actually appear in this course? In one printed equation, the inductance of a solenoid, at 13.4.A.1.iii:

The subscript is "core" rather than "0" precisely because the material inside the coil is doing the work. Put an iron core in a solenoid and its inductance rises by whatever factor $\mu_{\text{core}}/\mu_0$ happens to be under those conditions. The CED's own sample activity for Topic 13.5 has students measure exactly this: build a solenoid, measure its inductance from an LR time constant, then repeat with iron or steel in the core and observe the increase.

So the sequence to remember is: 12.1.C.1 defines it, 12.1.C.2 fixes it for vacuum and puts it on the sheet, 12.1.C.3 says it is not fixed for matter, and 13.4.A.1.iii is where the matter version is actually used. That is a clean four-link chain and the third link is the one questions probe.

How Topic 12.1 is tested

None of the fifteen sample multiple-choice questions or four sample free-response questions in the CED aligns to a 12.1 learning objective. That is not evidence the topic is unexamined, since the sample set is small and covers only a slice of the course, but it is a fair signal about how the topic behaves: 12.1 supplies vocabulary and one Maxwell equation that then get used elsewhere, rather than anchoring its own long question.

The four suggested skills tell you the shapes to expect.

1. Skill 1.A, create a diagram or schematic. Draw the field of a bar magnet with lines closing through the interior. Mark dots and crosses correctly for a field out of or into the page. Draw the field map of a dipole. The unit's own sample free-response question awards separate points for indicating consistent field directions in two regions and for writing "zero" in a third, so precision in a drawing carries marks on its own.
2. Skill 2.C, compare quantities between scenarios or locations. Compare the field magnitude at two distances from a dipole (12.1.B.1.iv, direction of the trend only). Compare the response of a ferromagnetic sample and a paramagnetic sample to the same external field, before and after it is removed.
3. Skill 3.B, apply a law or model to make a claim. Apply $\oint \vec{B} \cdot d\vec{A} = 0$ to claim that the net flux through a stated closed surface is zero, whatever is inside it.
4. Skill 3.C, justify a claim with evidence. Here the unit's exam-preparation page is blunt about what does not count. It warns that "simply referencing an equation, law or physical principle is not sufficient", and its own example is that stating an induced current is clockwise "because of 'Lenz's Law'" is not complete enough to earn credit. The same standard applies here: writing "because there are no monopoles" is a claim, not a justification. Say which surface you drew, which statement you are using, and what follows.

Three traps specific to this topic:

• Treating a magnetic pole as a magnetic charge. They behave alike for the attraction and repulsion rule at 12.1.B.1.iii and nowhere else. There is no Coulomb's law for poles in this course, and $\oint \vec{B} \cdot d\vec{A} = 0$ exists to say why not. Compare with Coulomb's law, which is the real thing the analogy is borrowing from.
• Assuming an inverse-square law for a dipole's field. 12.1.B.1.iv gives a direction of change and no exponent, and no exponent is printed anywhere.
• Calling permeability a material constant. 12.1.C.3 explicitly denies it, and lists temperature, orientation and field strength as factors.

Where this topic goes next: 12.1.A.1 names the three things a field acts on and the rest of the unit works through them, 12.2 for moving charges, 12.3 and 12.4 for currents. Then Unit 13 turns the flux idea from this page into the engine of induction.