Unit 9: Electric Potential

What the CED requires across Unit 9

Unit 9 of AP Physics C: Electricity and Magnetism is Electric Potential. 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 Units 12 and 13. Units 8 and 11 are heavier at 15 to 25%, and Unit 10 is lighter at 10 to 15%.

Three topics, which makes this the shortest unit in the course by topic count. It carries four learning objectives, because Topic 9.2 has two.

Do not read three topics as a light unit. Topic 9.2 spans three printed pages of the framework, carries the unit's only boundary statement, and holds both of the calculus relationships the unit exists to teach. On the CED's own sample exam, Topic 9.2's objectives appear in both a multiple-choice question and the free-response Mathematical Routines question.

All four objectives open with the task verb describe, which the CED says "encompasses the range of possible graphical, mathematical, or verbal skill applications", adding that students should be able to describe a physical concept graphically, mathematically, and verbally. In this unit the graphical version has a name: an isoline map.

The CED's framing is that students are introduced to the concept of electric potential, which is another way to describe interactions between charged systems, and that a thorough understanding of the relationship between electric potential and electrical energy will support the ability to analyze physical scenarios, such as the interaction between two charged objects, using the concept of energy.

The essential questions printed on the unit opener are all about batteries and voltage: what the difference is between electric potential and electric potential energy, why different voltage batteries exist, what the difference is between a 1.5 V AA battery and a 9 V battery, why a car battery is more dangerous than a 9 V battery, and why high voltage power lines are dangerous. The first of those is the question this unit is built to answer, and essential knowledge 9.2.A.4 is the CED's own nod to the rest: electric potential difference may also result from chemical processes that cause positive and negative charges to separate, such as in a battery.

One boundary statement, and it is Unit 8's with one phrase changed

Unit 9 prints exactly one boundary statement. It sits under Topic 9.2, attached to objective 9.2.A, and Topics 9.1 and 9.3 print none. Quoted whole:

"AP Physics C: Electricity & Magnetism only expects students to use calculus to find the electric potential resulting from the following charge distributions and locations: an infinitely long, uniformly charged wire or cylinder at a distance from its central axis, a thin ring of charge at a location along the axis of the ring, a semicircular arc or part of a semicircular arc at its center, and a finite wire or line charge at a point collinear with the line charge or at a location along its perpendicular bisector."

Set that beside the Topic 8.4 boundary statement and the two are the same sentence with one phrase swapped: 8.4 says "the electric field resulting from", 9.2 says "the electric potential resulting from". The list of four geometries and their location clauses is identical, word for word.

That is more useful than it sounds, for two reasons.

Every distribution you can integrate for a field, you can integrate for a potential, and the reverse. So there are only four geometries to practise, not eight, and each one is worth practising both ways. The CED's own sample Question 1 does exactly that: it takes a thin ring of charge and asks for the potential in one part and the field on the axis in another, and the scoring guidelines accept getting the field either by integrating for it directly or by differentiating the potential.

The potential integral is the easier of the two, every time. Statement 9.2.A.2.ii says the potential due to multiple point charges is found by scalar superposition. There are no components to resolve, no $\cos\theta$ factor to insert and no cancellation argument to make. When a problem offers both routes, the potential route is usually shorter, and $E_x = -dV/dx$ then hands you the field for the price of one derivative.

Notice too what the boundary statement does not restrict. It bounds the use of calculus to find a potential. Nothing bounds a question that hands you a potential as a function of position, or a graph of one, and asks for the field: that is objective 9.2.B, and it needs a derivative rather than an integral.

How the three topics build

9.1 Electric Potential Energy starts with work. Statement 9.1.A.1 defines the electric potential energy of a system of two point charges as the amount of work required for an external force to bring the point charges to their current positions from infinitely far away. Two things are packed into that sentence: the energy belongs to the system, not to either charge, and the zero of energy is at infinite separation. The exam conventions printed with the equation sheet make the second one explicit for potential: "The electric potential is zero at an infinite distance from an isolated point charge."

Statement 9.1.A.2 gives the general form,

and note that the charges enter with their signs, unlike Coulomb's law where the framework writes $\lvert q_1 q_2 \rvert$. Statement 9.1.A.3 then handles more than two charges: the total electric potential energy of a system is the sum of the electric potential energies of the individual interactions between each pair of charged objects. Pairs, not charges. Three charges make three pairs; four charges make six.

