Topic 12.3: Magnetism and Current-Carrying Wires

What Topic 12.3 requires

Topic 12.3 has two learning objectives, and the pair is the whole topic in miniature: a current makes a field, and a field pushes a current.

12.3.A, describe the magnetic field produced by a current-carrying wire. One essential knowledge statement, 12.3.A.1, carries five sub-statements.

• 12.3.A.1 states that a current-carrying wire produces a magnetic field.
• 12.3.A.1.i states that the magnetic field vectors around a long, straight, current-carrying wire are tangent to concentric circles centered on that wire, and that the field has no component toward, away from, or parallel to the long, straight, current-carrying wire.
• 12.3.A.1.ii states that at a point in space, the magnitude of the magnetic field due to a long, straight, current-carrying wire is proportional to the magnitude of the current in the wire and inversely proportional to the perpendicular distance from the central axis of the wire to the point, with the relevant equation $B = \dfrac{\mu_0}{2\pi}\dfrac{I}{r}$.
• 12.3.A.1.iii states that the direction of the magnetic field created by a current-carrying wire is determined with the right-hand rule.
• 12.3.A.1.iv states that the direction of the magnetic field at the center of a current-carrying loop is directed along the axis of the loop and can be found using the right-hand rule.
• 12.3.A.1.v states that the magnetic field at a location near two or more current-carrying wires can be determined using vector addition principles.

12.3.B, describe the force exerted on a current-carrying wire by a magnetic field. One essential knowledge statement, 12.3.B.1, carries two sub-statements.

• 12.3.B.1 states that a magnetic field may exert a force on a current-carrying wire.
• 12.3.B.1.i states that the magnitude of that force is proportional to the current, the length of the portion of the wire within the magnetic field, and the magnitude of the magnetic field, and also depends on the angle between the direction of the current in the wire and the direction of the magnetic field, with the relevant equation $F_B = I \ell B \sin\theta$.
• 12.3.B.1.ii states that the direction of the force exerted by the magnetic field on a current-carrying wire is determined by the right-hand rule.

Topic 12.3 prints no boundary statement. That is worth knowing precisely, because the topic next door does. Topic 12.2 carries the unit's only boundary statement: "Quantitative treatment of the magnitude of the magnetic force exerted by a magnetic field on a moving charge is limited to angles of 0, 90, and 180 degrees between the velocity and the magnetic field. Qualitative analysis of other angles is permitted." Read the subject of that sentence. It fences the force on a moving charge. Nothing in the CED extends it to the force on a wire, and Topic 12.3 sets no limit of its own, so treat $\sin\theta$ in $F_B = I \ell B \sin\theta$ as a factor you may be asked to evaluate.

The CED lists five suggested skills here, more than it lists for any other topic in the unit: 1.B, create quantitative graphs with appropriate scales and units, including plotting data; 2.A, derive a symbolic expression from known quantities by selecting and following a logical mathematical pathway; 2.D, predict new values or factors of change of physical quantities using functional dependence between variables; 3.A, create experimental procedures that are appropriate for a given scientific question; and 3.C, justify or support a claim using evidence from experimental data, physical representations, or physical principles or laws. Unit 12 is weighted at 12 to 15 percent of the multiple-choice section, with a suggested 10 to 14 class periods.

The field around a straight wire is circles, and nothing else (12.3.A.1.i)

Essential knowledge 12.3.A.1.i does two jobs. It says what the field is, then it says, at unusual length for a CED statement, what the field is not.

What it is: the field vectors are tangent to concentric circles centered on the wire. Pick a point near a long straight wire, draw the circle through it centered on the wire in the plane perpendicular to it, and the field there runs along that circle.

What it is not: the field has no component toward, away from, or parallel to the wire. Three exclusions, spelled out one by one. This is the sentence to reread, because the picture most students bring into Unit 12 is the electric field picture from the electric field guide, where the field around a line of charge points straight out from it. Magnetic field lines around a wire never point at the wire and never point away from it. They go around.

Three consequences follow, and each is a question someone can ask you.

• The field has the same magnitude everywhere on a given circle, because every point on it is the same perpendicular distance $r$ away. The direction changes as you go round; the size does not.
• A compass placed near a current-carrying wire lines up across the wire, not along it. The CED's sample instructional activity for this topic hands students a battery, wire, and a cheap compass and asks them to turn exactly this observation into evidence.
• There is no end to the field and no pole. A bar magnet has a north end and a south end; a straight current does not. That is the closed-loop property Topic 12.1 states for magnetic fields generally.

One more piece of care about $r$. Essential knowledge 12.3.A.1.ii calls it "the perpendicular distance from the central axis of the wire to the point": measure from the axis, not the surface.

