AP Physics C: E&M · Topic 8.1

Topic 8.1: Electric Charge and Electric Force

Unit 8: Electric Charges, Fields, and Gauss's Law15-25% of the multiple-choice section

Charge is a scalar property of matter, positive or negative, built from whole numbers of the elementary charge. Coulomb's law gives the force between two charged objects: proportional to each charge, inversely proportional to the square of their separation, and directed along the line joining them.

AP Physics: Unit 8 (topics 8.1 Electric Charge and Electric Force). AP Physics C: Electricity and Magnetism Unit 8, Topic 8.1. Three learning objectives: 8.1.A, describe the electric force that results from the interactions between charged objects or systems, with 8.1.A.1 (charge is a fundamental property of all matter) and its four sub-statements, 8.1.A.2 (Coulomb's law, with the printed two-form equation), 8.1.A.3 (direction depends on the signs and is along the line of separation) with two sub-statements, and 8.1.A.4 (contact forces as convenient nonfundamental descriptions); 8.1.B, describe the electric and gravitational forces that result from interactions between charged objects with mass, with 8.1.B.1 to 8.1.B.3; and 8.1.C, describe the electric permittivity of a material or medium, with 8.1.C.1 to 8.1.C.4 including 8.1.C.4.i and 8.1.C.4.ii, which is where the course defines conductors and insulators. The boundary statement reads: 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. Suggested skills are 1.A, 2.B, 2.D and 3.B. This topic is nearly identical to AP Physics 2 Topic 10.1; the differences are that 8.1.A.1.i adds that charge is a scalar quantity, 8.1.A.3 says along rather than parallel to the line of separation, the boundary statement adds the third sentence about charge distributions, and the suggested skill list carries 2.B where Physics 2 carries 2.A. The CED's sample multiple-choice Question 3 aligns to 8.1.A and essential knowledge 8.1.A.3 at skill 3.B, with answer B. Unit 8 is weighted 15 to 25% of the multiple-choice section over about 12 to 24 class periods.

What Topic 8.1 requires

Topic 8.1 carries three learning objectives, which is more than any other topic in Unit 8.

8.1.A, describe the electric force that results from the interactions between charged objects or systems.

8.1.A.1 states that charge is a fundamental property of all matter. Its four sub-statements say that charge is a scalar quantity and is described as positive or negative (8.1.A.1.i); that the magnitude of the charge of a single electron or proton, the elementary charge $e$ , can be considered to be the smallest indivisible amount of charge (8.1.A.1.ii); that the charge of an electron is $-e$ and the charge of a proton is $+e$ , and a neutron has no electric charge (8.1.A.1.iii); and that a point charge is a model in which the physical size of a charged object or system is negligible in the context of the situation being analyzed (8.1.A.1.iv).
8.1.A.2 states Coulomb's law as a proportionality: the electrostatic force between two charged objects is directly proportional to the magnitude of each of the charges and inversely proportional to the square of the distance between the objects. The relevant equation printed with it is the two-form line you will see again on the sheet.
8.1.A.3 states that the direction of the electrostatic force depends on the signs of the charges of the interacting objects and is along the line of separation between the objects, with same signs repelling (8.1.A.3.i) and opposite signs attracting (8.1.A.3.ii).
8.1.A.4 states that electric forces are responsible for some of the macroscopic properties of objects in everyday experiences, but that the large number of particle interactions makes it more convenient to treat everyday forces in terms of nonfundamental forces called contact forces, such as normal force, friction, and tension.

8.1.B, describe the electric and gravitational forces that result from interactions between charged objects with mass. Three statements: electrostatic forces can be attractive or repulsive while gravitational forces are always attractive (8.1.B.1); for any two objects that have mass and electric charge, the magnitude of the gravitational force is usually much smaller than the magnitude of the electrostatic force (8.1.B.2); and gravitational forces dominate at larger scales even though they are weaker than electrostatic forces, because systems at large scales tend to be electrically neutral (8.1.B.3).

