Unit 10 · Topic 10.1 · about 30 minutes · 7 parts · 14 scored checks, 3 of them multiple choice

AP Physics 2: Algebra-Based 10.1: Electric Charge and Electric Force

Q: What is the elementary charge?

The size of the charge on a proton or an electron, e = 1.60 10-19 C. Every net charge is a whole number of elementary charges.

Built by Tanner Crow from College Board's course framework · Every answer key independently checked

Unit 10: 15 to 18% of the multiple-choice section

What you'll be able to do

Describe the electric force between charged objects: its size from Coulomb's law, and its direction along the line between them.
Compare the electric and gravitational forces between objects that have both charge and mass, and explain why gravity dominates at large scales.
Describe electric permittivity, and how a material polarizes near charge depending on how easily its electrons rearrange.

Key vocabulary

Term	What it means	The exam trap
Elementary charge	The size of the charge on one proton or one electron, $e = 1.60 \times 10^{- 19}$ C.	Charges are often given with a prefix, such as 2.0 $μ$ C for $2.0 \times 10^{- 6}$ C, so convert with the prefix before you calculate.
Net charge	The total of all the positive and negative charges in an object.	A neutral object still contains charges, in equal positive and negative amounts, so neutral does not mean the object has no charges in it.
Coulomb	The SI unit of electric charge, written C.	Later in this unit a capital C also stands for capacitance, so check whether a C in a problem is a unit or a quantity.
Vacuum	A region of space with no matter in it, also called free space.	Air is not a vacuum, but electric forces in air are almost the same as in a vacuum, so problems set in air are usually worked as if in a vacuum.

Three point charges drawn to scale. Charge P, +2.0 microcoulombs, sits at the right-angle corner. Charge A, +3.0 microcoulombs, is 0.30 m to the left of P, and charge B, -4.0 microcoulombs, is 0.40 m straight above P. Dashed lines join P to A and P to B, with a right-angle mark at P. No forces are drawn. — The charges worked in Watch it done.

Part 1 Think first Two balloons, one push

Commit to a guess first. Being wrong now makes the right idea stick better.

Predict first

Mia rubs two small balloons on her hair. Balloon A picks up three times as much charge as balloon B, and both charges have the same sign. She hangs the balloons from threads a few centimeters apart, and they push each other away. Which balloon pushes harder on the other?

In short. Electric charge comes in whole numbers of the elementary charge $e$ , and Coulomb's law gives the size of the force between two point charges, $| {\vec{F}}_{E} | = k \frac{| q_{1} q_{2} |}{r^{2}}$ , directed along the line joining them.

Like charges repel and opposite charges attract, and the two charges exert forces of equal size on each other, however different the charges are.
Between charged particles the electric force is far stronger than gravity, but large objects are nearly neutral, so gravity dominates at large scales.
A material's permittivity measures how much it polarizes near charge, which depends on how easily its electrons rearrange.

Part 2 Learn it Charge and the force between charges

2a. Charge comes in whole units

Electric charge is a fundamental property of matter, in the way mass is. It comes in two kinds, described as positive and negative. A proton carries a charge of $+ e$ and an electron a charge of $- e$ , where $e = 1.60 \times 10^{- 19}$ C is the elementary charge. A neutron has no charge. The SI unit of charge is the coulomb, C.

An object's net charge is the total of all the charges in it. A neutral object has as many electrons as protons. One with 5 more electrons than protons has a net charge of $- 5 e$ , and one with 5 fewer has $+ 5 e$ . Since every proton and electron carries exactly $e$ , the elementary charge can be treated as the smallest indivisible amount of charge, and any net charge is a whole number of elementary charges: $3 e$ or $- 40 e$ , never $1.5 e$ .

Real charged objects have a size. When that size is negligible next to the distances in a situation, we model each one as a point charge, with all its charge at a single point. A bead 1 mm across, 30 cm from another bead, is a point charge.

Quick check · Calculate

A speck of dust has a net charge of $- 8.0 \times 10^{- 17}$ C. Take $e = 1.60 \times 10^{- 19}$ C. How many more electrons than protons does the speck carry? Give a whole number.

