AP Physics C: E&M · Topic 8.2

Topic 8.2: Conservation of Electric Charge and the Process of Charging

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

Charge is never created or destroyed, only moved. A system's net charge stays constant unless charge crosses its boundary, and charging almost always means electrons moving. Friction, contact and grounding all transfer charge; induced separation rearranges it without changing the total.

AP Physics: Unit 8 (topics 8.2 Conservation of Electric Charge and the Process of Charging). AP Physics C: Electricity and Magnetism Unit 8, Topic 8.2. One learning objective, 8.2.A, describe the behavior of a system using conservation of charge. Eight essential-knowledge statements: 8.2.A.1 (the net charge or charge distribution of a system can change in response to the presence of, or changes in, the net charge or charge distribution of other systems), 8.2.A.1.i (net charge can change due to friction or contact), 8.2.A.1.ii (induced charge separation occurs when the electrostatic force between two systems alters the distribution of charges within them, polarizing one or both), 8.2.A.1.iii (induced charge separation can occur in neutral systems), 8.2.A.2 (any change to a system's net charge is due to a transfer of charge between the system and its surroundings), 8.2.A.2.i (charging typically involves the transfer of electrons to and from the system), 8.2.A.2.ii (the net charge of a system will be constant unless there is a transfer of charge to or from the system), and 8.2.A.3 (grounding involves electrically connecting a charged object to a much larger and approximately neutral system, for example Earth). Topic 8.2 prints no boundary statement and no relevant equations; Unit 8's three boundary statements sit under Topics 8.1, 8.4 and 8.6. Suggested skills are 1.A, 2.C, 3.B and 3.C; 3.C is listed for only two topics in the unit, 8.2 and 8.4. This topic is identical to AP Physics 2 Topic 10.2 apart from one word: 8.2.A.3 says a charged object where 10.2.A.3 says a charged system. None of the CED's fifteen sample multiple-choice questions or four sample free-response questions for this course aligns to 8.2.A. Unit 8 is weighted 15 to 25% of the multiple-choice section over about 12 to 24 class periods.

What Topic 8.2 requires

Topic 8.2 is the smallest topic in Unit 8: one learning objective, eight essential-knowledge statements counting sub-statements, no relevant equations, and no boundary statement.

8.2.A, describe the behavior of a system using conservation of charge.

8.2.A.1 states that the net charge or charge distribution of a system can change in response to the presence of, or changes in, the net charge or charge distribution of other systems.
8.2.A.1.i the net charge of a system can change due to friction or contact between systems.
8.2.A.1.ii induced charge separation occurs when the electrostatic force between two systems alters the distribution of charges within the systems, resulting in the polarization of one or both systems.
8.2.A.1.iii induced charge separation can occur in neutral systems.
8.2.A.2 states that any change to a system's net charge is due to a transfer of charge between the system and its surroundings.
8.2.A.2.i the charging of a system typically involves the transfer of electrons to and from the system.
8.2.A.2.ii the net charge of a system will be constant unless there is a transfer of charge to or from the system.
8.2.A.3 states that grounding involves electrically connecting a charged object to a much larger and approximately neutral system, giving Earth as the example.

Topic 8.2 prints no boundary statement. Neither do Topics 8.3 and 8.5. Unit 8 carries exactly three boundary statements in total, under Topics 8.1, 8.4 and 8.6, and none of them constrains anything in 8.2.

The CED lists four suggested skills here: 1.A, create diagrams, tables, charts, or schematics to represent physical situations; 2.C, compare physical quantities between two or more scenarios or at different times and locations in a single scenario; 3.B, apply an appropriate law, definition, theoretical relationship, or model to make a claim; and 3.C, justify or support a claim using evidence from experimental data, physical representations, or physical principles or laws.

Skill 3.C is listed for exactly two of the six topics in Unit 8, this one and Topic 8.4. Take that as a hint about the shape of the questions: 8.2 is assessed by asking you to justify a claim, not by asking you to compute something.

