Electric Charge and Electric Force
Connect elementary charge, point-charge forces, and material polarization with stated model assumptions.
AP Physics 2 Topic 10.1 · College Board CED printed p. 43, 44, 45 / PDF p. 50, 51, 52 · See this topic in the course outline
A charged balloon can stick to a wall, and two charged beads can push one another without touching. Start with charge and the line joining the objects; then decide which direction each force points.
For another explanation, read Charge and Matter. Use the corrected charge and Coulomb-law explanations to compare signed charges and pair forces.
Before you start
- Signed quantities and scientific notation
- Vector components and Newton’s third law
- Inverse-square gravitational force
Charge, particles, and the point-charge model
Charge is a fundamental property of matter. Its sign is positive or negative; “neutral” means the positive and negative contributions sum to zero, not that the matter has no charged particles. A proton carries +e, an electron −e, and a neutron zero electric charge. For ordinary isolated objects, charge changes occur in whole multiples of the elementary charge: , where n is an integer and e = 1.60 × 10⁻¹⁹ C. Removing electrons leaves positive net charge; adding them makes negative net charge.
A point charge is a model of an object whose size is negligible compared with the distances in the problem. Two small beads several centimeters apart can often use this model. The same beads nearly touching may not: charge rearrangement and shape can matter. Draw a dot to represent the charge only after checking that scale.
Coulomb force: size first, direction second
For stationary point charges in vacuum, . The distance r is the center-to-center separation. Determine direction separately: like signs repel, unlike signs attract, and either force lies along the line of separation. The forces on the two objects are equal in magnitude and opposite in direction even if one has a larger charge or mass.
For multiple charges, draw each pair force acting on the chosen object and add those vectors, not their unsigned magnitudes. Component addition is . We calculate at most four interacting objects unless high symmetry makes a larger configuration manageable.
| Change | Force magnitude factor | Direction check |
|---|---|---|
| Double one charge | 2 | Unchanged if its sign stays the same |
| Double both charges | 4 | Unchanged if signs stay the same |
| Double separation | 1/4 | Still along the connecting line |
| Reverse one sign | 1 | Attraction and repulsion exchange |
Normal force, friction, and tension summarize immense numbers of electromagnetic interactions among atoms and molecules. The contact-force model remains useful because summing every microscopic Coulomb interaction in a table or rope is impractical.
Why electricity wins locally and gravity wins astronomically
Gravity between ordinary masses is attractive: . Electricity can attract or repel. For two charged particles, compare the magnitudes by taking a ratio: . The separation cancels, so a closer pair does not change this ratio within these models. The electric interaction is usually enormously stronger for individual charged particles.
Large objects normally contain nearly equal amounts of positive and negative charge. Electric contributions cancel extensively at large scales, while gravity from positive mass adds. That is why gravity organizes planets and stars even though electricity is stronger between many microscopic charged pairs. A charged dust grain can still be an exception; “usually” is not “always.”
Permittivity connects the microscopic rearrangement to the medium
An external electric field pushes negative charge one way and positive charge the other. Bound electron clouds can shift relative to nuclei, or existing molecular dipoles can partly align. This separation of charge within a material is polarization. The sample can remain neutral while its charge distribution changes.
Permittivity describes a material’s electric response to a field, including how readily it polarizes. Free space has ε₀ = 8.85 × 10⁻¹² F/m. For a simple uniform linear dielectric, , where κ is its dielectric constant. Composition and atomic or molecular arrangement determine how easily charge configurations respond; different media therefore have different permittivities. Under the simple uniform-medium point-charge model, replacing ε₀ by ε gives a smaller force for the same free charges when κ is larger. Interfaces and rearranging conductors can require a different model.
In a conductor, mobile carriers can travel through the material. In an insulator, carriers cannot travel easily across the sample, although bound charges can shift a little and produce polarization. Do not mistake polarization for a supply of new net charge.
Two component forces on a third bead
A +1.0 μC bead is at the origin. A +2.0 μC source is at (0.30 m, 0), and a −3.0 μC source at (0, 0.40 m). Find the net force on the origin bead. Ignore gravity.
Choose +x right and +y up. The right-hand positive source repels the origin bead left; the upper negative source attracts it up.
Why this step? The signs fix directions before any vector addition.
. Similarly, Fy = +0.169 N.
Why this step? Each source acts independently under the point-charge approximation.
. The angle is arctan(0.169/0.200) = 40.2° above the negative x direction.
