Unit 8: Electric Charges, Fields, and Gauss's Law

What the CED requires across Unit 8

Unit 8 of AP Physics C: Electricity and Magnetism is Electric Charges, Fields, and Gauss's Law. The course and exam description weights it at 15 to 25% of the multiple-choice section and suggests about 12 to 24 class periods. Unit 11 (Electric Circuits) carries the same 15 to 25% band and the same 12 to 24 periods. Units 9, 12 and 13 sit at 10 to 20%, and Unit 10 is the lightest at 10 to 15%. So Unit 8 and Unit 11 are the two heavyweight units of the course, and the sequence starts with one of them.

Six topics and nine learning objectives. Topic 8.1 carries three of the nine on its own, which is why it takes three printed pages of the framework while Topic 8.5 takes one.

Every one of those nine objectives opens with the task verb describe. The CED says that verb, used in nearly all learning objectives, "encompasses the range of possible graphical, mathematical, or verbal skill applications", and adds that within those multiple representations students should be able to describe a physical concept graphically, mathematically, and verbally. So describing an electric field can mean drawing a field map, writing the superposition integral, or saying in words what a test charge would feel.

The CED's own framing is that students will begin the study of electric force, which is exerted on all objects with a property called charge, and that the electric force, in contrast to gravitational force, is one of attraction or repulsion and therefore leads to different effects on objects. It says this knowledge will help students understand the role electrostatics plays in devices such as photocopiers, defibrillators, and printers, as well as television, radio, and radar industries.

The essential questions printed on the unit opener are the everyday versions: why hair stands up after brushing it with a plastic comb, how a charged rubber rod bends a stream of water, how a balloon can be made to stick to the wall, why cell phones do not work in concrete buildings, and why a bird can land safely on a high voltage wire. The last two are answered in Unit 10 rather than here, by electrostatic shielding and by a conductor sitting at a single electric potential.

Three boundary statements, quoted with their exception clauses

Boundary statements are how the CED fences off a treatment, and they are the only authority on how far the calculus goes. Unit 8 prints three of them, under Topics 8.1, 8.4 and 8.6. Topics 8.2, 8.3 and 8.5 print none. Here they are whole, because in the first one the sentences after the first reverse what the first one appears to say.

Topic 8.1, on how many charges you can be handed:

"AP Physics C: Electricity & Magnetism only expects students to make calculations of the electric force between four or fewer interacting charged objects or systems. The analysis of the resulting electric force from more charges is allowed in situations of high symmetry. Note that students are expected to calculate the electric fields of charge distributions, as described in Topics 8.4 and 8.6."

Stop after the first sentence and you would conclude that a five-charge arrangement is out of bounds. The second sentence allows it whenever there is high symmetry, and the third says outright that continuous charge distributions, which are infinitely many charges, are expected. The limit is on brute-force vector addition, not on the number of charges in the picture.

Topic 8.4, on which distributions you integrate:

"AP Physics C: Electricity & Magnetism only expects students to use calculus to find the electric field 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."

That is a closed list, and every geometry on it comes with a location clause that is part of the boundary rather than decoration. A ring is fair game on its axis, not off it. An arc is fair game at its center, not elsewhere. A finite line is fair game at a point collinear with it or on its perpendicular bisector, not at an arbitrary point in the plane.

Topic 8.6, on Gauss's law:

"AP Physics C: Electricity & Magnetism only expects students to quantitatively apply Gauss's law to point charges and charge distributions that have spherical, cylindrical, or planar symmetry."

Three symmetries, plus the point charge. Notice that the 8.4 and 8.6 lists interlock rather than duplicate. The infinitely long charged wire or cylinder appears on both, so its field is reachable either by integrating or by drawing a Gaussian surface. The ring and the arc appear only under 8.4, because neither has a symmetry that makes a Gaussian surface do any work. The plane appears only under 8.6, because a uniformly charged infinite sheet is one of the cleanest Gauss problems available and one of the nastiest integrals. Statement 8.6.A.4, quoted further down, explains why that list and no other.