This topic is genuinely the same in both courses

It is worth stating plainly rather than manufacturing a difference. AP Physics C Topic 8.2 and [AP Physics 2 Topic 10.2](/ap-physics-2/unit-10-electric-force-field-and-potential/10-2-conservation-of-electric-charge-and-the-process-of-charging) are the same topic, statement for statement. Same title. Same single learning objective with identical wording. The same eight essential-knowledge statements, numbered in parallel and worded the same. The same four suggested skills, 1.A, 2.C, 3.B and 3.C. Neither prints a boundary statement. Neither prints an equation.

One word differs across the whole topic. Statement 8.2.A.3 in the calculus-based course says grounding involves electrically connecting a charged object to a much larger and approximately neutral system. The algebra-based 10.2.A.3 says a charged system. Nothing follows from that.

So there is no calculus treatment of Topic 8.2 to learn. No derivative, no integral, no equation. If you are working through the C course looking for what changed, the answer for this topic is nothing, and the time is better spent on Topics 8.4, 8.5 and 8.6, which have no algebra-based counterpart at all.

Which page is for you. The Physics 2 page is the fuller treatment of the physics, written for the algebra-based course, and it will serve a C student perfectly well. This page is written for C students and spends its length on where the material reappears later in the C course, which is where the two courses actually part company.

Where the C course does push it. Not here, but in Unit 10, which opens with two topics that exist only in the calculus-based course: Electrostatics with Conductors, and Redistribution of Charge between Conductors. Those take the qualitative processes named in 8.2, grounding, induced separation, contact, and make them quantitative using the tools of Topics 8.6 and Unit 9. Everything you learn about tracking charge across a system boundary here is the prerequisite for that.

Conservation of charge is an accounting rule, and the boundary is the point

Statement 8.2.A.2 is the whole topic in one sentence: any change to a system's net charge is due to a transfer of charge between the system and its surroundings. Read it as bookkeeping. Charge is not made or destroyed anywhere in this course; it moves.

That makes every conservation question a two-step exercise:

Draw the system boundary. Decide what is inside and what is outside, and say so.
Ask whether anything crossed it. If nothing did, the net charge inside is the same before and after, by 8.2.A.2.ii.

Almost every mistake in this topic is a boundary that was never drawn. Two spheres touch and separate: if the system is one sphere, its charge changed and the surroundings supplied the difference; if the system is both spheres, the total is unchanged and all that happened was internal redistribution. Both descriptions are correct. Only one of them answers the question that was asked.

A worked table for the standard scenarios, with the system named explicitly:

Scenario	System	Did charge cross the boundary?	Net charge
Rod rubbed with cloth	the rod alone	yes, electrons moved to the cloth	changes
Rod rubbed with cloth	rod plus cloth	no	unchanged, still zero
Two spheres touch, then separate	one sphere	yes	changes
Two spheres touch, then separate	both spheres	no	unchanged
Charged rod held near a neutral sphere	the sphere	no	unchanged, still zero
Sphere briefly grounded with rod nearby	the sphere	yes, to or from Earth	changes
Sphere briefly grounded with rod nearby	sphere plus Earth	no	unchanged

The fifth row is the one students get wrong, and 8.2.A.1.iii is the statement that fixes it: induced charge separation can occur in neutral systems. A polarized neutral sphere has a positive end and a negative end and a net charge of exactly zero.

Electrons move, and that fixes the sign every time

Statement 8.2.A.2.i says the charging of a system typically involves the transfer of electrons to and from the system. In every scenario in this course, protons stay put: they are bound in nuclei, and nothing in an AP electrostatics problem moves them.

That single fact resolves the sign of every charging problem without any thought:

An object that ends up negative gained electrons.
An object that ends up positive lost electrons.
The other object in the interaction did the opposite, by exactly the same number.

And because charge comes in whole numbers of the elementary charge (statement 8.1.A.1.ii in the previous topic), the number of electrons that moved is always the net charge divided by $e = 1.60 \times 10^{-19} \ \mathrm{C}$ from the Table of Information.

The word "typically" is doing real work in 8.2.A.2.i. It is there because charge carriers are not always electrons in general physics. In an electrolyte or a plasma, ions carry the charge. Nothing in Unit 8 asks you to handle those, but the hedge is why the statement does not say "always".