8.1.C, describe the electric permittivity of a material or medium. Four statements: permittivity measures the degree to which a material or medium is polarized in the presence of an electric field (8.1.C.1); electric polarization can be modeled as the induced rearrangement of electrons by an external electric field, resulting in a separation of positive and negative charges within a material or medium (8.1.C.2); free space has a constant value of electric permittivity, $\varepsilon_0$ , that appears in physical relationships (8.1.C.3); and the permittivity of matter has a value different from that of free space that arises from the matter's composition and arrangement (8.1.C.4), determined by the ease with which electrons can change configurations within the material (8.1.C.4.i), with conductors made from electrically conducting materials in which charge carriers move easily and insulators made from electrically nonconducting materials in which charge carriers cannot move easily (8.1.C.4.ii).

The boundary statement, in full. "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."

That last sentence is the one to read twice. It fences the force calculation at four bodies and then, in the same breath, tells you the field calculation is not fenced at all.

The CED lists four suggested skills for Topic 8.1: 1.A, create diagrams, tables, charts, or schematics to represent physical situations; 2.B, calculate or estimate an unknown quantity with units from known quantities by selecting and following a logical computational pathway; 2.D, predict new values or factors of change of physical quantities using functional dependence between variables; and 3.B, apply an appropriate law, definition, theoretical relationship, or model to make a claim. Unit 8 is weighted 15 to 25 percent of the multiple-choice section over about 12 to 24 class periods.

How this differs from AP Physics 2 Topic 10.1, precisely

There is no point pretending otherwise: Topic 8.1 and [AP Physics 2 Topic 10.1](/ap-physics-2/unit-10-electric-force-field-and-potential/10-1-electric-charge-and-electric-force) are the same topic. Same title, same three learning objectives with the same wording, same eleven numbered essential-knowledge statements in the same order, same printed Coulomb's law. If you have read one carefully you have read most of the other.

Four things do differ, and all four are checkable against the two documents.

8.1.A.1.i adds two words. The calculus-based course says charge "is a scalar quantity and is described as positive or negative". The algebra-based course says only that charge "is described as positive or negative". The C course names the scalar explicitly, which matters once you are adding fields as vectors in Topic 8.3 and integrating over charge in Topic 8.4: $dq$ carries a sign but no direction.
8.1.A.3 changes one phrase. Physics C says the force "is along the line of separation between the objects". Physics 2 says it "is parallel to the line of separation". Same physics, and worth knowing only so that a quotation from one document does not look wrong beside the other.
The boundary statement gains a sentence. Both courses cap force calculations at four or fewer interacting charged objects and both allow more charges in situations of high symmetry. Only the C boundary statement adds the pointer to Topics 8.4 and 8.6.
One suggested skill is swapped. Physics 2 lists 1.A, 2.A, 2.D and 3.B for Topic 10.1. Physics C lists 1.A, 2.B, 2.D and 3.B. The calculus course asks for the numerical computational pathway here and saves symbolic derivation (2.A) for Topics 8.3 through 8.6, where it is listed for all four.

So the honest summary is: nothing about Coulomb's law itself changes when you take the calculus-based course. The change arrives one topic later. Do not spend extra time on 8.1 hunting for a calculus treatment that the framework does not ask for. Spend it on Topic 8.4 and Topic 8.6, which have no algebra-based counterpart at all.

Which page is for you. The Physics 2 page is written for students in the algebra-based course, and it is the better read if you want the concept without the framework comparison. This page is written for students in AP Physics C: Electricity and Magnetism, and it spends its length on where the C framework diverges.

Coulomb's law, and the two forms the sheet prints

The AP Physics C: E&M equation sheet opens with Coulomb's law, written on a single line in two equivalent forms:

\left| \vec{F}_E \right| = \frac{1}{4\pi\varepsilon_0} \frac{|q_1 q_2|}{r^2} = k \frac{|q_1 q_2|}{r^2}

Read what is actually printed. The left side is a magnitude, and the charges appear inside absolute value bars. The equation as printed cannot give you a direction and cannot give you a negative force. Direction comes from 8.1.A.3, which you supply by reasoning, not by algebra.