How the six topics build

8.1 Electric Charge and Electric Force sets up the objects. Statement 8.1.A.1 makes charge a fundamental property of all matter, 8.1.A.1.i makes it a scalar, 8.1.A.1.ii calls the elementary charge $e$ the smallest indivisible amount of charge, and 8.1.A.1.iv defines the point-charge model as one in which the physical size of the object is negligible in the context being analyzed. Then 8.1.A.2 gives Coulomb's law and 8.1.A.3 puts its direction along the line of separation between the objects. Objective 8.1.B compares the two long-range forces: 8.1.B.2 says that for any two objects with both mass and charge the gravitational force is usually much smaller in magnitude, and 8.1.B.3 explains why gravity nonetheless dominates at large scales, because large systems tend to be electrically neutral. Objective 8.1.C is permittivity: 8.1.C.1 defines it as a measurement of the degree to which a material or medium is polarized in the presence of an electric field, and 8.1.C.4.ii distinguishes a conductor from an insulator by how easily charge carriers move. That is the seed of Unit 10's dielectric constant.

8.2 Conservation of Electric Charge and the Process of Charging is the only topic in the unit that prints no equation at all. Statement 8.2.A.1.ii defines induced charge separation as what happens when the electrostatic force between two systems alters the distribution of charges within them, and 8.2.A.1.iii adds that it can occur in neutral systems, which is the balloon on the wall. Statement 8.2.A.2.ii is conservation stated plainly: the net charge of a system will be constant unless there is a transfer of charge to or from the system. Statement 8.2.A.3 defines grounding as electrically connecting a charged object to a much larger and approximately neutral system, giving Earth as the example.

8.3 Electric Fields introduces the field as a ratio. Statement 8.3.A.2 defines the electric field at a point as the ratio of the electric force on a test charge at that point to the charge of the test charge, with 8.3.A.2.i requiring the test charge to be small enough that its presence does not significantly affect the field in its vicinity. Objective 8.3.B then does charged conductors and insulators: 8.3.B.1 puts the excess charge of a conductor in electrostatic equilibrium on its surface with zero field inside, 8.3.B.1.ii says the field outside an isolated sphere with a spherically symmetric charge distribution matches that of a point charge of the same net charge at the center, and 8.3.B.2 says an insulator's excess charge is spread through its interior as well as its surface and the field inside it may be nonzero.

8.4 Electric Fields of Charge Distributions is the calculus topic. Statement 8.4.A.1 says expressions for the field of specified charge distributions can be found using integration and the principle of superposition:

Statement 8.4.A.2 adds that symmetry considerations can simplify the analysis. That sentence carries more than it looks: on every geometry in the 8.4 boundary list, symmetry is what kills one or two components of the integral so that only one survives.

8.5 Electric Flux defines the quantity Gauss's law counts. Statement 8.5.A.1 says flux describes the amount of a given quantity that passes through a given area. Statement 8.5.A.2 gives $\Phi_E = \vec{E} \cdot \vec{A}$ for a field constant across the area, with 8.5.A.2.i fixing the area vector as perpendicular to the surface and outward from a closed surface, and 8.5.A.2.ii pointing out that the sign of the flux is just the sign of that dot product. Statement 8.5.A.3 gives the general case as a surface integral.

8.6 Gauss's Law ties the two together. Statement 8.6.A.1 relates the flux through a Gaussian surface to the charge enclosed by that surface, and 8.6.A.6 names Gauss's law as Maxwell's first equation. The four statements in between are procedure rather than physics, and they are worth reading as a method, which the next section does.

Read in order, the unit is one sentence with six clauses: name the charge, conserve it, turn force into field, integrate the field over a distribution, count the field crossing a surface, then use the count to get the field back.

The Unit 8 equations, and which ones are printed

Eight distinct equations appear in Unit 8's required content. Six of the eight are printed on the AP Physics C: E&M formula sheet, and the two that are missing are the two simple ones.