Mass is transferred along with the charge, and it is negligible in a way worth seeing once. Move enough electrons to give a rod $32 \ \mathrm{nC}$ and you have moved $2.0 \times 10^{11}$ electrons, which is $1.8 \times 10^{-19} \ \mathrm{kg}$ . No balance can see that. Charge and mass are both conserved here; only one of them is measurable.

The processes the CED actually names

Statement 8.2.A.1.i names friction and contact as ways the net charge of a system can change. Statement 8.2.A.1.ii names induced charge separation. Statement 8.2.A.3 names grounding. Those are the four processes in the framework, and the important structural fact is that they are not four of a kind.

Friction (rubbing). Two neutral materials in contact, then separated. One material holds electrons more tightly, so it ends up negative and the other ends up positive, with equal magnitudes. The CED's own sample instructional activity for this topic is the sticky-tape experiment: press tape onto a desk, add a second and a third layer, peel them off together and then apart, and predict which pieces are charged and whether their charges match. It is a friction-and-contact experiment with no equipment.

Contact (conduction). A charged object touches a conductor and the excess charge redistributes over the combined conductor. For two identical conductors the split is even, which is what makes the arithmetic in the first worked example below work. For conductors of different size it is not even, and the rule that governs it is a Unit 10 topic, Redistribution of Charge between Conductors, not this one. Do not assume an even split unless the objects are stated to be identical.

Induced charge separation (polarization). No charge crosses the boundary. The electrostatic force from a nearby charged object rearranges the charge already present, so one end of the object goes positive and the other negative while the total stays put. This is why a charged rod attracts a neutral scrap of paper or a thin stream of water: the near end of the neutral object carries the opposite sign, so it is closer and its attraction beats the repulsion of the far end. The CED lists that stream-of-water question as one of the unit's essential questions.

Grounding. Statement 8.2.A.3 defines it as connecting a charged object to a much larger, approximately neutral system such as Earth. The consequence is that charge can flow in or out freely, so the object can end up with whatever net charge the rest of the arrangement demands. The reason Earth works as a ground is the "much larger": any charge it absorbs or supplies changes its own state negligibly.

Only the first, second and fourth of those change the net charge of the object. The third does not, and mixing that up is the single most common error in the topic.

Justification questions, which is how 8.2 is actually assessed

Skill 3.C, justify or support a claim using evidence from experimental data, physical representations, or physical principles or laws, is listed for this topic. On the free-response section, science practice 3 as a whole carries a 30 to 35 percent weighting, and skill 3.C specifically carries 5 to 10 percent on the multiple-choice section.

A justification for a charging question is not a description of what happened. It has to name the principle. The shape that earns the point:

Name the system you are talking about.
State what crossed the boundary, or that nothing did.
Cite the principle: the net charge of a system is constant unless charge is transferred to or from it.
Give the consequence in signs or numbers.

A worked example of that in prose. Claim: after a negatively charged rod is brought near an isolated neutral metal sphere, the sphere is attracted to the rod even though its net charge is zero. Justification: the system is the sphere alone; no charge crossed its boundary, so its net charge is still zero; the rod's field induces charge separation within the sphere, pushing electrons to the far side and leaving the near side positive; the attraction on the near, positive side is stronger than the repulsion on the far, negative side because the electrostatic force falls off with distance; so the net force is attractive.

Every clause there is doing work, and the third and fourth are the ones that get skipped.

None of the CED's fifteen sample multiple-choice questions or four sample free-response questions for this course aligns to learning objective 8.2.A. That is not evidence that the topic is untested on the real exam; the CED says its sample questions do not represent the full range and distribution of items. It does say something about proportion. Topic 8.2 is one of six topics in a unit weighted 15 to 25 percent, it has one objective where 8.1 has three, and it prints no equations. Learn it properly and quickly, then move on.

Where Topic 8.2 goes wrong

Saying protons moved. Statement 8.2.A.2.i says electrons. A positively charged rod lost electrons; it did not gain protons.

