Topic 12.4: Ampère's Law

What Topic 12.4 requires

Topic 12.4, which the CED titles Ampère's Law, closes Unit 12. It has one learning objective, and it is the only objective in the whole unit that does not open with the task verb "describe".

12.4.A, use Ampère's law to describe the magnetic field created by a moving charge carrier.

• 12.4.A.1 Ampère's law relates the magnitude of the magnetic field to the current enclosed by a closed imaginary path called an Amperian loop. The relevant equation is

• 12.4.A.1.i Ampère's law can be used to determine the magnetic field near a long, straight current-carrying wire. The derived equation is $B_{\text{wire}} = \dfrac{\mu_0}{2\pi}\dfrac{I}{r}$.
• 12.4.A.1.ii 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.
• 12.4.A.1.iii Ampère's law can be used to determine the magnetic field inside of a long solenoid. The derived equation is $B_{\text{sol}} = \mu_0 n I$.
• 12.4.A.2 an Amperian loop is a closed path around a current-carrying conductor.
• 12.4.A.3 the principle of superposition can be used to determine the net magnetic field at a point in space created by various combinations of current-carrying conductors, or conducting loops, segments, or cylinders.
• 12.4.A.4 Maxwell's equations are the collection of equations that fully describe electromagnetism. Maxwell's fourth equation is Ampère's law with Maxwell's addition; it states that magnetic fields can be generated by electric current (Ampère'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). The relevant equation is

The suggested skills are 1.A, create diagrams, tables, charts, or schematics to represent physical situations; 2.A, derive a symbolic expression from known quantities by selecting and following a logical mathematical pathway; 2.B, calculate or estimate an unknown quantity with units from known quantities by selecting and following a logical computational pathway; and 3.B, apply an appropriate law, definition, theoretical relationship, or model to make a claim.

Two boundary statements sit under this topic, quoted whole later on this page. Note the verb in the objective. The CED says "describe", used in nearly all learning objectives, "encompasses the range of possible graphical, mathematical, or verbal skill applications". Objective 12.4.A opens with use instead, and that is a signal about what the topic is: a tool you apply, not a phenomenon you characterise.

Note also the phrase "created by a moving charge carrier" in the objective. It reaches back to statement 12.2.A.1, that a single moving charged object produces a magnetic field. The whole arc of the unit is that one moving charge makes a field, many of them make a current, and Ampère's law is the shortcut for finding the field of a current when the arrangement is symmetric enough.

Ampere's law is Gauss's law with a loop instead of a surface

If you have already met Gauss's law in Unit 8, you already own the method here. The CED says so itself: its Developing Understanding page for Unit 12 states that knowledge from previous units helps students make connections between electric fields and magnetic fields "as well as between Gauss's Law and Ampère's Law."

That connection is structural, not decorative. Both laws have the same three-part shape, and both are used the same way.

The last row is the one that separates people who understand these laws from people who have memorised them. A charge outside your Gaussian surface still makes an electric field at every point on that surface; its net flux through the surface is zero because as much goes in as comes out. In exactly the same way, a current outside your Amperian loop still makes a magnetic field at every point on that loop; its net circulation around the loop is zero because the loop does not link it. So $I_{\text{enc}}$ genuinely means enclosed, and it is not a claim that outside currents make no field.

Two more points of structure worth keeping:

Both are always true, and both are only sometimes useful. Every closed surface satisfies Gauss's law and every closed path satisfies Ampère's law. Neither hands you a field unless the symmetry lets you take the field magnitude out of the integral. Choosing the boundary is the entire skill in both cases, and choosing it badly does not give a wrong answer, it gives an equation you cannot solve.

The dimension of the boundary differs because the sources differ. Electric field lines start and end on charges, so counting what is inside a closed surface counts charge. Magnetic field lines close on themselves, which is Topic 12.1 and Maxwell's second equation $\oint \vec{B} \cdot d\vec{A} = 0$, so a closed surface counts nothing at all for magnetism. What you count instead is how much current threads through a closed loop. The move from a surface to a loop is forced by the fact that there are no magnetic monopoles.