Two constants come from the Table of Information at the front of the booklet:

Constant	Printed value
Coulomb constant $k = \dfrac{1}{4\pi\varepsilon_0}$	$9.0 \times 10^9 \ (\mathrm{N \cdot m^2})/\mathrm{C^2}$
Vacuum permittivity $\varepsilon_0$	$8.85 \times 10^{-12} \ \mathrm{C^2}/(\mathrm{N \cdot m^2})$
Elementary charge $e$	$1.60 \times 10^{-19} \ \mathrm{C}$

Those first two are not exactly consistent with each other. Compute $1/(4\pi\varepsilon_0)$ from the printed $\varepsilon_0$ and you get $8.99 \times 10^9$ , not $9.0 \times 10^9$ , a difference of about 0.1 percent. At the two or three significant figures an AP answer carries it never matters, and the sheet's own scoring notes accept answers in terms of either $\varepsilon_0$ or $k$ . Pick whichever makes your algebra shorter and say which you used.

The $\varepsilon_0$ form is not decoration. Every later equation in Unit 8 is written with $\varepsilon_0$ rather than $k$ : the field of a charge distribution, the flux integral and Gauss's law all carry $1/(4\pi\varepsilon_0)$ or $\varepsilon_0$ on their own. Getting comfortable moving between the two forms in Topic 8.1 saves algebra in Topic 8.6.

Superposition is assumed, not printed. Coulomb's law is a two-body statement. For three or four charges you compute each pairwise force separately, then add them as vectors. Nothing on the sheet tells you that, and nothing in 8.1 states it as an essential-knowledge item either; the vector-sum statement lives in Topic 8.3 as 8.3.A.3.i, for fields.

Direction, and the four-charge ceiling

Statement 8.1.A.3 does the work the equation cannot. The force on one charge from another lies along the line joining them, pointing away if the signs match and toward if they do not. That single sentence, plus a diagram, settles every direction question in this topic.

The working habit that survives contact with a hard problem:

Draw the charges to scale and mark the line between each pair.
Draw one force arrow per pair, on the charge you care about, using the sign rule. Skill 1.A is listed first for this topic for exactly this reason.
Resolve into components against axes you declare on the diagram.
Add the components, then recombine into a magnitude and an angle.
Sanity check the direction against the picture before you write the answer.

Steps 1 and 5 are where the marks actually go missing. A sign dropped in step 4 shows up immediately in step 5 if you drew the arrows first.

The ceiling. The boundary statement caps you at four or fewer interacting charged objects or systems, and then reopens the door: analysis with more charges is allowed in situations of high symmetry. That exception is not a technicality. A ring of eight equal charges, or a square with a charge at each corner and one at the centre, is fair game precisely because symmetry cancels most of the vector sum before you compute anything. When you meet more than four charges, the intended first move is to look for the cancellation, not to start adding.

What the ceiling does not cap. It caps the force calculation. The same boundary statement then says students are expected to calculate the electric fields of charge distributions as described in Topics 8.4 and 8.6, and a continuous distribution is infinitely many charges. Physics 2's Topic 10.1 boundary statement stops after the symmetry exception, because that course has nowhere to send you next.

Electric force against gravity (8.1.B)

Objective 8.1.B is three sentences about a comparison, and the third one is the interesting one.

The first two are the familiar facts. Gravity only attracts; the electrostatic force does both. For two objects that have both mass and charge, the gravitational force is usually much smaller. "Usually" is the CED's own hedge, and it is well placed: the comparison depends on the charge-to-mass ratio of the specific objects, so it is a statement about ordinary matter rather than a law.

The third, 8.1.B.3, is the one that answers the question students actually ask. Gravity dominates at large scales not because it is strong, but because large systems are close to electrically neutral. Both forces fall off as $1/r^2$ , so distance does not decide it. What decides it is that a star has essentially zero net charge and a very large mass, so the electric term cancels itself out while the gravitational term keeps adding.