B = mu0 I over 2 pi r, and the 1 over r that is not an inverse square (12.3.A.1.ii)

This is printed on the AP Physics 2 equation sheet in the Magnetism group, so you do not have to recall it. The constant $\mu_0$ is the vacuum permeability, and the Table of Information gives $\mu_0 = 4\pi \times 10^{-7} \ (\mathrm{T \cdot m})/\mathrm{A}$. Because the $4\pi$ in $\mu_0$ and the $2\pi$ underneath share a factor, the prefactor collapses to a number worth carrying:

So $B = (2 \times 10^{-7})\,I/r$ in tesla, with $I$ in amperes and $r$ in meters. That one simplification removes most of the arithmetic from this topic. Where $\mu_0$ belongs conceptually is Topic 12.1, whose essential knowledge 12.1.C.2 introduces it as the constant value of magnetic permeability that free space has.

Now the part that gets marked wrong. The dependence is $1/r$, not $1/r^2$. Essential knowledge 12.3.A.1.ii says "inversely proportional to the perpendicular distance", with no square anywhere in it. Every inverse-square law you have met so far pulls the other way: Coulomb's law goes as $1/r^2$, the field of a point charge goes as $1/r^2$, gravity goes as $1/r^2$. A long straight wire is not a point, and its field does not.

Skill 2.D, predicting new values or factors of change using functional dependence, is listed for this topic, and this equation is what it is listed for. Read the factor changes straight off the two proportionalities.

That last row is not in the equation, which gives only a magnitude. Direction is a separate question, answered by the right-hand rule below.

A note on where the equation is safe. It describes a long, straight wire, and the CED says so in both statements that mention it. It is not the field at the center of a loop. For the loop, 12.3.A.1.iv gives a direction and no formula, which tells you what can be asked.

The right-hand rule for a wire, and for a loop (12.3.A.1.iii and iv)

Two right-hand rules live in this topic, and they are one rule applied to two shapes.

For a long straight wire. Point the thumb of your right hand along the current. Your fingers curl the way the field circles the wire. If the current runs to the right across the page, the field is into the page above the wire and out of the page below it.

For a loop. Curl your fingers the way the current runs around the loop, and your thumb points along the axis, in the direction of the field at the loop's center. That is essential knowledge 12.3.A.1.iv. Notice the two rules trade roles: for the straight wire the thumb is the current and the fingers are the field, and for the loop the fingers are the current and the thumb is the field.

One convention decides whether your answer is right or backwards, and it is printed on the exam rather than left to you. The conventions list on the AP Physics 2 Table of Information includes: "Current is conventional current." Conventional current runs the way positive charge would move, opposite to the drift of the electrons actually carrying it in a metal wire. Point your thumb along the conventional current. Using electron flow flips every field and every force on the page.

A loop of current, seen from far off, behaves like a magnetic dipole: out of one face, round the outside, back into the other. Essential knowledge 12.1.B.1 states that magnetic dipoles result from the circular or rotational motion of electric charges, and that in magnetic materials this can be the motion of electrons. A bar magnet and a current loop are one phenomenon at two scales.

When you draw this, use the page notation and say which you mean: a dot for a vector out of the page toward the reader, a cross for one into the page.

Two or more wires: add the fields as vectors (12.3.A.1.v)

Essential knowledge 12.3.A.1.v lets you use something you already have: the field near two or more current-carrying wires is found using vector addition principles. No new equation. Compute each wire's contribution from $B = \mu_0 I / 2\pi r$, get each direction from the right-hand rule, and add them as you would any vectors, as in Topic 1.5.

The standard setup is two long parallel wires with a point between them.

1. Directions first. At the midpoint between two parallel wires carrying current in the same direction, the two fields point opposite ways, because you are above one circle and below the other. Between wires carrying current in opposite directions, the two fields point the same way.
2. Magnitudes second. Then add or subtract.

So the midpoint between two equal currents running the same way is a point of zero field, and the midpoint between two equal opposing currents is where the field along that line is largest. Students often expect the reverse, on the intuition that "same direction" should reinforce. It does not, because the field is a circulation rather than an arrow pointing away from the wire.

A check on any answer here: if two currents run the same way but are unequal, the null point where the fields cancel lies between them, closer to the smaller current. Set $\mu_0 I_1 / 2\pi r_1 = \mu_0 I_2 / 2\pi r_2$ and the constants drop out, leaving $I_1/r_1 = I_2/r_2$. That is skill 2.A doing real work: the answer is a ratio, and no value of $\mu_0$ is substituted.

The force a field exerts on a wire: F = I l B sin theta (12.3.B)

The second half of the topic reverses the first. Now the wire is the thing being pushed. Essential knowledge 12.3.B.1.i names four dependencies, and each is something a question can vary: the current, the length of the portion of the wire within the magnetic field, the magnitude of the magnetic field, and the angle between the direction of the current in the wire and the direction of the magnetic field.

Read the second one again. The length of the portion of the wire within the magnetic field. Not the length of the wire. If a 2.0 m wire crosses a magnet gap only 0.05 m wide, $\ell = 0.05$ m, because the rest of the wire sits where there is no field and feels nothing. That wording is in the CED for a reason, and it is the quiet way to lose the question.

And $\theta$ is the angle between the current direction and the field direction, not an angle measured from a surface or from the vertical.