Your answer electrons

2b. Coulomb's law

Two point charges exert electric forces on each other. Coulomb's law gives the size of each force:

| {\vec{F}}_{E} | = \frac{1}{4 π ε_{0}} \frac{| q_{1} q_{2} |}{r^{2}} = k \frac{| q_{1} q_{2} |}{r^{2}}

Here $r$ is the distance between the charges and $k = 9.0 \times 10^{9} N \cdot m^{2} / C^{2}$ . This lesson uses that value of $k$ , and every charge in it is in a vacuum.

The force is proportional to each charge, so doubling one charge doubles it. It is inversely proportional to the square of the distance, so doubling the distance makes it a quarter as large, not half.

There is one force size for the pair, and the formula shows why: swap $q_{1}$ and $q_{2}$ and nothing changes. Each charge pushes or pulls on the other, and by Newton's third law those two forces are equal in size and opposite in direction, however different the charges are.

Each force points along the line joining the two charges. Charges of the same sign repel, so each force points away from the other charge. Charges of opposite sign attract, so each points toward the other.

Four rows, one pair of point charges in each. Pair 1: charges +q and +q, a distance r apart. Pair 2: +2q and +q, a distance r apart. Pair 3: +2q and -2q, a distance 2r apart. Pair 4: +q and -q, a distance 2r apart. The distances are drawn to scale. — The four pairs ranked in the check below.

Quick check · Rank it

Four pairs of point charges are each alone in a vacuum. In pair 1, charges $+ q$ and $+ q$ are a distance $r$ apart. In pair 2, $+ 2 q$ and $+ q$ are $r$ apart. In pair 3, $+ 2 q$ and $- 2 q$ are $2 r$ apart. In pair 4, $+ q$ and $- q$ are $2 r$ apart. Rank the pairs by the size of the force each charge exerts on the other.

Largest force first

Give each one a place, 1 for first. Give two the same place if they are equal.

Pair 1
Pair 2
Pair 3
Pair 4

Your ranking: none yet

2c. Electric force and gravity

Any two objects with mass also attract each other by gravity, $| {\vec{F}}_{g} | = G \frac{m_{1} m_{2}}{r^{2}}$ , with $G = 6.67 \times 10^{- 11} N \cdot m^{2} / {kg}^{2}$ . Coulomb's law has the same inverse square form, but the two forces differ in two ways.

Gravitational forces are always attractive. Electric forces can attract or repel.

And between charged particles, the electric force is usually far larger. Two protons $1.0 \times 10^{- 15}$ m apart, as in an atomic nucleus, repel with an electric force of about 230 N, while their gravitational attraction is about $1.9 \times 10^{- 34}$ N.

So why does gravity, not the electric force, hold the Moon in orbit around Earth? Large objects are very nearly neutral. Earth and the Moon each hold enormous amounts of positive and negative charge in almost equal measure, so their electric forces on each other nearly cancel. Mass has no negative kind, so gravity never cancels, and at large scales it dominates.

At everyday scales, electric forces are everywhere. The normal force of a floor, friction and the tension in a rope all come from electric forces among huge numbers of atoms. There are so many that we treat them instead as contact forces.

Quick check · Multiple choice

Between two protons, the electric force is far larger than the gravitational force. Yet the force that keeps Earth in orbit around the Sun is gravitational. Which explains this?

2d. Polarization and permittivity

Materials differ in how their electrons behave. In a conductor, such as a metal, some electrons are free to move through the whole object, so charge carriers move easily. In an insulator, such as glass, plastic or dry wood, charge carriers cannot move easily, and each electron stays with its own atom or molecule.

Bring a charged object near either kind and its electric force pushes on the material's electrons. In a conductor, free electrons shift across the whole object. In an insulator, the electrons of each molecule shift a little toward one side. Either way, the result is an induced rearrangement of electrons that leaves positive and negative charge separated within the material: polarization.

Electric permittivity measures how much a material or medium polarizes near charge. It depends on what the material is made of and how it is arranged: the more easily its electrons change configuration, the greater its permittivity. Free space has a constant permittivity, $ε_{0} = 8.85 \times 10^{- 12} C^{2} / (N \cdot m^{2})$ , the constant in $k = \frac{1}{4 π ε_{0}}$ . Matter has a different value.

Quick check · Match

Tap a term, then the statement that describes it.