Both inverse-square laws sit on the same equation sheet, which makes the parallel easy to see. The C: E&M sheet reprints the whole Mechanics table, so Newton's law of universal gravitation is available to you on an E&M exam:

\left| \vec{F}_g \right| = G \frac{m_1 m_2}{r^2} \qquad \left| \vec{F}_E \right| = k \frac{|q_1 q_2|}{r^2}

with $G = 6.67 \times 10^{-11} \ \mathrm{N \cdot m^2 / kg^2}$ from the Table of Information. Note the direction of the reprint: mechanics equations are fair game on the E&M exam, and the reverse is not true.

A structural warning about ratios. The ratio $F_E/F_g$ for a fixed pair of particles is independent of $r$ , because the $r^2$ cancels. That is why the ratio is quotable as a single number for a given pair and why it is a favourite functional-dependence item (skill 2.D). If a question changes the separation and asks what happens to the ratio, the answer is nothing.

Permittivity, the objective students skip (8.1.C)

Objective 8.1.C has four essential-knowledge statements and no equation, which is exactly why it gets skipped and exactly why it is easy to test.

Read 8.1.C.1 as a definition rather than a constant: electric permittivity measures how much a material polarizes in the presence of an electric field. It is a property of stuff, not a number in a formula. Statement 8.1.C.2 then says what polarization is at the microscopic level: an external field induces a rearrangement of electrons, separating positive and negative charge within the material.

Free space has one fixed value, $\varepsilon_0$ (8.1.C.3). Matter has a different value, set by its composition and arrangement (8.1.C.4), and specifically by how easily electrons can change configurations inside it (8.1.C.4.i).

Then 8.1.C.4.ii lands the definition that half of Unit 8 depends on: conductors are materials in which charge carriers move easily; insulators are materials in which charge carriers cannot move easily. That sentence, and not anything in Topic 8.2 or 8.3, is where the C framework defines the two words. If a question asks you to justify why the excess charge on a conductor sits on its surface, the chain starts here.

The payoff arrives in Unit 10. There, the dielectric constant $\kappa = \varepsilon / \varepsilon_0$ appears on the sheet, along with $C = \kappa \varepsilon_0 A / d$ , and the exam's own convention list states that capacitors are air-filled with $\kappa = 1.0$ unless told otherwise. Permittivity is the quantity those equations are built from, and 8.1.C is where the course introduces it.

For the plain-language version of the conductor and insulator distinction, the conductor vs insulator comparison covers it without the framework numbering.

How Topic 8.1 is tested

Of the fifteen sample multiple-choice questions the CED prints for this course, four align to Unit 8 learning objectives, and one of those four is 8.1: sample Question 3, aligned to 8.1.A and essential knowledge 8.1.A.3, at skill 3.B.

It is worth seeing what the item actually looks like, because it is not a plug-in. Two small spheres of charge $+Q$ each are fixed at two corners of an equilateral triangle. Point A is the third corner, Point B is the midpoint of the segment between the spheres. A positive test charge is moved from A to B at constant velocity by an external force, and the question asks for the signs of the work done by the external force and the work done by the electrostatic forces. The answer key gives choice B: the external work is positive, the electrostatic work is negative.

Nothing in that requires a number. It requires 8.1.A.3, applied twice, and the observation that constant velocity means the two works must sum to zero. Skill 3.B is "apply an appropriate law, definition, theoretical relationship, or model to make a claim", and that is what it is asking for.

The patterns worth rehearsing for this topic:

Compute a net force on one of three or four charges (skill 2.B). Components, then recombine.
Predict a factor of change when a charge or a separation changes (skill 2.D).
Argue a direction or a sign from 8.1.A.3 without computing (skill 3.B), as in the sample question above.
Find an equilibrium position or an unknown charge from a force balance, usually with gravity or a string tension in the picture.
Compare electric and gravitational forces for a named pair of particles (8.1.B).

On the multiple-choice section, skill 2.B carries a 20 to 25 percent weighting and skill 3.B carries 15 to 25 percent, and both are listed for this topic. On free response, science practice 2 as a whole carries 40 to 45 percent, the largest of the three.

For step-by-step drill on the two-charge case, the Coulomb's law guide works the routine and the Coulomb's law calculator checks arithmetic. This page covers the framing; those cover the procedure.