Four things in that table are worth carrying into the exam.

Unit 8 has no Derived Equations. The CED's Required Equations page explains that not all equations in the framework appear on the equation sheet, that many are provided for reference and guidance or to demonstrate the final results of derivations expected of students on the exam, and that those are denoted "Derived Equations". Unit 10 labels three of its equations that way and Unit 13 labels four. Unit 8 labels none: every labelled equation in it reads "Relevant equation". Nothing in this unit is a canned result you are expected to reproduce, because in this unit the setting up of the integral is the work.

Gauss's law is printed in one form and stated in two. Statement 8.6.A.1 gives both $\Phi_E = q_{\text{enc}}/\varepsilon_0$ and the closed-surface integral as relevant equations. Only the integral form is on the sheet. If a question hands you a flux and asks for the enclosed charge, you are using the first form, and you got there yourself.

The plain dot-product flux is not printed either. The sheet gives you the surface integral and nothing simpler, exactly as it does for magnetic flux in Unit 13. Collapsing $\int \vec{E} \cdot d\vec{A}$ to $EA\cos\theta$ for a uniform field is a step you take, and on a free-response question it is a step you justify.

Coulomb's law is printed with $k$ as well as $\varepsilon_0$. The sheet gives $k = 1/(4\pi\varepsilon_0) = 9.0 \times 10^9\ \mathrm{N \cdot m^2/C^2}$ and prints $\varepsilon_0 = 8.85 \times 10^{-12}\ \mathrm{C^2/(N \cdot m^2)}$ separately. The two are rounded independently, since $1/(4\pi \times 8.85 \times 10^{-12})$ is closer to $8.99 \times 10^9$, so the same problem worked both ways can differ in the third significant figure. Pick one and stay on it. The scoring guidelines for the CED's own sample Question 1 make this explicit, noting that the answer can be in terms of either $\varepsilon_0$ or $k$.

One resource this site's formula page does not carry but the official Table of Information does: alongside the geometry and trigonometry tables, the appendix prints a calculus box with the power rule, the chain rule, and the standard derivatives and integrals, including $\int x^n dx$ and $\int dx/(x+a) = \ln\lvert x + a\rvert$. For a unit built on integrals, knowing the integral table travels with you is worth as much as knowing any single formula on it.

What calculus changes, and where AP Physics 2 stops

AP Physics C: Electricity and Magnetism is a calculus-based, college-level course, equivalent to the second course in an introductory college sequence in calculus-based physics. Its prerequisites say students should have taken or be concurrently taking calculus. Unit 8 is where that prerequisite starts earning its place, and the difference from the algebra-based treatment is not cosmetic.

• The field of a distribution is an integral, not a sum. AP Physics 2 adds the fields of a handful of point charges as vectors. This course integrates $dq$ over a wire, a ring or an arc, which is the only way to get a field when the charge is spread out rather than lumped.
• Flux is a surface integral, not a product. A field that varies across the surface, or a surface that curves through a varying field, gives a flux you have to integrate for.
• Gauss's law does not exist in AP Physics 2. The phrases electric flux, Gaussian surface and Gauss's law appear nowhere in that course's framework, which reaches flux only as magnetic flux, in its Unit 12. Topics 8.5 and 8.6, a third of this unit, have no algebra-based counterpart at all.
• The charge itself can be a function of position. Statement 8.6.A.5 puts a charge density $\rho$ inside an integral, so "find the total charge" becomes a calculus question before the field question even starts.

The nearest algebra-based page on this site is the AP Physics 2 Unit 10 hub, Electric Force, Field, and Potential. Its Topics 10.1, 10.2 and 10.3 carry the same three titles as this unit's Topics 8.1, 8.2 and 8.3, word for word. Those pages are for AP Physics 2 students; this unit is for AP Physics C students, and they are not the same material at two reading levels. If you are in the algebra-based course you want Topic 10.1, Electric Charge and Electric Force and Topic 10.3, Electric Fields, and you can stop before the integral. If you are in Physics C you want those three titles plus three topics the other course never reaches.