Common mistakes

The mistake	Why it is wrong	Do this instead
Thinking the object with more charge exerts the larger force on the other.	The two forces are a third-law pair. Each has the size $k \frac{\| q_{1} q_{2} \|}{r^{2}}$ , the same for both charges.	Give both forces the same size and opposite directions, and use the charges only to find that size.
Halving the force when the distance doubles.	Coulomb's law has $r^{2}$ in the denominator, so the force goes as the inverse square of the distance.	Square the distance factor: twice as far gives a quarter of the force, three times as far a ninth.
Adding the sizes of two forces on one charge when they point in different directions.	Forces are vectors, and two forces at an angle partly point different ways.	Break each force into components, add the components, and only then find the size of the net force.

Part 3 Watch it done The net force from two charges

Predict first, then watch each step, then finish the problem yourself.

In the picture at the top of this lesson, point charges sit in a vacuum at the corners of a right triangle. Charge P, $+ 2.0 μ$ C, is at the right angle. Charge A, $+ 3.0 μ$ C, is 0.30 m to the left of P, and charge B, $- 4.0 μ$ C, is 0.40 m directly above P. Take $k = 9.0 \times 10^{9} N \cdot m^{2} / C^{2}$ . What is the size of the net electric force on P?

Predict first

Before you open the first step, predict: which way does the net force on P point?

Step 1 · Define the system and choose axes
The system is charge P. Take $+ x$ to the right and $+ y$ up. Two forces act on P, one exerted by A and one exerted by B, and each points along the line joining P to the charge exerting it.
Step 2 · The force exerted by A
A and P are both positive, so they repel: the force on P points away from A, to the right, along $+ x$ . Its size is
$F_{A} = k \frac{| q_{A} q_{P} |}{r^{2}} = (9.0 \times 10^{9}) \frac{(3.0 \times 10^{- 6}) (2.0 \times 10^{- 6})}{(0.30)^{2}} = 0.60 N$
Step 3 · The force exerted by B, and the plan
B is negative and P is positive, so they attract: the force on P points toward B, straight up, along $+ y$ . Its size uses the same law with B's charge and the 0.40 m distance. The two forces are at right angles, so the net force has components $F_{x} = F_{A}$ and $F_{y} = F_{B}$ , and its size is $\sqrt{F_{x}^{2} + F_{y}^{2}}$ .
Why this step?
Why not just add the sizes of the two forces?
The forces point in different directions, one along $+ x$ and one along $+ y$ . Adding their sizes would treat them as pointing the same way. Each component of the net force is the sum of the forces' components along that axis.
Your step · Calculate

Finish it. Point charges sit in a vacuum at the corners of a right triangle. Charge P, $+ 2.0 μ$ C, is at the right angle. Charge A, $+ 3.0 μ$ C, is 0.30 m to the left of P, and charge B, $- 4.0 μ$ C, is 0.40 m directly above P. Take $k = 9.0 \times 10^{9} N \cdot m^{2} / C^{2}$ . What is the size of the net electric force on P? Give your answer to the nearest 0.01 N.

Your answer N

Part 4 Try it Draw the forces, then weigh them against gravity

The ball in the picture below hangs at rest. Before you build its force diagram, say which way you expect each force to point, then check each one against what exerts it.

A small ball marked with a plus sign hangs from a thread tied to a support at the top. The thread hangs 37 degrees from a dashed vertical line, tilted to the right, away from a larger sphere marked with plus signs. The sphere sits on an insulating stand to the left of the ball, level with it. No forces are drawn. — The ball hangs at rest, its thread 37 degrees from vertical.

Try it · Draw the forces

A small ball with a positive charge hangs at rest from an insulating thread. A sphere with a positive charge, on an insulating stand, is level with the ball and to its left, and the thread hangs 37 degrees from vertical, tilted away from the sphere. Build the force diagram for the ball: pick each force on it, give its direction, and give its length compared with the gravitational force.

The dot is the ball.

Pick each force exerted on the ball, then choose the way it points and how long its arrow is compared with the others. The arrows are drawn for you.

Now compare the electric force with gravity for two objects you could hold in your hand. Predict first: is one of them larger than the other by a factor of about ten, about a thousand, or more than a million?

Try it · Decide and justify

Two small spheres, each of mass 2.0 g and charge $+ 5.0$ nC, are 0.10 m apart in a vacuum. Take $k = 9.0 \times 10^{9} N \cdot m^{2} / C^{2}$ and $G = 6.67 \times 10^{- 11} N \cdot m^{2} / {kg}^{2}$ . Is the gravitational force between the spheres smaller than the electric force between them?