Where Topic 8.1 goes wrong

Putting signs into the equation. The printed form has absolute value bars on both charges and gives a magnitude. Feed it $-3.0 \ \mu\mathrm{C}$ and you get a negative number that means nothing. Compute the magnitude, then set the direction from 8.1.A.3.

Treating charge as a vector because force is. Statement 8.1.A.1.i is explicit that charge is a scalar. Charges add as signed numbers, never as arrows. Fields and forces are the vectors.

Adding force magnitudes. Two forces of 0.9 N do not make 1.8 N unless they happen to be parallel. Resolve first.

Assuming the bigger charge feels the bigger force. Newton's third law applies here in full, and 8.1.A.2 is symmetric in $q_1$ and $q_2$ . A $+60 \ \mathrm{nC}$ sphere and a $-20 \ \mathrm{nC}$ sphere pull on each other with forces of equal magnitude. Their accelerations differ, by the ratio of their masses, and that is the distinction the third worked example is built on.

Using constant-acceleration kinematics on a released charge. The force depends on separation, so the acceleration changes as the objects move. The kinematic equations on the sheet all assume constant acceleration and none of them apply. In the calculus-based course this is not a dead end, but the route through it is energy, which is Unit 9, not a kinematics equation.

Rounding the point-charge model away. Statement 8.1.A.1.iv makes the point-charge model conditional on the object's size being negligible in the context. Two spheres almost touching are not point charges. AP items signal the licence by telling you the spheres are "small", and $r$ is then a centre-to-centre distance, not a surface-to-surface gap.

Two charged pith balls on strings, and how few electrons it takes

Two small pith balls, each of mass $0.60 \ \mathrm{g}$ , hang from the same point on insulating threads of length $0.25 \ \mathrm{m}$ . Each ball is given the same charge $q$ , and the threads settle at $12^\circ$ from the vertical on either side. (a) Find $q$ . (b) Find how many electrons were removed from each ball. (c) Taking the CED's own modelling assumption of one electron for every $3.3 \times 10^{-27} \ \mathrm{kg}$ of mass, find what fraction of each ball's electrons that is. Use $g = 9.8 \ \mathrm{m/s^2}$ .

Set the geometry first. Each thread makes $12^\circ$ with the vertical, so the horizontal offset of each ball from the pivot is $L\sin\theta$ and the separation between the balls is $r = 2L\sin\theta = 2(0.25)\sin 12^\circ = 2(0.25)(0.20791) = 0.10396 \ \mathrm{m}$ .
Draw the free-body diagram for one ball: weight $mg$ down, thread tension $T$ along the thread, and the Coulomb repulsion $F_E$ horizontal, pushing the balls apart. Skill 1.A, and it is what makes the next line obvious.
Equilibrium in components. Vertically $T\cos\theta = mg$ ; horizontally $T\sin\theta = F_E$ . Divide the second by the first and the tension leaves: $F_E = mg\tan\theta$ .
Numbers. $mg = (6.0 \times 10^{-4})(9.8) = 5.88 \times 10^{-3} \ \mathrm{N}$ and $\tan 12^\circ = 0.21256$ , so $F_E = (5.88 \times 10^{-3})(0.21256) = 1.2498 \times 10^{-3} \ \mathrm{N}$ , which is $1.25 \ \mathrm{mN}$ .
(a) Now Coulomb's law with two equal charges: $F_E = kq^2/r^2$ , so $q = r\sqrt{F_E/k}$ . With $r^2 = (0.10396)^2 = 1.0807 \times 10^{-2} \ \mathrm{m^2}$ : $q^2 = F_E r^2 / k = (1.2498 \times 10^{-3})(1.0807 \times 10^{-2})/(9.0 \times 10^9) = 1.5007 \times 10^{-15} \ \mathrm{C^2}$ .
$q = \sqrt{1.5007 \times 10^{-15}} = 3.87 \times 10^{-8} \ \mathrm{C}$ , so about $39 \ \mathrm{nC}$ on each ball. The sign is not determined: the balls repel, so the charges match, but they could both be positive or both negative.
(b) Divide by the elementary charge from the Table of Information, $e = 1.60 \times 10^{-19} \ \mathrm{C}$ : $N = (3.87 \times 10^{-8})/(1.60 \times 10^{-19}) = 2.42 \times 10^{11}$ elementary charges, about 240 billion electrons removed from each ball if the balls are positive.
(c) Total electrons on a ball, on the stated assumption: $(6.0 \times 10^{-4})/(3.3 \times 10^{-27}) = 1.82 \times 10^{23}$ . The fraction removed is $(2.42 \times 10^{11})/(1.82 \times 10^{23}) = 1.3 \times 10^{-12}$ .
Read that last number before moving on. Stripping about one electron in a trillion produces a force about a fifth as big as the ball's own weight, since the force-to-weight ratio here is just $\tan 12^\circ = 0.21$ . That is 8.1.B.2 made concrete, and it is why 8.1.B.3 has to invoke neutrality rather than strength.

