Unit 10: Electric Force, Field, and Potential

What Unit 10 is really about

Unit 10 describes one interaction, the push or pull between charges, in four different languages: force, field, energy, and potential. Learn it as four names for one thing and the unit is short. Learn it as seven unrelated topics and you will spend the exam guessing at formulas.

The CED says the unit begins the study of electrostatic phenomena at a fundamental level and introduces the model of field forces. It adds that the electric force, like the forces introduced in AP Physics 1, can be described with Newton's laws, and it encourages you to apply AP Physics 1 principles to fields. There is practical instruction in that: a charged ball on a string is still a free-body diagram problem, and a charge released between two plates is still an energy problem.

Unit 10 carries 15 to 18 percent of the multiple-choice section and the CED estimates about 14 to 21 class periods for it. Three units share that 15 to 18 percent band: Unit 9 Thermodynamics, Unit 10, and Unit 11 Electric Circuits. The remaining four units sit at 12 to 15 percent each.

The essential questions on the unit opener are everyday ones: how to suspend a charged water droplet in the air, why a charged rubber balloon sticks to a wall, where the safest place is in a lightning storm, and how to protect electronics from an EMP.

Force, field, energy, potential: the four-way confusion

This is the mistake that spans every topic in the unit: four quantities with similar names that get swapped constantly. Learn this table before anything else in Unit 10.

The split in that last column is what the unit is built on. Field and potential are properties of the space around a charge configuration; they are there whether or not anything sits at the point you are asking about. Force and potential energy only exist once a second charge shows up.

Two definitions carry you between the halves, and both are on the sheet. Field is force per unit charge (EK 10.3.A.2); potential difference is change in potential energy per unit charge (EK 10.5.A.3):

Divide by the charge to go from what a charge feels to what the space holds; multiply by it to come back. The sheet prints that second one already rearranged, as $\Delta U_E = q \Delta V$.

Three checks that catch the error before it costs you a point:

• Check the unit in the answer. Newtons or N/C puts you on the vector side, so contributions add by components. Joules or volts puts you on the scalar side, so they add as signed numbers with no angles resolved. The CED calls that scalar superposition (EK 10.5.A.2).
• Check whether a test charge appears. If the question never puts a charge at the point, it cannot be asking for a force or a potential energy.
• Check that zero means what you think. The field can be zero where the potential is not, and the reverse. The electric field and potential guide works both cases with numbers.

The sheet's symbol keys are worth one read for this reason: in the Electricity column $V$ is electric potential, and in the Thermal Physics column the same letter is volume.

What the CED requires across the unit

Seven topics carry eleven learning objectives, and every one of them begins with the same verb, describe:

• 10.1.A the electric force that results from interactions between charged objects or systems. 10.1.B the electric and gravitational forces that result from interactions between charged objects with mass. 10.1.C the electric permittivity of a material or medium.
• 10.2.A the behavior of a system using conservation of charge.
• 10.3.A the electric field produced by a charged object or configuration of point charges. 10.3.B the electric field generated by charged conductors or insulators.
• 10.4.A the electric potential energy of a system.
• 10.5.A the electric potential due to a configuration of charged objects. 10.5.B the relationship between electric potential and electric field.
• 10.6.A the physical properties of a parallel-plate capacitor.
• 10.7.A changes in energy in a system due to a difference in electric potential between two locations.

Five of the seven topics carry a boundary statement, and one number runs through four of them. Topic 10.1 says AP Physics 2 only expects calculations of the electric force between four or fewer interacting charged objects or systems, then allows analysis of the resulting force from more charges in situations of high symmetry. Topic 10.3 repeats the cap for the electric field, keeps the same high-symmetry exception, and adds that students will only be expected to perform qualitative analysis of electric fields within insulators.

Topics 10.4 and 10.5 give the reason for the cap: the methods needed for extended charge distributions are outside the scope of the course. So 10.4 requires electric potential energy only for four or fewer point charges, and 10.5 requires electric potential only for four or fewer particles, or more in situations of high symmetry.

Topic 10.6 is bounded differently: other shapes can also separate charges, but only the analysis and description of parallel-plate capacitors is required, and edge effects will be ignored unless explicitly stated otherwise. Topics 10.2 and 10.7 carry no boundary statement.

Read those five together and the shape of the unit appears: no integrating over a charged rod, but plenty of summing four contributions correctly, as vectors for the field and as signed scalars for the potential.