• $\theta = 90^{\circ}$: current across the field, $\sin\theta = 1$, force at maximum.
• $\theta = 0^{\circ}$ or $180^{\circ}$: current along or against the field, $\sin\theta = 0$, no force at all, however large the current.
• In between, scale by $\sin\theta$. The Table of Information prints trigonometric values for 0, 30, 37, 45, 53, 60, and 90 degrees, so $\sin 30^{\circ} = 1/2$ and $\sin 53^{\circ} = 4/5$ come off the sheet exactly.

The direction is given by the right-hand rule, per 12.3.B.1.ii, and is perpendicular to both the current and the field. Whatever hand motion you use, sanity-check the result: the force can never point along the wire, and never along the field.

This is the same physics as Topic 12.2, not a second law. There, $F_B = qvB\sin\theta$ acts on one moving charge. A current is charge moving through a wire, so the wire equation is the charge equation summed over the carriers in the length $\ell$, with the same $\sin\theta$ and the same right-hand rule. The practical difference is the boundary statement quoted above, which restricts quantitative work to 0, 90, and 180 degrees for the moving charge and says nothing about the wire.

This force is also why a motor turns. Current runs one way along one side of a loop and the other way along the opposite side, so the two forces point opposite ways, and a pair of opposite forces offset from an axis is a torque. The unit's essential questions include "How does an electric motor work?"

Two parallel wires, a derivation rather than a printed rule (skill 2.A)

The force between two parallel current-carrying wires is a standard Unit 12 question, and its status is worth stating precisely: it is not a separate essential knowledge statement. The CED prints no parallel-wire formula in Topic 12.3, and neither does the equation sheet. It is 12.3.A and 12.3.B chained, which is the kind of thing skill 2.A, deriving a symbolic expression by following a logical mathematical pathway, is listed here to test.

The chain has two links.

1. Wire 1 makes a field at wire 2's location of magnitude $B_1 = \dfrac{\mu_0 I_1}{2\pi d}$, where $d$ is the separation. It is tangent to a circle round wire 1, so at wire 2 it is perpendicular to wire 2.
2. Wire 2 sits in that field carrying $I_2$, so a length $\ell$ of it feels $F = I_2 \ell B_1 \sin 90^{\circ} = I_2 \ell B_1$.

Substitute, and the force per unit length comes out clean:

Swap the labels and the expression is unchanged, because $I_1 I_2$ is symmetric. So the two wires push on each other equally hard, which they must: they are a Newton's third law pair, and the third worked example checks that numerically from both ends.

Which way? Apply the right-hand rule twice, and the result inverts the electrostatic habit.

• Currents in the same direction: the wires attract.
• Currents in opposite directions: the wires repel.

Compare Coulomb's law, where like charges repel. Here, like currents attract, because the magnetic field circulates rather than radiates and the geometry setting the sign is a different geometry.

Note the dependence too: $1/d$, inherited straight from the field. Double the separation and the force per meter halves. It does not quarter.

Designing the experiment, plotting the line, justifying the claim (skills 3.A, 1.B, 3.C)

Three of this topic's five suggested skills are about evidence rather than algebra, and the CED's own sample activities show what that looks like.

Skill 3.A, create experimental procedures appropriate for a given scientific question. The CED's sample activity for Topic 12.3 gives groups a battery, three light bulbs, wire, and a cheap compass, and asks them to develop an experiment showing that magnetic field strength decreases with distance from the wire, and another showing that the magnetic field increases with current in the wire. Notice what the three bulbs are for: they are the current control. Adding bulbs in series changes the current without changing anything else, which is how the second experiment isolates one variable.

Skill 1.B, create quantitative graphs with appropriate scales and units, including plotting data. $B = \mu_0 I / 2\pi r$ has two variables, so it gives two straight lines, and knowing which axes straighten it is the skill.

The third row earns the marks. When data will not lie on a line, plot against the reciprocal, and the slope becomes a quantity you can compare with a prediction. A straight graph of $B$ against $1/r$ is evidence for the $1/r$ dependence specifically. Both lines should pass through the origin; an intercept that does not is a signal of a background field, most often Earth's.

Skill 3.C, justify or support a claim. The unit's "Preparing for the AP Exam" page is blunt about this, and it is worth taking literally. It says that when writing or identifying justifications for claims, simply referencing an equation, law, or physical principle is not sufficient, and gives as its example that stating "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, it says, should clearly and concisely explain the steps that lead from the equation, law or physical principle to the justification of their claim.

So a full-credit direction answer has steps in it:

1. The current in wire 1 is out of the page, so by the right-hand rule its field at wire 2 points upward in the plane of the page.
2. Wire 2 carries current out of the page in that upward field.
3. Applying the right-hand rule to wire 2's current in that field gives a force directed toward wire 1, so the wires attract.

Three sentences, each one a step, with no gap where "by the right-hand rule" does the work alone. What a changing field produces is the subject of Topic 12.4, which carries the same right-hand rule into Faraday's law.