There is one asymmetry worth flagging. Gauss's law for electricity has a partner, Gauss's law for magnetism, and both are on the sheet. Ampère's law has a partner too, Faraday's law, and it is Maxwell's third equation, printed on the sheet as $\mathcal{E} = \oint \vec{E} \cdot d\vec{\ell} = -\dfrac{d\Phi_B}{dt}$ and taught in Unit 13. Look at the two closed line integrals side by side and the symmetry of the subject is visible: a current or a changing electric field circulates $\vec{B}$, and a changing magnetic field circulates $\vec{E}$.

Choosing the Amperian loop

Statement 12.4.A.2 defines it in one line: an Amperian loop is a closed path around a current-carrying conductor. It is imaginary, it can be any shape, and it does not have to lie in any physical object. What it must do is make the left-hand side computable.

Four conditions to aim for, in order of importance:

1. Along the part of the loop where the field is nonzero, $\vec{B}$ is parallel to $d\vec{\ell}$. Then $\vec{B} \cdot d\vec{\ell} = B\,d\ell$ and the cosine disappears.
2. Along that same part, $B$ has constant magnitude. Then it comes out of the integral, leaving $B \oint d\ell = B \times (\text{length of that part})$.
3. Along any remaining part, either $\vec{B}$ is perpendicular to $d\vec{\ell}$, so the dot product is zero, or $\vec{B}$ is itself negligible. Statement 12.4.A.1.ii is what licenses the second option for a solenoid.
4. The current enclosed is easy to state. For a thin wire it is the whole current; for a thick conductor you need the fraction of the cross-section your loop encloses.

Get conditions 1 and 2 and the whole left-hand side becomes $B\ell$, one symbol times one length, and the law is a one-line equation for $B$.

Argue the symmetry before you use it. For a long straight wire, statement 12.3.A.2 already says the field vectors are tangent to concentric circles centred on the wire, with no component toward, away from, or parallel to it. That is the justification for choosing a circle: on it $\vec{B}$ is tangent, so parallel to $d\vec{\ell}$, and every point on it is equivalent by symmetry, so $B$ is the same all the way around. Writing that sentence earns credit under skill 3.B.

The right-hand rule for a loop. Curl the fingers of your right hand around the loop in the direction you are traversing it, and the thumb points in the positive direction for current through it. Current the other way counts negative. That sign convention is what makes a coaxial cable's outside field vanish.

Two shapes cover almost everything on this exam: a circle concentric with a long straight wire or a cylindrical conductor, of path length $2\pi r$, and a rectangle with one side inside a solenoid and the opposite side outside it. A third appears occasionally, a conductive slab carrying a current density, named in the first boundary statement, where the loop is a rectangle straddling the slab and two of its four sides contribute.

What not to do. Do not draw an Amperian loop around a single circular current loop or a finite wire segment and expect an answer. The field magnitude is not constant along any convenient path there, condition 2 fails, and the equation is true but unsolvable. Those geometries are Biot-Savart problems, and the first boundary statement below says as much by listing what Ampère's law will actually be applied to.

The two results the sheet withholds, and how to produce them

Both of the derived equations in this topic are things you are expected to generate. One of them is on the sheet anyway; the other is not.

The long straight wire, 12.4.A.1.i. Draw a circle of radius $r$ centred on the wire, in the plane perpendicular to it. By 12.3.A.2, $\vec{B}$ is tangent to that circle everywhere and has the same magnitude at every point on it. So

and the loop encloses the whole current, so $B(2\pi r) = \mu_0 I$ and

Three lines, and this equation is not printed on the AP Physics C: E&M equation sheet. Check the sheet's magnetism column and it runs from the resistor and capacitor combination rules through $\omega_{LC} = 1/\sqrt{LC}$ with no field-of-a-wire anywhere in it. What is printed is Ampère's law itself, so the derivation above is the intended route every time you need the result. Contrast AP Physics 2, which prints the same expression as a relevant equation at its statement 12.3.A.1.ii and puts it on its own sheet.