The Unit 10 equations on the sheet

The AP Physics 2 sheet holds 129 entries in seven tables, one of them Electricity. That table has 20 entries: eleven belong to Unit 10, and the other nine (current, resistivity, power, Ohm's law, equivalent resistance and capacitance in series and parallel, and the RC time constant) are Unit 11 material.

Four things to notice.

Use the sheet's Coulomb constant. The Constants table prints $k = \frac{1}{4\pi\varepsilon_0} = 9.0 \times 10^9\ \text{N}\cdot\text{m}^2/\text{C}^2$, and the force, field and potential-energy entries are each printed twice over, once with $\frac{1}{4\pi\varepsilon_0}$ and once with $k$. Use $9.0 \times 10^9$, not 8.99. Vacuum permittivity is given too, $\varepsilon_0 = 8.85 \times 10^{-12}\ \text{C}^2/(\text{N}\cdot\text{m}^2)$, and the capacitance entries need it.

The bars are not decoration. The force entry wraps the charges in absolute-value bars, $\lvert q_1 q_2 \rvert$, so it returns a magnitude and you take the direction from the signs instead: like charges repel, opposite charges attract (EK 10.1.A.3). The potential energy entry has no bars, so the sign survives: two like charges give a positive $U_E$, an attracting pair a negative one.

Superposition is printed for potential and not for field. The sheet hands you $V = \frac{1}{4\pi\varepsilon_0}\sum_i \frac{q_i}{r_i}$, a ready-made sum. There is no matching entry for adding fields, even though EK 10.3.A.3.i requires the net field to be the vector sum of the individual fields.

Two exam conventions ship with the sheet. The Table of Information page states that unless otherwise noted, electric potential is zero at an infinite distance from an isolated point charge, and capacitors are air-filled with $\kappa = 1.0$. The first makes the work described in EK 10.4.A.1, bringing charges in from infinitely far away, a well-defined quantity. The second means the dielectric constant disappears from most capacitor questions. The full sheet is at the AP Physics 2 formula sheet.

How the seven topics build

The order is not arbitrary: charge, force, field, energy, potential, a device that stores it, then conservation.

1. 10.1 Electric Charge and Electric Force sets the foundations: charge is a fundamental property of all matter, and the elementary charge $e$ can be considered the smallest indivisible amount of charge. A point charge is the model for when the size of a charged object does not matter. Coulomb's law arrives here, and so does the comparison with gravity: electric forces attract or repel while gravitational forces only attract, and although gravity is weaker it dominates at large scales because large systems tend to be electrically neutral (EK 10.1.B.3). The routine lives in the Coulomb's law guide.
2. 10.2 Conservation of Electric Charge and the Process of Charging is the bookkeeping topic. Net charge changes only when charge transfers between a system and its surroundings, usually as electrons. Friction and contact change net charge; induced separation does not, because it only rearranges charge inside a system, and it can happen in a neutral one. Grounding connects a charged system to a much larger, approximately neutral system such as Earth.
3. 10.3 Electric Fields introduces the model the unit is named for. A test charge is small enough that it does not significantly disturb the field it measures. Field points away from isolated positive charges and toward isolated negative ones. Field maps use vectors; field-line diagrams are the simplified version and still give relative magnitude and direction. The conductor rules matter: in electrostatic equilibrium the excess charge on a solid conductor sits on the surface, the field inside is zero, and the field is perpendicular to the surface. Outside an isolated sphere with a spherically symmetric charge distribution, the field matches a point charge of the same net charge at the center. Insulators differ: excess charge sits through the interior too, and the field inside may be nonzero.
4. 10.4 Electric Potential Energy defines $U_E$ for a pair of point charges as the work an external force must do to bring them from infinitely far apart to where they are. For more than two charges, total the potential energies of the individual pairs.
5. 10.5 Electric Potential divides that energy by the charge: potential is electric potential energy per unit charge at a point, and multiple charges combine by scalar superposition. Potential difference can also come from chemical processes that separate charge, which is what a battery does, and conductors in electrical contact redistribute electrons until their surfaces sit at the same potential. The second half connects potential back to field: equipotential lines, also called isolines, run perpendicular to the field vectors, the field points toward decreasing potential, and there is no field component along an isoline.
6. 10.6 Capacitors is the device topic. Two parallel conducting surfaces hold equal and opposite charge; capacitance relates that charge to the potential difference the separation creates, and it depends only on the physical properties of the capacitor, not on how much charge you put on it. The field between the plates is constant in magnitude and direction except near the edges, so a charged particle there undergoes constant acceleration and its motion shares characteristics with projectile motion near Earth's surface. A dielectric changes the capacitance and induces a field in itself opposing the field between the plates.
7. 10.7 Conservation of Electric Energy closes the loop. Move a charged object between two points at different potentials and the potential energy changes by $\Delta U_E = q \Delta V$, with the kinetic energy change following from conservation of energy. Treat it as an AP Physics 1 conservation of energy problem with one extra term.