Which reason supports your answer?

Part 5 Explain it Does distance change the winner?

On the exam this is handwritten, so answer on paper first. Nothing you type here leaves this page. Then check your answer against the model and each scoring point.

Free response is 50% of the exam score. This one is shaped like Question 1, Mathematical Routines.

Free-response practice · Question 1 style

Two identical small particles, each of mass $m$ and charge $+ q$ , are a distance $r$ apart in a vacuum, far from anything else.

(a) Derive an expression for the ratio of the electric force to the gravitational force between them, $\frac{F_{E}}{F_{g}}$ , in terms of $q$ , $m$ , $k$ and $G$ .

(b) Calculate the ratio for two electrons, each with $q = 1.60 \times 10^{- 19}$ C and $m = 9.11 \times 10^{- 31}$ kg. Use $k = 9.0 \times 10^{9} N \cdot m^{2} / C^{2}$ and $G = 6.67 \times 10^{- 11} N \cdot m^{2} / {kg}^{2}$ .

(c) A student claims that if the two electrons were moved far enough apart, gravity would become the larger force, because the electric force weakens faster with distance. Use your expression from (a) to justify whether the claim is correct.

Your answer stays on this page. It is never sent or saved.

Model answer

(a) Coulomb's law gives $F_{E} = \frac{k q^{2}}{r^{2}}$ and Newton's law of gravitation gives $F_{g} = \frac{G m^{2}}{r^{2}}$ , so dividing one by the other, $\frac{F_{E}}{F_{g}} = \frac{k q^{2}}{G m^{2}}$ . The $r^{2}$ in each cancels.

(b) $\frac{F_{E}}{F_{g}} = \frac{(9.0 \times 10^{9}) (1.60 \times 10^{- 19})^{2}}{(6.67 \times 10^{- 11}) (9.11 \times 10^{- 31})^{2}} = 4.2 \times 10^{42}$ . The electric force is more than $10^{42}$ times the gravitational force.

(c) Incorrect. Both forces are proportional to $\frac{1}{r^{2}}$ , so the ratio in (a) does not contain $r$ at all. Moving the electrons twice as far apart makes each force a quarter as large, and the electric force stays about $4 \times 10^{42}$ times the gravitational force at every distance. The electric force does not weaken faster than gravity; it weakens at exactly the same rate, so it is the larger force at any separation.

Score your own answer. This is a self-check, not a grade. Tick each point your answer makes.

Writes Coulomb's law and Newton's law of gravitation for the pair, each with $r^{2}$ in the denominator.
Derives $\frac{F_{E}}{F_{g}} = \frac{k q^{2}}{G m^{2}}$ , with $r$ cancelled.
Calculates a ratio of about $4 \times 10^{42}$ for two electrons.
States that the claim is incorrect and justifies it from the expression: the ratio does not depend on $r$ , because both forces fall off as $\frac{1}{r^{2}}$ .

Part 6 Check yourself Three charges on a line

Both case questions use the figure.

Three point charges sit on a straight line in a vacuum. Take $k = 9.0 \times 10^{9} N \cdot m^{2} / C^{2}$ .

Three point charges on a straight horizontal line. Charge X, +8.0 microcoulombs, is on the left. Charge Y, -1.0 microcoulombs, is 0.30 m to its right, and charge Z, +1.0 microcoulombs, is another 0.30 m further right, 0.60 m from X. — X is $+ 8.0 μ$ C, Y is $- 1.0 μ$ C and Z is $+ 1.0 μ$ C, with 0.30 m between neighbors.

Case question 1 · Calculate

Using the figure of three charges on a line: X, $+ 8.0 μ$ C, Y, $- 1.0 μ$ C, and Z, $+ 1.0 μ$ C, sit in that order, 0.30 m apart, so X and Z are 0.60 m apart. What is the size of the net electric force on Z? Give your answer to the nearest 0.01 N.

Your answer N

Case question 2 · Multiple choice

Using the figure of three charges on a line: X carries $+ 8.0 μ$ C and Y carries $- 1.0 μ$ C, 0.30 m apart. How does the force exerted by X on Y compare with the force exerted by Y on X?