Why this step? Perpendicular components combine by the Pythagorean relation; quadrant information comes from the sketch.
Answer and check
0.261 N, 40.2° above left. Its magnitude lies between the larger single force and the sum of magnitudes, as a perpendicular-vector result should.
A proton pair and a change of medium
Two protons each have mass 1.67 × 10⁻²⁷ kg and charge +e. Compare their electric and gravitational forces. Then use an ideal uniform dielectric with κ = 4 to compare electric forces for fixed free charges and separation.
Divide the inverse-square expressions: .
Why this step? The common r² factor cancels; the electric force is repulsive and gravity attractive.
The ratio is (8.99 × 10⁹)(1.60 × 10⁻¹⁹)² / [(6.67 × 10⁻¹¹)(1.67 × 10⁻²⁷)²] = 1.24 × 10³⁶.
Why this step? Use the given proton values rather than a claim about all macroscopic charged objects.
For the stipulated uniform dielectric, ε = 4ε₀, so the electric-force magnitude becomes one fourth of its vacuum value.
Why this step? Polarization of the medium reduces the net interaction under this specified model.
Answer and check
The electric force exceeds gravity by about 1.24 × 10³⁶ in vacuum; the dielectric result is FE,vacuum/4. Large neutral systems behave differently because their charge contributions cancel.
Neutral matter still has the charge property
Charge is a fundamental property of all matter. A piece of matter can have zero net charge; zero is a value of charge, not evidence that the charge property is absent. Ordinary neutral matter contains positive protons and negative electrons whose contributions cancel. A proton has +e, an electron −e, and a neutron zero electric charge, with e = 1.60 × 10⁻¹⁹ C. A neutron’s zero charge does not exclude it from matter.
For ordinary object charging, adding or removing electrons changes net charge by integer multiples of e. An electron deficit produces positive net charge; an excess produces negative net charge. A smaller claimed charge increment than e is inconsistent with this ordinary elementary-charge model. The large counts in macroscopic examples are integers even when written with rounded scientific notation.
Permittivity measures how a medium polarizes
Apply an electric field to a neutral insulating material. Bound electron clouds can shift relative to positive nuclei, or molecular dipoles can partly align, separating positive and negative contributions without transporting net charge through the whole sample. The field induced by this polarization opposes the applied field in the simple capacitor model.
Permittivity quantifies the medium’s polarization response. Composition and atomic or molecular arrangement affect how easily electrons can change configuration, so materials respond differently. Free-space permittivity is the constant ε₀ = 8.85 × 10⁻¹² F/m. For a uniform linear dielectric, ε = κε₀. With the same fixed free charges and separations in the stipulated uniform-medium point-charge model, a larger ε reduces the electrostatic force. Do not extend that replacement rule blindly to interfaces or rearranging conductors. An insulator can polarize while remaining unable to conduct charge freely across the sample.
A charge ledger for a neutral atom and an ion
An ordinary atom contains 8 protons, 8 electrons and 8 neutrons. It loses two electrons without a nuclear change. Use e = 1.60 × 10⁻¹⁹ C.
- Before loss, Q = 8(+e) + 8(−e) + 8(0) = 0. The atom has charged constituents and zero net charge.
- After loss, it has 8 protons, 6 electrons and 8 neutrons: Q = (8 − 6)e = +2e = +3.20 × 10⁻¹⁹ C.
- The two emitted electrons carry −2e; the ion-plus-electrons total remains zero. The matter in both states has the fundamental charge property.
Answer and check
Neutral atom: 0 C; ion: +3.20 × 10⁻¹⁹ C. Proton, electron and neutron charges are +e, −e and 0 respectively.
Compare two microscopic responses under controlled conditions
Two neutral uniform linear insulating media fill otherwise identical point-charge setups. Medium A has κ = 2 and medium B κ = 5. Free charges are +1.0 nC and −2.0 nC, 0.10 m apart. Use k_vac = 8.99 × 10⁹ N·m²/C². Free-space permittivity is the constant ε₀ = 8.85 × 10⁻¹² F/m.
- The vacuum force magnitude is k_vac|q₁q₂|/r² = 1.798 × 10⁻⁶ N, attractive because the signs differ.
- ε_A = 1.77 × 10⁻¹¹ F/m and ε_B = 4.425 × 10⁻¹¹ F/m. Under the specified uniform-medium model, F_A = 0.899 μN and F_B = 0.3596 μN.