The solenoid, 12.4.A.1.iii. Draw a rectangle with one long side of length $L$ inside the solenoid and parallel to its axis, the opposite long side outside, and two short sides crossing the winding. Then:

• the inside side contributes $BL$, because there $\vec{B}$ is uniform and along the axis, so parallel to $d\vec{\ell}$;
• the outside side contributes nothing, because 12.4.A.1.ii says the field outside a long solenoid is negligible;
• the two crossing sides contribute nothing, because $\vec{B}$ is along the axis and $d\vec{\ell}$ is perpendicular to it, so each dot product is zero.

The enclosed current is one $I$ for every turn the rectangle encloses, and a length $L$ of a solenoid with $n$ turns per unit length holds $nL$ turns. So $BL = \mu_0 (nL) I$ and

The $L$ cancels, and that cancellation is the physics. It means the answer does not depend on where along the axis you drew the rectangle or how long you made it, which is precisely the claim in 12.4.A.1.ii that the field inside is uniform. So the assumption you used at the start comes back out as a result at the end, which is a good sign the derivation is consistent.

This one is printed on the sheet, with the variable list defining $n$ as the number of loops per unit length. It is the only derived equation in Unit 12 that also appears on the sheet, and the label still says the derivation is expected of you.

The single most common solenoid error is using $N$, the total number of turns, where $n$, the turns per unit length, belongs. The sheet's variable list distinguishes them: $n$ is the number of loops per unit length and $N$ is the number of loops. They differ by a factor of the solenoid's length, and a solenoid 0.5 m long with 1000 turns has $n = 2000$ per metre, not 1000.

What the enclosed current really means

$I_{\text{enc}}$ is the current that passes through any surface bounded by your loop. Three cases cover the course.

A thin wire, entirely inside the loop. $I_{\text{enc}} = I$. The simple case.

A thick conductor, with the loop inside it. Now only part of the current threads the loop. For a solid cylinder of radius $a$ carrying total current $I$ uniformly, an Amperian circle of radius $r < a$ encloses the fraction of the cross-section inside it:

so $B(2\pi r) = \mu_0 I r^2/a^2$ and $B = \dfrac{\mu_0 I r}{2\pi a^2}$, which grows linearly with $r$ inside the conductor. Outside it falls as $1/r$. The two expressions agree at the surface, so the field peaks at $r = a$ and nowhere else. Sketching that, rising straight then falling as a hyperbola, is a standard skill 1.A task.

If the current density is not uniform, the enclosed current is an integral, and the sheet prints the equation for it in its electricity column: $I = \int \vec{J} \cdot d\vec{A}$. With a radially varying $J(r)$ the area element is an annulus, $dA = 2\pi r\,dr$. The first boundary statement below names cylindrical conductors carrying a current density explicitly, so this is a live case rather than an exotic one.

More than one current through the loop. Add them with signs, using the right-hand rule for the traversal direction. Currents one way are positive, the other way negative. This is what makes a coaxial cable interesting: enclose both the inner conductor and the return shell and $I_{\text{enc}} = I - I = 0$, so the circulation vanishes, and combined with the circular symmetry that forces $B = 0$ everywhere outside the cable.

Notice the two steps in that last sentence, because leaving one out is a real error. Zero circulation on its own does not mean zero field; it means the field's contributions cancel around the loop. It is the symmetry, which says $B$ has the same magnitude all the way around and is tangent, that upgrades zero circulation to zero field. Gauss's law has the identical subtlety, and it is the reason a symmetry argument is a required part of an answer rather than a courtesy.

Superposition, and the CED's own example of it

Statement 12.4.A.3 says the principle of superposition can be used to determine the net magnetic field at a point in space created by various combinations of current-carrying conductors, or conducting loops, segments, or cylinders.

The method is the same as for electric fields, with one extra piece of care.