The rest stand on their own and do not use the case.

Check yourself · Fill the statement

Tap a blank, then an answer.

Two charged beads, each 2 mm across, are 40 cm apart in a vacuum, so each can be modeled as a . Moving them to 80 cm apart makes the force between them as large. Swapping one bead for a bead with three times the charge, at the same distance, makes the force as large.

Check yourself · Sort it

Each card names a force between two objects. Tap a card, then whether that force attracts, repels, could do either, or is not there at all.

Attracts

Repels

Could do either

No force at all

Check yourself · Decide and justify

Blocks of two kinds of plastic, P and Q, sit near the same charged rod. In plastic P, the electrons of each molecule shift easily toward or away from the rod. In plastic Q, they barely shift at all. Does P have the greater electric permittivity?

Which reason supports your answer?

Part 7 Keep it Bring back earlier ideas

Gas pressure and entropy come from the unit before this one, and the pushes of gas atoms on their container are electric forces at the scale of atoms. Pulling those ideas back up is what makes them last.

Bring it back · Calculate

A gas in a cylinder pushes on a piston whose face has an area of $0.050 m^{2}$ . The perpendicular components of the forces the gas's atoms exert on the piston add to $6.0 \times 10^{3}$ N on average. What is the pressure of the gas on the piston? Give your answer in kilopascals, to the nearest 1 kPa.

Your answer kPa

Bring it back · Multiple choice

A sealed, insulated box is split into two halves by a thin metal wall. One half holds hot gas and the other cold gas, and nothing enters or leaves the box. What happens to the total entropy of the gas in the box?

In three lines

Charge comes in whole numbers of $e = 1.60 \times 10^{- 19}$ C: a proton carries $+ e$ , an electron $- e$ , and a neutron none.
Coulomb's law, $| {\vec{F}}_{E} | = k \frac{| q_{1} q_{2} |}{r^{2}}$ , gives one force size for the pair, along the line between the charges: like charges repel, opposite charges attract.
Electric forces dwarf gravity between particles, but large objects are nearly neutral, so gravity rules at large scales; permittivity measures how much a medium polarizes.

Exam strategy for this topic

Section I, Multiple Choice: 42 questions in 85 minutes.
Section II, Free Response: 4 questions in 95 minutes.
This lesson's Explain it is shaped like free-response question 1, Mathematical Routines.
A four-function, scientific, or graphing calculator is allowed on both sections of the exam.

When two or more charges act on one, find each force's size from Coulomb's law with the sizes of the charges, give it a direction from their signs, then add the forces by components. Add the sizes directly only when the forces point the same way.

Questions students ask

What is Coulomb's law?

It gives the size of the electric force between two point charges, $| {\vec{F}}_{E} | = k \frac{| q_{1} q_{2} |}{r^{2}}$ , with $k = 9.0 \times 10^{9} N \cdot m^{2} / C^{2}$ . The force points along the line between the charges.

Does the larger charge exert the larger force?

No. Each charge exerts a force of the same size on the other, a Newton's third law pair, however different the two charges are.

What is the elementary charge?

The size of the charge on a proton or an electron, $e = 1.60 \times 10^{- 19}$ C. Every net charge is a whole number of elementary charges.

Why does gravity matter more than electric force for planets?

Planets and stars are almost exactly neutral, so their electric forces on each other nearly cancel, while gravity always attracts and adds up.

What is electric permittivity?

A measure of how much a material or medium polarizes near charge. It depends on how easily the material's electrons rearrange, and free space has the constant value $ε_{0}$ .

Beyond the AP Exam

College Board sets this limit on what the AP Exam asks about this topic. Anything past it is not on the exam.

AP Physics 2 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.

Course alignment, for teachers

AP Physics 2: Algebra-Based topic 10.1, Unit 10: Electric Force, Field, and Potential.

Skill 1.A: Create diagrams, tables, charts, or schematics to represent physical situations.
Skill 2.A: Derive a symbolic expression from known quantities by selecting and following a logical mathematical pathway.
Skill 2.D: Predict new values or factors of change of physical quantities using functional dependence between variables.
Skill 3.B: Apply an appropriate law, definition, theoretical relationship, or model to make a claim.

This lesson maps all 19 essential knowledge statements of the topic to the part that teaches each one: 19 are checked by a scored question. The map is in the lesson file.