Treating polarization as charging. An induced separation leaves the net charge exactly where it was, by 8.2.A.1.iii. If a question asks for the net charge after a rod is brought near, and no contact or grounding happened, the answer is whatever it was before.

Splitting charge evenly between unlike conductors. The even split is a consequence of the two objects being identical. Nothing in 8.2 licenses it otherwise.

Forgetting the ground wire has to be removed first. In induction charging, the order matters: bring the rod near, ground the object, remove the ground, then remove the rod. Remove the rod before the ground and the induced charge simply flows back and the object ends up neutral. This is a sequencing question as much as a physics one, and skill 2.C, comparing quantities at different times in a single scenario, is listed for the topic.

Reporting a charge that is not a whole number of elementary charges. Charge is quantized (8.1.A.1.ii). If your answer works out to $2.5 \times 10^{-19} \ \mathrm{C}$ , something is wrong, because that is $1.6$ electrons.

Confusing conductor with charged. A conductor is a material in which charge carriers move easily, defined in statement 8.1.C.4.ii of the previous topic. A neutral conductor is still a conductor. The conductor vs insulator comparison separates the two ideas.

Three conducting spheres, touched in sequence

Three identical isolated metal spheres carry charges $q_A = +12 \ \mathrm{nC}$ , $q_B = 0$ and $q_C = -4.0 \ \mathrm{nC}$ . Sphere $A$ is touched to sphere $B$ and then separated. Sphere $A$ is then touched to sphere $C$ and separated. Find the final charge on each sphere, and check the total.

Record the total first, because it is the thing that cannot change: $q_{\text{total}} = +12 + 0 - 4.0 = +8.0 \ \mathrm{nC}$ . The three spheres together are an isolated system, nothing crosses that boundary, so by 8.2.A.2.ii this number holds at every stage.
First contact, $A$ with $B$ . The two spheres are identical, so while they are in contact they form one conductor and the excess charge shares evenly. Total available: $+12 + 0 = +12 \ \mathrm{nC}$ . Each takes half: $q_A = q_B = +6.0 \ \mathrm{nC}$ .
Check the intermediate state: $+6.0 + 6.0 - 4.0 = +8.0 \ \mathrm{nC}$ . Still correct.
Second contact, $A$ with $C$ . Total available for this pair: $+6.0 + (-4.0) = +2.0 \ \mathrm{nC}$ . Sharing evenly gives $q_A = q_C = +1.0 \ \mathrm{nC}$ . Sphere $B$ is not involved and keeps $+6.0 \ \mathrm{nC}$ .
Final: $q_A = +1.0 \ \mathrm{nC}$ , $q_B = +6.0 \ \mathrm{nC}$ , $q_C = +1.0 \ \mathrm{nC}$ . Total $= +8.0 \ \mathrm{nC}$ , unchanged. That check is the answer to the last part and it is also how you catch an arithmetic slip.
Two things to notice. First, sphere $C$ started negative and finished positive: electrons left it during the contact, which is a transfer across its own boundary, entirely allowed by 8.2.A.2. Second, the order of the touches matters. Touching $A$ to $C$ first would give $q_A = q_C = +4.0 \ \mathrm{nC}$ , then $A$ with $B$ gives $q_A = q_B = +2.0 \ \mathrm{nC}$ , for a final state of $+2.0$ , $+2.0$ , $+4.0 \ \mathrm{nC}$ . Same total, different distribution.
One caution the CED supports. The even split is licensed only because the spheres are stated to be identical. For conductors of different sizes the split is uneven, and the rule that sets it is Unit 10 material, not Topic 8.2.

$q_A = +1.0 \ \mathrm{nC}$ , $q_B = +6.0 \ \mathrm{nC}$ , $q_C = +1.0 \ \mathrm{nC}$ . The total is $+8.0 \ \mathrm{nC}$ before, during and after, because no charge ever left the three-sphere system. Reversing the order of the two contacts gives a different distribution with the same total.