- The larger κ indicates the larger polarization response in this linear model: an applied field induces bound-electron rearrangement and separates positive and negative charge contributions locally. Differences in composition and arrangement affect ease of reconfiguration. This does not imply that B supplies new charge or conducts through the sample.
- F_B/F_A = κ_A/κ_B = 2/5. No precise molecular structure can be uniquely inferred from these two coefficients.
Answer and check
Attractive forces: 0.899 μN in A and 0.360 μN in B. A stronger stipulated polarization response gives the smaller interaction for the same fixed free charges.
Try it from memory
Write your answer and reason before opening the worked answer. Your self-assessment is a study decision; it is not an automatically graded score.
Question 1
An object gains 5.0 × 10⁶ electrons. What is its charge change, and what changes if the distance to a fixed source doubles?
Worked answer and reason
Δq = −8.0 × 10⁻¹³ C. Its electric force from that source becomes one fourth as large.
Electron transfer has a negative charge sign; inverse-square scaling uses the square of distance.
Question 2
A neutral plastic strip is attracted to a charged rod. Does that prove the strip conducts? Explain the microscopic model.
Worked answer and reason
No. Bound electrons can shift within atoms or molecules, polarizing an insulator while its net charge stays zero. Mobile carriers would be needed for conduction through the strip.
Polarization requires local charge rearrangement; conduction requires charge transport over the sample.
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Catch the tempting mistake
- A larger charge feels the larger force in a two-object interaction.
- Both forces have magnitude k|q₁q₂|/r²; differing accelerations can come from different masses.
- Neutral material has no electric forces.
- Neutral matter contains charged constituents; microscopic electric interactions also underlie contact forces.
Put it to work
Give the physical reason as well as the calculation. The worked solutions are on a separate page.
- Problem 1 · charge ledger and force vectors
- Problem 2 · symbolic comparison and microscopic explanation
- Problem 16 · capacitance experiment and controlled comparison
Explain and apply the model
Try these problems before opening the solutions. Show the physical model, your calculation or sketch, and the reason your conclusion follows.
Does zero net charge remove the charge property?
A sample contains N initially neutral atoms, each with 6 protons, 6 electrons and 6 neutrons. It later loses exactly 3.0 × 10⁹ electrons, with no nuclear change. N is large enough to supply that count. Use e = 1.60 × 10⁻¹⁹ C.
(a) Evaluate the claim “charge is a fundamental property only of matter with nonzero net charge.” Explain the role of zero net charge for this sample and state whether neutral matter and neutrons are excluded from the fundamental charge description.
(b) State the signed charge of one electron, one proton and one neutron. Use all three values to calculate the initial charge of one atom.
(c) Calculate the sample’s final net charge and the signed charge carried away. Explain why the change is quantized and why a claim that one isolated atom loses a charge of +0.4e by losing electrons is inconsistent with this charging model.
Written working workspace
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Worked solution and rubric →Revisit the explanation →Revisit the explanation →
Polarization or conduction?
A neutral insulator has larger permittivity than another in their common linear-response range. Neither conducts net charge freely across its sample.
(a) Which statement best explains the difference?
Worked solution and rubric →Revisit the explanation →Revisit the explanation →
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For the teacher
- Ask students to explain force direction without using a negative magnitude.
- Have students state the point-charge and medium assumptions before using Coulomb’s law.
- Diagnose scale reasoning with a neutral planet versus a charged dust grain.
Source objectives and scope
- 10.1.A — Describe the electric force that results from the interactions between charged objects or systems.
- 10.1.B — Describe the electric and gravitational forces that result from interactions between charged objects with mass.
- 10.1.C — Describe the electric permittivity of a material or medium.
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.
Suggested skills are teaching guidance. The course applies all science practices across its objectives.
Science references for this teaching
- College Board AP Physics 2 Course and Exam Description, Unit 10 (PDF pages 50–61) — Official topic order, learning objectives, all essential-knowledge clauses, algebra-based scope boundaries; equations visually inspected in local source pages 50,54,56–61.
- OpenStax University Physics Volume 2, 8.5 Molecular Model of a Dielectric — Independent verification that bound polarization leaves a dielectric net neutral and its induced field opposes the free-charge field. All lesson prose and problems are original.
- Corrected original reading: Charge and Matter — Use the corrected charge and Coulomb-law explanations to compare signed charges and pair forces.
- Corrected original reading: Conductors and Insulators — Use the corrected charge-mobility discussion to explain contact, polarization, and grounding with a system boundary.
- Corrected original reading: The Electric Field — Use the corrected point-source and vector-superposition discussion to practice fields and forces. General quantitative fields within insulators are outside the AP core.