9.2 Electric Potential is the unit. Objective 9.2.A defines electric potential as the potential energy per unit charge at a point in space (9.2.A.1), gives the superposition integral $V = \frac{1}{4\pi\varepsilon_0}\int \frac{dq}{r}$ for a distribution (9.2.A.2), specialises it to a single point charge (9.2.A.2.i) and to a scalar sum over several (9.2.A.2.ii), and defines potential difference as the change in potential energy per unit charge when a test charge is moved between two points (9.2.A.3).

Objective 9.2.B is the pair of calculus relationships, and it is the reason this course exists as a separate course. Statement 9.2.B.1 says the value of an electric field component in any direction at a given location is equal to the negative of the spatial rate of change in electric potential at that location:

Statement 9.2.B.2 runs it the other way: the change in electric potential between two points can be determined by integrating the dot product of the electric field and the displacement along the path connecting the points, written $\Delta V = V_b - V_a = -\int_a^b \vec{E} \cdot d\vec{r}$.

Then 9.2.B.3 supplies the graphical version. Equipotential lines represent lines of equal potential and are also called isolines (9.2.B.3.i); isolines are perpendicular to electric field vectors, and each kind of map can be constructed from the other (9.2.B.3.ii); an electric field vector points in the direction of decreasing potential (9.2.B.3.iii); and there is no component of an electric field along an isoline (9.2.B.3.iv). Those last two are 9.2.B.1 restated without symbols, and they answer most of the qualitative questions this unit can ask.

9.3 Conservation of Electric Energy cashes the potential in for motion. Statement 9.3.A.1 gives $\Delta U_E = q\,\Delta V$, and 9.3.A.2 says the movement of a charged object between two points at different potentials results in a change in kinetic energy of the object consistent with the conservation of energy. That is the whole topic: one equation and one sentence, both of which reduce a field problem to an energy problem you already knew how to solve in mechanics.

The Unit 9 equations, and which ones are printed

Eight distinct equations appear in Unit 9's required content. Five are printed on the AP Physics C: E&M formula sheet, one is printed in a shortened form, and two are not printed at all.

Four observations.

The potential of a single point charge is not on the sheet. The sheet gives you the integral $V = \frac{1}{4\pi\varepsilon_0}\int \frac{dq}{r}$ and expects you to collapse it. Neither is the scalar sum over several point charges, even though it carries the "Relevant equation" label in the framework. In a unit where nearly every problem starts by adding potentials, the two most-used lines are yours to remember.

Coulomb's law is printed twice over; the potential energy is printed once. The sheet gives the force as both $\frac{1}{4\pi\varepsilon_0}\frac{\lvert q_1 q_2\rvert}{r^2}$ and $k\frac{\lvert q_1 q_2\rvert}{r^2}$, but prints the potential energy only in the permittivity form, with no $k$ version beside it. The framework's 9.1.A.2 does show both. If you work in $k$, that conversion is on you.

Unit 9 has no Derived Equations. The CED's Required Equations page reserves that label for equations "provided for reference and guidance, or to demonstrate the final results of derivations expected of students on the exam". Unit 10 carries three such labels and Unit 13 carries four. Unit 9 carries none, which does not mean no derivations are expected here. It means the expected derivations in this unit end in an expression specific to the geometry you were handed, not in a named result the CED could print.

The sheet trims 9.2.B.2. The framework writes $\Delta V = V_b - V_a = -\int_a^b \vec{E}\cdot d\vec{r}$; the sheet prints $\Delta V = -\int_a^b \vec{E}\cdot d\vec{r}$. Same equation, but the framework's version is the one that tells you which endpoint is which, and sign errors in this unit are almost always endpoint errors.

The sheet also reprints the whole C: Mechanics table, which matters here because Topic 9.3 is a mechanics problem wearing an electrical hat. $K = \frac{1}{2}mv^2$, $\Delta K = \sum W_i$ and $F_x = -dU(x)/dx$ are all on the same page. That last one is worth staring at next to $E_x = -dV/dx$: they are the same mathematical statement, once about energy and force, once about potential and field.