Counting the electrons, and weighing them

A plastic rod and a wool cloth are both neutral. They are rubbed together, and afterward the rod carries a net charge of $+32 \ \mathrm{nC}$ . (a) How many electrons moved, and in which direction? (b) What is the charge on the cloth? (c) By how much did the rod's mass change? Take the electron mass as $9.11 \times 10^{-31} \ \mathrm{kg}$ from the Table of Information.

(a) The rod ended positive, so it lost electrons, by 8.2.A.2.i. The number is the net charge divided by the elementary charge: $N = (32 \times 10^{-9})/(1.60 \times 10^{-19})$ .
Do the division in two parts: $32/1.60 = 20$ , and $10^{-9}/10^{-19} = 10^{10}$ . So $N = 20 \times 10^{10} = 2.0 \times 10^{11}$ electrons, moved from the rod to the cloth.
(b) The system of rod plus cloth started neutral and nothing crossed its boundary, so its total is still zero. The cloth therefore carries $-32 \ \mathrm{nC}$ . Statement 8.2.A.2.ii is the whole argument, and this is the step a justification question wants written out rather than assumed.
(c) Mass lost by the rod: $\Delta m = N m_e = (2.0 \times 10^{11})(9.11 \times 10^{-31}) = 1.822 \times 10^{-19} \ \mathrm{kg}$ , so about $1.8 \times 10^{-19} \ \mathrm{kg}$ .
Put that in perspective. A laboratory balance reading to a milligram resolves $10^{-6} \ \mathrm{kg}$ , which is thirteen orders of magnitude larger. The mass transfer is unmeasurable, which is why "charge is conserved" is a useful statement while "mass is conserved during charging" is technically true and useless.
A sign check worth building into the habit. Nothing in the problem says which material holds electrons more tightly, and nothing needs to: the stated final charge on the rod fixes the direction of the transfer on its own.

(a) $2.0 \times 10^{11}$ electrons moved from the rod to the cloth. (b) The cloth carries $-32 \ \mathrm{nC}$ , because the rod-plus-cloth system started neutral and no charge crossed its boundary. (c) The rod's mass fell by $1.8 \times 10^{-19} \ \mathrm{kg}$ , which no balance can detect.

Two touching spheres charged by induction, without ever touching the rod

Two identical neutral metal spheres rest on insulating stands, touching each other. A rod carrying a large negative charge is brought close to the left sphere but never touches it. While the rod is held in place, the two spheres are separated. The rod is then taken away. Afterward the left sphere is found to carry $+7.2 \ \mathrm{nC}$ . (a) State the charge on the right sphere and justify it. (b) How many electrons moved between the spheres? (c) What would the final charges be if the rod had been removed before the spheres were separated?

Set up the stages and name the system at each one. System: the two spheres together. The rod is outside it and never touches, so no charge ever crosses the system boundary. By 8.2.A.2.ii the pair's net charge is zero at every stage.
Stage 1, rod brought near. The spheres are in contact, so they behave as one conductor. The rod's negative charge repels electrons in that conductor toward the far end, which is the right sphere. This is induced charge separation, 8.2.A.1.ii, occurring in a neutral system, 8.2.A.1.iii. The combined object is polarized: left end positive, right end negative, net charge still zero.
Stage 2, spheres separated while the rod is still there. The separation cuts the conductor at a point where the two halves already carry opposite charge, and now there is no conducting path between them. Each sphere is trapped with what it had.
Stage 3, rod removed. Charge on each sphere is now free to redistribute over that sphere's own surface, but it cannot get back across the gap. The magnitudes are locked in.
(a) The left sphere is $+7.2 \ \mathrm{nC}$ , so the right sphere is $-7.2 \ \mathrm{nC}$ . Justification: the two-sphere system started neutral, no charge was transferred to or from it at any stage, so its net charge is still zero, and the two charges must therefore be equal in magnitude and opposite in sign.
(b) Electrons moved from left to right, and the number is $N = (7.2 \times 10^{-9})/(1.60 \times 10^{-19})$ . Split it: $7.2/1.60 = 4.5$ and $10^{-9}/10^{-19} = 10^{10}$ , so $N = 4.5 \times 10^{10}$ electrons.
(c) Both spheres end neutral. With the rod gone before the cut, the conducting path between them is still intact, so the electrons that had been pushed to the right simply return and the polarization vanishes. Nothing is trapped and nothing was transferred, so both spheres go back to zero.
That last part is the whole point of the sequence. In induction charging, what fixes the final state is which connection is broken while the external charge is still present.