The split also runs the other way. AP Physics 2 packs force, field, potential energy, potential and capacitors into one seven-topic unit. Physics C spreads the same ground over three units: Unit 8 for charge and field, Unit 9 for energy and potential, Unit 10 for conductors and capacitors. If you are coming from Physics 2, the reorganisation is as much of an adjustment as the calculus.

Flux and Gaussian surfaces: the sentences that do the work

Four of Topic 8.6's six essential-knowledge statements are procedure rather than physics. Read them as a method and the unit's hardest problems get shorter.

8.6.A.2: a Gaussian surface is a three-dimensional, closed surface. So a disc is not one, a hemisphere is not one, and a cube face is not one. If a question asks for the flux through an open surface, Gauss's law does not apply to it directly; you either integrate, or you close the surface yourself and subtract what you added.

8.6.A.3: the total flux is independent of the size of the surface if the enclosed charge is unchanged. This is what lets you shrink or grow a sphere around a point charge at will, and it is why a flux answer usually contains no radius at all. It is also the statement most naturally tested as a comparison question, which fits skill 2.C being listed for Topic 8.5.

8.6.A.4: Gaussian surfaces are typically constructed such that the field is either perpendicular or parallel to different regions of the surface, resulting in a simplified surface integral. That is the whole trick in one sentence. Perpendicular regions contribute $E \, dA$ with the cosine equal to one; parallel regions contribute nothing; nothing in between is ever needed. Which is why the boundary statement restricts you to spherical, cylindrical and planar symmetry: those are the three shapes where a closed surface can be built out of only those two kinds of region.

8.6.A.5: a charge density given as a function of position is integrated over length, area or volume. The CED prints the volume case as $Q_{\text{total}} = \int \rho(\vec{r})\, dV$ with a vector argument; the equation sheet prints the same line with a plain $r$. Either way, the enclosed charge in Gauss's law is itself sometimes an integral, and for a spherically symmetric $\rho(r)$ the volume element you want is a shell, $dV = 4\pi r^2 dr$. The Gauss's law guide walks the routine end to end.

One sentence from Topic 8.3 belongs in this list even though it sits elsewhere. Statement 8.3.B.1 says the field inside a conductor in electrostatic equilibrium is zero. Combine that with Gauss's law and you get the result that carries most of Unit 10: any Gaussian surface drawn wholly inside conducting material encloses zero net charge, because the flux through it is zero.

Traps that span more than one topic

A charge distribution is not a point charge at its center, except when it is. Statement 8.3.B.1.ii grants that licence to one shape only: an isolated sphere with a spherically symmetric charge distribution, viewed from outside. A ring, a rod, a disc or a cube gets no such licence, and treating a semicircular arc of total charge $Q$ as a point charge at distance $R$ overstates the field at the center by a factor of $\pi/2$.

Zero field does not mean zero enclosed charge, and zero enclosed charge does not mean zero field. Gauss's law equates enclosed charge to the flux through the whole closed surface, not to the field at any one point on it. A surface enclosing a dipole has zero net enclosed charge and zero net flux, while the field is nonzero everywhere on it.

Charge outside the surface changes the field on the surface and contributes nothing to the flux. Both halves of that sentence are true at once, and mixing them up is the most common Gauss error. Statement 8.6.A.1 says the flux is set by $q_{\text{enc}}$. It does not say the field is.

The area vector points outward from a closed surface, always. Statement 8.5.A.2.i fixes it, and 8.5.A.2.ii makes the sign of the flux the sign of the dot product. Field entering the surface gives negative flux; field leaving gives positive. On an open surface the CED gives you no convention, so declare the direction you chose before you use it.

Symmetry kills components; it does not kill the integral. On the arc, the ring and the finite line, the boundary statement's location clause is exactly the position where one component cancels. Off that position nothing cancels and the problem leaves the course, so if you find yourself integrating two components at once, check whether you are still at the location the boundary statement named.