1. Find each source's field at the point separately, using whichever tool suits that source. A long wire gets Ampère's law; a loop or an arc gets Biot-Savart. Different sources in the same problem can need different tools.
2. Get each direction right, with the right-hand rule, one source at a time. This is the extra care. Electric fields point along the line from the charge; magnetic fields point around the source, so the direction changes as you move around it and two identical wires can give opposite contributions at the same point.
3. Add as vectors, in components.

The CED's own sample multiple-choice Question 4 is built on this and is worth studying. Three long wires lie parallel to each other and to the $x$-axis in the $xy$-plane, at $y = +4d$, $y = +2d$ and $y = -d$, carrying $4I$, $2I$ and $2I$ respectively, with the third in the opposite direction to the other two. The question asks for the magnitude and direction of the field at the origin, and it is tagged to learning objective 12.4.A and essential knowledge 12.4.A.1 with skill 2.A. Working through it, each wire contributes a field in the same direction at the origin, and the three add to $2\mu_0 I/\pi d$. That the contributions all point the same way despite one current running backwards is the whole trap: the wire below the origin carries its current the other way, so the reversal of the current and the reversal of the side cancel each other out.

Two facts to carry from the structure of these problems:

• A point between two parallel wires with currents in the same direction always has the two fields opposing. So there is a null point between them, and it sits closer to the weaker current. Outside the pair, both fields point the same way and never cancel, so that is the only null.
• With currents in opposite directions, it is the reverse. The fields add between the wires and can cancel outside, on the side of the weaker current.

Working out which of those you are in, before computing anything, saves the most common error in this topic.

The two boundary statements, quoted whole

First, under 12.4: "AP Physics C: Electricity & Magnetism only expects quantitative application of Ampère'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 Ampère's law will be applied on the AP Physics C: Electricity & Magnetism Exam."

Unlike the Topic 12.3 boundary statement, this one does not say "such as". It says these are the types of shapes, which is a closed list of four:

1. long straight wires,
2. long solenoids carrying currents,
3. conductive slabs,
4. cylindrical conductors carrying a current density.

The fourth item is the one to prepare for beyond the obvious, because it brings Unit 11 back into play: getting $I_{\text{enc}}$ inside a thick conductor with a stated current density means integrating $\vec{J}$ over the part of the cross-section your loop encloses. The third, a conductive slab, is the least familiar shape and is worth sketching once so it is not a surprise.

Second, under 12.4: "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."

Both sentences ship together. The first is a limit on computation and the second is a requirement of understanding, and dropping the second turns a boundary into a licence to ignore the physics, which it is not.

Note that Topic 12.4 carries no boundary statement fencing off the symmetry argument itself. What is bounded is the list of shapes, not the level of reasoning about them. You can be asked to justify why a chosen loop works, to sketch a field against radius, or to compare two radii inside a conductor, and none of that is limited by either statement.

Maxwell's fourth equation, and what you owe it

Statement 12.4.A.4 prints this as a relevant equation and describes it in plain words: Maxwell's fourth equation is Ampère's law with Maxwell's addition, and it states that magnetic fields can be generated by electric current (Ampère'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).

It is not printed on the equation sheet. The sheet carries $\oint \vec{B} \cdot d\vec{\ell} = \mu_0 I_{\text{enc}}$ and stops there. The extra term appears only in the framework, and the second boundary statement explains why: you are not expected to compute with it.

What you are expected to do, per that statement's second sentence, is understand that a changing electric field generates a magnetic field. That is worth being able to say in a sentence and to illustrate with one situation.

The illustration to have ready is a charging capacitor. Between the plates there is no current at all, yet a magnetic field exists there while the capacitor is charging. The reason is the second term: the electric flux between the plates is growing as charge accumulates, and $d\Phi_E/dt$ takes the place of the missing current. Without Maxwell's addition, Ampère's law would give two different answers for the same loop depending on which surface you chose to count current through, which is not allowed. The addition is what makes the law consistent.