(a) $q = 3.9 \times 10^{-8} \ \mathrm{C}$ , about 39 nC, the same sign on both balls. (b) About $2.4 \times 10^{11}$ electrons transferred per ball. (c) About $1.3 \times 10^{-12}$ of the ball's electrons, roughly one in a trillion. Electric forces are enormous per unit charge, which is why almost everything you meet is almost exactly neutral.

Four charges at the corners of a square, which is the ceiling exactly

Four small charged spheres sit at the corners of a square of side $0.20 \ \mathrm{m}$ . Label the corners $A(0,0)$ , $B(0.20, 0)$ , $C(0.20, 0.20)$ and $D(0, 0.20)$ , in metres. The charges are $q_A = q_B = q_C = +2.0 \ \mu\mathrm{C}$ and $q_D = -2.0 \ \mu\mathrm{C}$ . Find the magnitude and direction of the net electrostatic force on the sphere at $A$ .

Declare the axes before anything else: $x$ to the right, $y$ upward, angles measured counterclockwise from the $+x$ axis. Four interacting charged objects is exactly the ceiling the Topic 8.1 boundary statement sets, so no symmetry shortcut is needed and none is allowed to be assumed.
Three pairwise forces act on $A$ . Compute each magnitude with $F = k|q_1 q_2|/r^2$ and set each direction with 8.1.A.3.
From $B$ , at distance $0.20 \ \mathrm{m}$ : $F_{AB} = (9.0 \times 10^9)(2.0 \times 10^{-6})(2.0 \times 10^{-6})/(0.20)^2 = (3.6 \times 10^{-2})/(4.0 \times 10^{-2}) = 0.90 \ \mathrm{N}$ . Like signs, so it is repulsive: $A$ is pushed directly away from $B$ , in the $-x$ direction.
From $D$ , also at $0.20 \ \mathrm{m}$ : same magnitude, $F_{AD} = 0.90 \ \mathrm{N}$ . Opposite signs, so it is attractive: $A$ is pulled toward $D$ , in the $+y$ direction.
From $C$ , across the diagonal at $r = 0.20\sqrt{2} = 0.28284 \ \mathrm{m}$ , so $r^2 = 8.0 \times 10^{-2} \ \mathrm{m^2}$ : $F_{AC} = (3.6 \times 10^{-2})/(8.0 \times 10^{-2}) = 0.45 \ \mathrm{N}$ . Like signs, repulsive, directed away from $C$ along the diagonal, that is along $(-1,-1)/\sqrt{2}$ .
Resolve the diagonal one: $F_{AC,x} = -0.45/\sqrt{2} = -0.31820 \ \mathrm{N}$ and $F_{AC,y} = -0.31820 \ \mathrm{N}$ .
Add components. $F_x = -0.90 - 0.31820 = -1.2182 \ \mathrm{N}$ . $F_y = +0.90 - 0.31820 = +0.5818 \ \mathrm{N}$ .
Magnitude: $F = \sqrt{(1.2182)^2 + (0.5818)^2} = \sqrt{1.4840 + 0.3385} = \sqrt{1.8225} = 1.35 \ \mathrm{N}$ .
Direction: $\theta = \arctan(0.5818 / -1.2182)$ in the second quadrant, which is $180^\circ - 25.5^\circ = 154.5^\circ$ from the $+x$ axis. In words, up and to the left, $25.5^\circ$ above the direction pointing from $B$ back toward $A$ .
Check it against the picture. Two of the three forces push or pull $A$ leftward and upward, and the diagonal repulsion partly cancels the upward pull. A net force up and to the left, with the leftward part the larger, is what the diagram says. Good.