Permittivity is a property of the material, not a universal constant. Statement 8.1.C.3 gives free space one constant value $\varepsilon_0$, and 8.1.C.4 says the permittivity of matter differs from it. Coulomb's law with $\varepsilon_0$ in it is the free-space law, and that distinction is what makes Unit 10's dielectric constant mean anything.

Charge is a scalar; force and field are vectors. Statement 8.1.A.1.i says so about charge. That is why potentials add arithmetically and fields do not, which is the most productive single idea in Unit 9.

How Unit 8 is assessed

The exam is 3 hours long. Section I is 42 multiple-choice questions in 85 minutes for 50% of the score. Section II is 4 free-response questions in 95 minutes for the other 50%, always one of each type in a fixed order: Mathematical Routines (10 points), Translation Between Representations (12 points), Experimental Design and Analysis (10 points), and Qualitative/Quantitative Translation (8 points). A four-function, scientific, or graphing calculator is allowed on both sections. The course also requires that 25 percent of instructional time be spent on hands-on laboratory work.

The unit's own Preparing for the AP Exam note points at the first free-response question. It says the Mathematical Routines question focuses on assessing students' ability to create and use mathematical models and representations, that students will be required to derive an expression for a physical quantity which may culminate in a numerical calculation, and that while Unit 8 offers content perfect for practicing the MR question, the MR question can pull content from any of the six units of the course. Read the second half of that as carefully as the first: nothing guarantees Unit 8 appears in a given year's free-response section. The reliable figure is the multiple-choice weighting, 15 to 25%. The unit's AP Classroom Progress Check runs about 18 multiple-choice questions and 4 free-response questions, one of each type.

On the science practices, the unit opener names skills 1.A, 1.C and 2.A as the ones Unit 8 develops. Across the exam, skill 2.A, deriving a symbolic expression by following a logical mathematical pathway, carries a 25 to 30% weighting on the multiple-choice section, the largest of any single skill there, and Practice 2 as a whole carries 40 to 45% of the free-response section. Practice 1 is not assessed on the multiple-choice section at all. Four of the six topics list 2.A among their suggested skills; the two that do not are 8.1 and 8.2.

The CED prints a sample exam with an alignment table, and four of its fifteen sample multiple-choice questions align to Unit 8 objectives: 8.1.A on the signs of the work done by an external force and by the electrostatic forces as a test charge moves between two points near two fixed charges, 8.3.A on where along a line the field of two spheres carrying $-3Q$ and $+4Q$ is greatest, 8.5.A on the total flux through a cube sitting in a field $\vec{E} = (bx)\hat{i}$, and 8.6.A on the net field at radius $R_S/2$ inside a uniformly charged sphere with a point charge at its center. That last one needs enclosed charge scaling as $r^3$, superposition, and no integration at all.

The sample free-response Question 1, the Mathematical Routines question, is worth 10 points and aligns to learning objectives 8.4.A, 9.2.A, 9.2.B and 9.3.A, using skills 2.A, 3.B and 3.C. It puts a thin nonconducting ring of radius $R$ and charge $+Q$ in a plane with a small sphere of charge $-Q$ on the axis, then asks where the net potential can be zero, for a derivation of the sphere's speed from conservation of energy, and for a derivation of the field due to the ring at a point on the axis. Its scoring guidelines award one point for a multistep derivation that starts with an equation for the electric field of a charge distribution, and a separate point for indicating that only the axial component need be considered, for instance by including a $\cos\theta$ term in the integral. They then print an alternate solution that gets the field by differentiating the potential, and award it the same three points. Method earns points on its own, and more than one method is accepted.

The five optional sample instructional activities sit at the front of the unit, two on Topic 8.1 and one each on 8.2, 8.3 and 8.6. Topics 8.4 and 8.5 get none, which is a fair signal that the integral and the flux are taught with a pencil rather than an experiment.