(a) $-7.2 \ \mathrm{nC}$ , because the two-sphere system started neutral and no charge ever crossed its boundary, so the totals must cancel. (b) $4.5 \times 10^{10}$ electrons, moved from the left sphere to the right. (c) Both spheres would end up neutral, since removing the rod first lets the induced separation relax through the contact that is still in place.

Frequently asked questions

What does conservation of electric charge mean in AP Physics C?

It means charge is never created or destroyed, only transferred. Essential knowledge 8.2.A.2 states that any change to a system's net charge is due to a transfer of charge between the system and its surroundings, and 8.2.A.2.ii states that the net charge of a system will be constant unless there is a transfer of charge to or from the system. In practice you apply it by choosing a system boundary and asking whether anything crossed it. If nothing did, the total charge inside is the same before and after, however much the charge moved around internally.

Is AP Physics C Topic 8.2 different from AP Physics 2 Topic 10.2?

No, apart from one word. The two topics share a title, a single learning objective with identical wording, eight essential-knowledge statements numbered in parallel, and the same four suggested skills of 1.A, 2.C, 3.B and 3.C. Neither prints a boundary statement and neither prints an equation. The only textual difference is that the Physics C statement 8.2.A.3 describes grounding as connecting a charged object to a much larger and approximately neutral system, where Physics 2 says a charged system. There is no calculus treatment of this topic in the C course.

Are there any equations for AP Physics C Topic 8.2?

None. Topic 8.2 is the only one of the six topics in Unit 8 whose essential knowledge prints no relevant equation, and nothing on the AP Physics C: Electricity and Magnetism equation sheet belongs specifically to it. The arithmetic you do in charging problems comes from elsewhere: the elementary charge, from essential knowledge 8.1.A.1.ii and the Table of Information, converts a net charge into a number of electrons, and even sharing between identical conductors follows from symmetry rather than from a printed formula.

What does grounding actually do?

Grounding connects a charged object to a much larger, approximately neutral system such as Earth, which is essential knowledge 8.2.A.3. Because that system is so much larger, it can absorb or supply as much charge as the object needs without measurably changing its own state. The practical effect is that the grounded object is free to reach whatever net charge the rest of the arrangement calls for. If a negatively charged rod is held nearby, electrons in the object are repelled and leave through the ground connection, so the object ends up positive once the ground is removed.

Can an object be charged without touching it?

Yes, by induction. Bring a charged rod near a conductor, connect the conductor briefly to ground, remove the ground while the rod is still in place, then remove the rod. The conductor ends up with a net charge opposite in sign to the rod, and the rod never touched it. The order matters: if the rod is removed before the ground connection, the induced charge simply flows back and the object ends up neutral. The same trick works with two conductors touching each other in place of the ground, which is the third worked example on this page.

Does induced charge separation change an object's net charge?

No. Induced charge separation rearranges the charge already inside an object without any charge crossing the object's boundary, so the net charge is exactly what it was. Essential knowledge 8.2.A.1.iii makes the point explicitly by noting that induced charge separation can occur in neutral systems: a polarized neutral object has a positive end and a negative end and a total of zero. That is why a charged rod attracts a neutral scrap of paper. The near end of the paper carries the opposite sign and is closer, so its attraction outweighs the repulsion of the far end.

Do protons ever move when an object is charged?

Not in this course. Essential knowledge 8.2.A.2.i states that the charging of a system typically involves the transfer of electrons to and from the system. Protons are bound in nuclei and nothing in an AP electrostatics problem moves them. That fixes the sign of every charging problem without further reasoning: an object that ends up negative gained electrons, and an object that ends up positive lost them. The word typically is in the statement because charge carriers in other settings, such as electrolytes, can be ions, but Unit 8 never asks you to handle those.