$F = 1.35 \ \mathrm{N}$ at $154.5^\circ$ from the $+x$ axis, that is up and to the left. Components $F_x = -1.22 \ \mathrm{N}$ and $F_y = +0.58 \ \mathrm{N}$ . Note that the diagonal charge is twice as far away and therefore contributes a quarter of the force per unit charge pair, which is the inverse square doing its job.

Two spheres released from rest, and the graph that is not a straight line

Two small metal spheres float at rest in space, $0.10 \ \mathrm{m}$ apart. The heavy one has mass $0.020 \ \mathrm{kg}$ and charge $+60 \ \mathrm{nC}$ ; the light one has mass $0.0050 \ \mathrm{kg}$ and charge $-20 \ \mathrm{nC}$ . They are released. (a) Find the force on each and the initial acceleration of each. (b) Find the acceleration of the light sphere when the separation has halved. (c) The spheres are identical in size, so on contact they share charge and then separate. Find the force at $0.10 \ \mathrm{m}$ after contact, and compare it with the force at $0.10 \ \mathrm{m}$ before contact.

(a) One magnitude serves both spheres. $F = k|q_1 q_2|/r^2 = (9.0 \times 10^9)(60 \times 10^{-9})(20 \times 10^{-9})/(0.10)^2 = (1.08 \times 10^{-5})/(1.0 \times 10^{-2}) = 1.08 \times 10^{-3} \ \mathrm{N}$ .
Opposite signs, so the forces are attractive and point at each other along the line of separation, per 8.1.A.3.ii. They are equal in magnitude by Newton's third law, and 8.1.A.2 is symmetric in the two charges, so nothing about the $60$ against the $20$ changes that.
Accelerations from $\vec{F}_{net} = m\vec{a}$ . Heavy sphere: $a_1 = (1.08 \times 10^{-3})/(0.020) = 0.054 \ \mathrm{m/s^2}$ . Light sphere: $a_2 = (1.08 \times 10^{-3})/(0.0050) = 0.216 \ \mathrm{m/s^2}$ .
The ratio is exactly $4$ , the inverse ratio of the masses, and it stays $4$ for the whole approach because the force stays common. That is the part of the motion that is simple.
(b) At $r = 0.050 \ \mathrm{m}$ the separation has halved, so the force quadruples: $F = 4(1.08 \times 10^{-3}) = 4.32 \times 10^{-3} \ \mathrm{N}$ , and $a_2 = (4.32 \times 10^{-3})/(0.0050) = 0.864 \ \mathrm{m/s^2}$ .
So the acceleration is not constant, and none of the constant-acceleration kinematic equations on the sheet apply to this motion. An acceleration-against-time graph for the approach curves upward and steepens without bound as the gap closes; a velocity-against-time graph is concave up, not a straight line.
(c) On contact the two identical spheres share the total charge equally. Conservation of charge, which is Topic 8.2, gives each $(+60 - 20)/2 = +20 \ \mathrm{nC}$ .
New force at the same $0.10 \ \mathrm{m}$ : $F' = (9.0 \times 10^9)(20 \times 10^{-9})(20 \times 10^{-9})/(0.10)^2 = (3.6 \times 10^{-6})/(1.0 \times 10^{-2}) = 3.6 \times 10^{-4} \ \mathrm{N}$ .
Compare: $F/F' = (1.08 \times 10^{-3})/(3.6 \times 10^{-4}) = 3$ , which is just the ratio of the products of charges, $(60)(20)$ against $(20)(20)$ . And the sign has flipped: like charges now, so the spheres repel and fly apart.