Where the four Maxwell equations sit in this course, since they are scattered and the CED never lists them in one place:

Three of the four statements that name Maxwell's equations, at 12.1.A.3.i, 12.4.A.4 and 13.2.A.3, open with the same sentence: Maxwell's equations are the collection of equations that fully describe electromagnetism. That repetition is the CED signalling that they belong together.

Unit 13 then repeats the pattern set here. Statement 13.2.A.4 gives the speed of light as a derived equation, $c = 1/\sqrt{\varepsilon_0 \mu_0}$, and the single Unit 13 boundary statement says the course does not expect students to derive it mathematically from Maxwell's equations, calling it an indication of further applications students may study in more advanced courses. Know what the equations say and what they imply, and compute only with the ones the sheet prints.

How Topic 12.4 is tested

Unit 12 appears twice in the CED's four sample free-response questions, and objective 12.4.A is tagged on both of them, more than any other Unit 12 objective. One of the fifteen sample multiple-choice questions is tagged to 12.4.A as well.

• Sample Question 2, Translation Between Representations, 12 points, aligned to 12.4.A, 13.1.A and 13.2.A. Part A puts a small conducting loop inside a long solenoid and asks students to mark the field direction in three square regions: one centred on the axis, one inside the solenoid but off the axis, and one outside it. Its scoring guideline awards one point for indicating consistent fields in the two interior regions, one for getting the direction right in the first, and one for writing "zero" next to the exterior region. Those three points are essential knowledge 12.4.A.1.ii being tested directly, and they are available before any calculation happens. Part B is worth four points and asks for a derivation of an induced emf that requires substituting the solenoid field, so 12.4.A.1.iii is load-bearing there too.
• Sample Question 3, Experimental Design and Analysis, 10 points, aligned to 11.3.B, 12.3.B and 12.4.A. Two parallel wires and a force sensor, with the vacuum permeability extracted from the slope of force against current squared. Its scoring guideline uses the derived equation from 12.4.A.1.i as one of its two ingredients, which is Topic 12.3 and this topic working together.
• Sample multiple-choice Question 4, skill 2.A, aligned to 12.4.A and essential knowledge 12.4.A.1, is the three-parallel-wires superposition problem described above.

Three of the five optional sample instructional activities for Unit 12 are attached to this topic, and all three are estimation tasks: how far from a high-tension power line you must stand before its field equals Earth's, designing a solenoid to match an MRI machine's field, and using a compass deflection next to a current-carrying strip of aluminium foil to determine Earth's magnetic field. Skill 2.B, calculate or estimate an unknown quantity with units, is listed for this topic and for no other topic in Unit 12, and those three activities are exactly what it looks like.

The four question shapes.

1. Derive a field from Ampère's law (skill 2.A). State the loop, argue the symmetry, evaluate the left-hand side, state $I_{\text{enc}}$, solve. All five steps carry credit.
2. Compute or estimate a field (skill 2.B), including comparisons against Earth's field, which is of order $10^{-5}$ T.
3. Draw the field (skill 1.A). Dots and crosses in named regions, or a graph of $B$ against $r$ for a thick conductor.
4. Justify a claim (skill 3.B). Why the field outside a coaxial cable is zero, or why a chosen loop does or does not work.

Four traps.

• Using $N$ instead of $n$ in the solenoid formula.
• Believing outside currents make no field. They contribute nothing to the circulation around your loop and plenty to the field at points on it. Only the total circulation is unaffected.
• Concluding zero field from zero circulation without a symmetry argument. Both steps are needed.
• Reaching for Ampère's law where the symmetry is not there. A single circular loop or a finite segment is a Biot-Savart problem. The first boundary statement lists the four shapes this law is applied to on the exam, and neither of those is among them.

Where this goes next. The solenoid field from 12.4.A.1.iii and the wire field from 12.4.A.1.i are the two fields that get differentiated in Unit 13, because every induction problem starts from a magnetic flux and a flux needs a field. Sample Question 2 is that handoff happening inside a single question.