What calculus changes, and where AP Physics 2 stops

AP Physics C: Electricity and Magnetism is equivalent to the second course in an introductory college sequence in calculus-based physics, and its prerequisites say students should have taken or be concurrently taking calculus. In Unit 9 the calculus does three specific jobs.

• The potential of a distribution is an integral. AP Physics 2 adds the potentials of a few point charges. This course integrates $dq/r$ over a rod, a ring or an arc.
• The field is the derivative of the potential. AP Physics 2 has a matching objective, 10.5.B, but its essential knowledge 10.5.B.1 says only that the average electric field between two points equals the potential difference divided by the distance between them. That is a finite difference giving an average. Statement 9.2.B.1 gives the field at a point, so a potential that varies non-linearly is fair game here and out of reach there.
• The potential difference is a line integral. Statement 9.2.B.2 replaces that ratio of differences with $-\int_a^b \vec{E}\cdot d\vec{r}$, which is what lets a question put a non-uniform field between the two points.
• A graph becomes a calculation. Given $V$ against $x$, the field is a slope, so a graph question and an algebra question become the same question. Skill 1.B, creating quantitative graphs, is listed for Topic 9.2, and skill 1.C, sketching qualitative graphs, for Topic 9.1.

The nearest algebra-based pages on this site sit inside AP Physics 2 Unit 10, Electric Force, Field, and Potential: Topic 10.4, Electric Potential Energy, Topic 10.5, Electric Potential and Topic 10.7, Conservation of Electric Energy. All three of those titles match this unit's three topics word for word and in the same order, which is exactly the situation where two pages on one site can end up competing instead of adding up. Those pages are for AP Physics 2 students; this unit is for AP Physics C students. If you are in the algebra-based course you want finite differences, uniform fields and the electron volt, and you can stop before the integral. If you are in Physics C you want the same three ideas with the calculus versions of objective 9.2.B on top, plus a potential integral that course never sets up.

Four quantities, and the questions that separate them

The unit's first essential question is what the difference is between electric potential and electric potential energy. That is really a question about four quantities, and the CED defines each one precisely enough to tell them apart.

Read down the third column and the pattern is clean: field and potential are properties of the arrangement of source charges, while force and potential energy need a second charge to exist. Read the fourth column and you have the reason potentials are easier than fields.

The two rows in the middle are linked by a derivative and an integral (9.2.B.1 and 9.2.B.2). The two rows on the right are linked by multiplication by a charge (9.2.A.3 and 9.3.A.1). And the two columns are linked by the mechanics result the sheet reprints, $F_x = -dU(x)/dx$, which stands in exactly the same relation to $U$ as $E_x = -dV/dx$ does to $V$.

One consequence is worth stating on its own, because it is the most-tested idea in the unit. A zero field does not mean a zero potential, and a zero potential does not mean a zero field. At the midpoint between two equal positive charges the field is zero and the potential is a maximum. At the midpoint between equal and opposite charges the potential is zero and the field is not. Statement 9.2.B.3.iv is the general version: there is no component of the field along an isoline, but the isoline itself can sit at any value, including zero.

Traps that span more than one topic

The potential energy belongs to the system. Statement 9.1.A.1 says "the electric potential energy of a system of two point charges", and the CED's sample multiple-choice questions keep the language: one of them writes $\Delta U_{\text{ps}}$ for a particle-and-sphere system. Saying the charge has potential energy is the habit that leads to double-counting when a third charge arrives.

Pairs, not charges. Statement 9.1.A.3 sums over the individual interactions between each pair. Three charges give three terms, four give six, five give ten. Writing one term per charge is the single most common arithmetic error in Topic 9.1.

Signs go into the potential energy; they come out of the force. The framework writes Coulomb's law with $\lvert q_1 q_2 \rvert$ and then writes $U_E$ with plain $q_1 q_2$. So a repulsive pair has positive $U_E$ and an attractive pair has negative $U_E$, and dropping the sign turns an attraction into a repulsion silently.

The minus sign in $E_x = -dV/dx$ is a direction, not a decoration. Statement 9.2.B.3.iii is the sanity check: the field points in the direction of decreasing potential. If your algebra gives a field pointing uphill in potential, you dropped the sign.