(a) $1.08 \times 10^{-3} \ \mathrm{N}$ attractive on each sphere; $a_1 = 0.054 \ \mathrm{m/s^2}$ and $a_2 = 0.216 \ \mathrm{m/s^2}$ , in the ratio 4 set by the masses, not by the charges. (b) $a_2 = 0.864 \ \mathrm{m/s^2}$ , four times larger, because the force goes as $1/r^2$ . (c) After contact each sphere carries $+20 \ \mathrm{nC}$ and the force at the same separation is $3.6 \times 10^{-4} \ \mathrm{N}$ , one third as large and now repulsive.

Frequently asked questions

Is AP Physics C Topic 8.1 the same as AP Physics 2 Topic 10.1?

Almost exactly. The two topics share a title, three learning objectives with identical wording, eleven essential-knowledge statements in the same order, and the same printed form of Coulomb's law. Four differences exist. The Physics C course says charge is a scalar quantity where Physics 2 only says it is described as positive or negative. Physics C says the force is along the line of separation where Physics 2 says parallel to it. The Physics C boundary statement adds a sentence directing students to calculate the electric fields of charge distributions in Topics 8.4 and 8.6. And Physics C lists suggested skill 2.B where Physics 2 lists 2.A. No calculus is introduced in this topic in either course.

What value of k should I use on the AP Physics C E and M exam?

Use 9.0 times 10 to the ninth newton metres squared per coulomb squared, which is what the Table of Information prints for the Coulomb constant. The same table prints the vacuum permittivity as 8.85 times 10 to the minus twelfth. Those two are not perfectly consistent with each other, since one over four pi epsilon zero with the printed permittivity works out to 8.99 times 10 to the ninth, but the 0.1 percent gap never shows at exam precision. The scoring guidelines for the CED's own sample question state that an answer may be expressed in terms of either epsilon zero or k.

How many charges can an AP Physics C question make you combine?

Four or fewer, for a force calculation. The Topic 8.1 boundary statement says the course only expects students to make calculations of the electric force between four or fewer interacting charged objects or systems, and then adds that analysis of the resulting force from more charges is allowed in situations of high symmetry. That ceiling applies to discrete charges only. The same boundary statement notes that students are expected to calculate the electric fields of charge distributions, which are continuous, as described in Topics 8.4 and 8.6.

Is electric charge a vector or a scalar?

A scalar. Essential knowledge 8.1.A.1.i in AP Physics C states directly that charge is a scalar quantity and is described as positive or negative. The sign is not a direction: it is a label that tells you whether the force on another charge is attractive or repulsive. Charges combine by ordinary signed addition, so plus 6 nanocoulombs and minus 4 nanocoulombs make plus 2 nanocoulombs regardless of where the objects sit. The vectors in this topic are the forces and, in Topic 8.3, the fields.

What is electric permittivity in AP Physics C?

Electric permittivity measures the degree to which a material or medium becomes polarized in the presence of an electric field, which is essential knowledge 8.1.C.1. Polarization here means the induced rearrangement of electrons by an external field, separating positive and negative charge inside the material. Free space has a fixed value, epsilon zero, printed on the equation sheet as 8.85 times 10 to the minus twelfth. Matter has a different value set by its composition and arrangement, specifically by how easily its electrons can change configuration, and that same ease is what distinguishes conductors from insulators.

Why is the electric force so much stronger than gravity if gravity runs the universe?

Because large systems are electrically neutral, not because gravity wins at long range. Both forces fall off as one over distance squared, so distance does not favour either one. Essential knowledge 8.1.B.3 states that gravitational forces dominate at larger scales even though they are weaker than electrostatic forces, because systems at large scales tend to be electrically neutral. A star has an enormous mass and essentially no net charge, so the gravitational contributions all add while the electric ones cancel.

Do you plug negative charges into Coulomb's law?

No. The form printed on the AP Physics C equation sheet has absolute value bars around both charges and a magnitude sign on the force, so it returns a positive number and nothing else. Compute the magnitude from the sizes of the charges, then decide the direction from essential knowledge 8.1.A.3: the force lies along the line joining the two objects, pushing them apart if the signs match and pulling them together if the signs differ. Putting a minus sign into the formula produces a negative force magnitude, which is not a meaningful quantity.