Traps that span more than one topic

Topic pages cover their own pitfalls. These show up where two topics meet, which is where the exam likes to sit.

• Vector sum versus scalar sum on the same diagram. Fields add as vectors, so contributions cancel by pointing opposite ways. Potentials add as signed scalars, so they cancel only when the signs differ. Between two equal and opposite charges, the potentials can cancel while the fields reinforce.
• Field direction versus force direction. The field is defined so that the force on a positive test charge points along it (EK 10.3.A.2.iii). Put a negative charge in the same field and the force points the other way. Nothing about the field changed.
• Grounding does not neutralize everything. Grounding connects a system to a much larger, approximately neutral one and allows charge to transfer. It does not automatically drive every charge to zero, and induced separation can persist while the connection is in place.
• Capacitance is fixed by geometry. $C = Q/\Delta V$ is a definition, not a dependence. Doubling the charge on a given capacitor doubles the potential difference and leaves $C$ alone. What changes $C$ is plate area, plate separation, and the dielectric.

How Unit 10 is assessed

The AP Physics 2 exam runs 3 hours: Section I is 42 multiple-choice questions in 85 minutes for 50 percent of the score, Section II is 4 free-response questions in 95 minutes for the other 50 percent, and a four-function, scientific, or graphing calculator is allowed on both. Unit 10 supplies 15 to 18 percent of Section I.

The four free-response questions are fixed by type: Question 1 is Mathematical Routines (10 points, suggested 20 to 25 minutes), Question 2 is Translation Between Representations (12 points, suggested 25 to 30 minutes), Question 3 is Experimental Design and Analysis, and Question 4 is Qualitative/Quantitative Translation.

The CED's Unit 10 opener flags Question 2 specifically. It describes a Translation Between Representations question in which a student might sketch equipotential lines around a small positively charged sphere, then create energy bar charts for a system containing a small point charge released from rest near that sphere, then explain how the two representations are consistent. The opener says Unit 10 content provides especially good practice for that question type, and that content from any unit may appear in it. So practice the translation; do not assume the question will be electrostatics. The representations it names for this unit are pictures, motion diagrams, graphs, energy bar charts, electric field vector diagrams, and equipotential diagrams.

The Unit at a Glance table lists suggested skills topic by topic. Skill 2.A, derive a symbolic expression from known quantities, is suggested for five of the seven topics. Skills 1.A (create diagrams and schematics), 2.C (compare quantities across scenarios), 3.B (apply a law or model to make a claim) and 3.C (justify a claim using evidence) each appear for four. Topic 10.6 Capacitors is the only Unit 10 topic listing 3.A, create experimental procedures for a given scientific question, and the only one carrying five suggested skills rather than four.

How to study Unit 10

Work in the CED's order, but front-load the vocabulary.

1. Write the four-quantity table from memory before you touch a problem set: symbol, vector or scalar, unit, and whether it needs a charge sitting at the point. Redo it each session until it is automatic.
2. Get Coulomb's law fully reliable. Magnitude from the equation, direction from the signs, never both from the equation. The Coulomb's law guide has the routine; the Coulomb's law calculator checks your arithmetic while you build speed.
3. Do field and potential side by side. For each arrangement of charges, answer both questions before moving on. The electric field and potential guide works the point-charge cases in detail.
4. Practice representations in both directions. Build an isoline map from a field vector map and a field map from an isoline map. Sketch energy bar charts for a charge released in a field. That is the Question 2 skill and it is cheap to drill.
5. Treat 10.7 as an energy problem. Set up the system, list the energy terms, then add $\Delta U_E = q \Delta V$. Everything in conservation of energy still applies.
6. Keep [the AP Physics 2 formula sheet](/formulas/ap-physics-2) open so you learn which eleven Electricity entries are yours and stop deriving what is printed.
7. Then go straight into circuits. Unit 11 Electric Circuits reuses $\Delta V$ everywhere and brings capacitors back in RC circuits, so revise the two together. Full course map: AP Physics 2.