Potential is not the potential energy per unit charge of the charge that made it. Statement 9.2.A.1 says it is the potential energy per unit charge at a point in space, meaning per unit of the charge you would put there. A source charge does not have a potential at its own location in this model.

A path integral does not mean the path matters. Statement 9.2.B.2 integrates along the path connecting the points, and for an electrostatic field the answer is the same for every path between the same endpoints. That is why $\Delta V$ can be quoted between two points with no route specified, and it is what makes the energy method in Topic 9.3 legal.

An electron volt is a unit of energy. The sheet prints $1\ \mathrm{eV} = 1.60 \times 10^{-19}$ J and $e = 1.60 \times 10^{-19}$ C. A charge of one elementary charge moved through one volt gains one electron volt of energy, which is $\Delta U_E = q\,\Delta V$ with convenient units, not a separate formula.

How Unit 9 is assessed

The exam is 3 hours long: 42 multiple-choice questions in 85 minutes for 50%, then 4 free-response questions in 95 minutes for the other 50%, one of each type in a fixed order, with a calculator allowed throughout. Unit 9 contributes 10 to 20% of the multiple-choice section.

The unit's Preparing for the AP Exam note points at the second free-response question, the Translation Between Representations question. It says the TBR requires students to create graphical and verbal models of scenarios as well as compare these representations of the same situation, and that in the TBR a student might be asked to sketch an equipotential diagram from an electric field map, or to create an energy diagram for a point charge moving inside the region with an electric field, and then to make connections between the two representations, justifying how they are consistent with each other. It closes with the standard caveat: while Unit 9 content provides especially good practice for the TBR, content from any unit may be included in that question. The unit's AP Classroom Progress Check runs about 18 multiple-choice questions and 4 free-response questions, one of each type.

The unit opener names skills 1.A, 1.C, 2.A and 3.C as the ones Unit 9 develops, which is the widest spread of the three units in this part of the course. Two of the three topics list a Practice 3 skill twice over, and Topic 9.3 is the only topic in Units 8 through 10 that lists 3.A, creating experimental procedures, alongside 3.C.

On the CED's sample exam, two of the fifteen sample multiple-choice questions align to Unit 9. One aligns to 9.2.B and essential knowledge 9.2.B.2, and it is the line-integral idea reduced to arithmetic: a proton is moved 5 m at 37 degrees to the $x$-axis in a uniform 1000 V/m field pointing along $+y$, and the answer is the change in potential energy of the proton-field system in electron volts. Only the displacement component along the field counts, and the CED's own trigonometric table gives $\sin 37^\circ = 3/5$. The other aligns to 9.3.A and 9.3.A.1, and it is a justification question: a positive particle is moved from $2R_S$ to the center of a uniformly charged sphere, and the credited answer is that the change in potential energy is positive because the motion is opposite to the direction of the field.

The sample free-response Question 1 is the Mathematical Routines question, worth 10 points, aligned to learning objectives 8.4.A, 9.2.A, 9.2.B and 9.3.A, with skills 2.A, 3.B and 3.C. Three of its four objectives are from this unit. Part A asks where on the axis of a charged ring a small sphere of opposite charge could sit so that the net potential at a labelled point is zero, and requires a justification: the credited reasoning is that because potential is a scalar quantity and the two charges have opposite signs, there are locations where the potentials cancel. Part A then asks for a derivation of the sphere's speed, starting from conservation of energy, and awards separate points for the energy statement, for correct potential energy expressions at two locations, for using the correct locations, and for the kinetic energy substitutions. Part B asks for the field due to the ring and accepts either integrating for the field or differentiating the potential.

That scoring breakdown is the most useful thing the CED prints about this unit. Of the seven points in Part A, five are for setting the problem up correctly and one is for the final answer. Writing $E_{\text{total}} = U + K = \text{constant}$ before anything else earns a point on its own.

The unit prints only two optional sample instructional activities, both on Topic 9.2 and both about isoline maps: one using a charge-and-field simulation to investigate field and potential in two- and three-charge systems, and one connecting two electrodes to a 9 V battery in a shallow pan of water and probing the potential with a voltmeter to construct an isoline map and estimate the field strength at various locations. Topics 9.1 and 9.3 get none.