- Corrected original reading: The Electric Potential and Potential Energy — Use the corrected algebraic point-charge energy discussion to connect configuration changes with external and electric work.
- Corrected original reading: Calculating Potentials — Use the corrected point-charge scalar sums to practice potential.
- Corrected original reading: Equipotentials, Field lines, and Gradients — Use the corrected contour maps and algebraic field estimates here; calculus-gradient material is an optional extension beyond the AP core.
- Corrected original reading: Capacitors — Use the corrected parallel-plate geometry discussion to compare C, Q, and voltage.
- Corrected original reading: Stored Energy and Dielectrics — Use the corrected energy and dielectric discussion with an explicit fixed-Q or fixed-voltage constraint.
Glossary and deeper reading · 17 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Force
A force describes an interaction that can change an object’s momentum. It is a vector measured in newtons, so both size and direction matter when forces are added. For example, a rope can pull a cart while the floor exerts friction in another direction.
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Electric charge
Electric charge is a conserved property that determines electric interactions. It can be positive or negative and is measured in coulombs; ordinary net charges arise from an imbalance of charged particles. For example, transferring electrons between initially neutral objects leaves one negatively charged and the other positively charged.
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Vector component
A vector component describes the part of a vector along a chosen axis. Its signed value changes when the axes change, even though the physical vector remains the same. For example, a diagonal launch velocity can be resolved into horizontal and vertical components that share the same elapsed time.
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Gravity
Gravity is the attractive interaction between masses in the Newtonian model. For spherical bodies or point-mass approximations, its strength decreases with the square of the separation between their centers. For example, Earth’s gravity provides the inward force that keeps a satellite in orbit.
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Elementary charge
The elementary charge is the magnitude of the charge carried by a proton or electron, approximately 1.60 × 10⁻¹⁹ coulomb. A proton carries positive elementary charge and an electron carries negative elementary charge. For example, removing one electron from a neutral atom leaves it with a net charge of one positive elementary charge.
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Magnitude
Magnitude is the nonnegative size of a vector quantity. It does not specify direction, and it differs from a signed component. For example, a westward velocity can have a negative x component while its speed, the velocity magnitude, stays positive.
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Vector
A vector is a quantity with a magnitude and a direction, represented by an arrow or by components along chosen axes. Vectors add by adding corresponding components rather than simply adding their magnitudes. For example, an eastward displacement followed by a northward displacement produces a resultant pointing northeast. The zero vector has zero magnitude and no unique direction.
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Normal force
A normal force is a contact force perpendicular to the local surface of contact. Its magnitude is set by the motion and the other forces, so it need not equal the object’s weight. For example, a wall’s normal force on a leaning ladder points horizontally rather than upward.
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Tension
Tension is a pulling force transmitted through a stretched rope, string, or similar connector. It acts along the connector away from the object it pulls, and an ideal massless rope has the same tension throughout an uninterrupted segment. For example, a rope supporting an accelerating load can have a tension different from the load’s weight.
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Electric field
An electric field assigns the electric force per unit positive test charge to each location. The field is produced by the sources and is a vector, while the force on a negative charge points opposite the field. For example, the electric field between ideal parallel capacitor plates can accelerate an electron opposite the field arrows.
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Electric polarization
Electric polarization is a separation or rearrangement of positive and negative charge within otherwise neutral matter. It can create induced dipoles without requiring a net charge to cross the object’s boundary. For example, a nearby charged rod can slightly shift bound charges in an insulating object and produce attraction.
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Permittivity
Permittivity describes a medium’s electric response and relates electric fields to charge in the chosen model. A material’s polarization can make its response differ from that of vacuum. For example, placing a dielectric between capacitor plates can change the field and capacitance because the material polarizes.
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Dielectric
A dielectric is an insulating material that polarizes in an electric field. In a capacitor it changes the field response and capacitance; charge or voltage changes depend on whether the battery remains connected. For example, inserting a dielectric into an isolated charged capacitor can reduce its voltage while its free plate-charge magnitude remains fixed.
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Electrical conductor
An electrical conductor contains charge carriers that can move through the material. In electrostatic equilibrium, mobile charges redistribute until the electric field within the conducting material is zero. For example, excess charge on an isolated metal object resides on its surface after the charges settle.
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Electrical insulator
An electrical insulator strongly restricts the movement of charge carriers through its bulk. Its bound charges can still rearrange slightly to polarize in an applied electric field. For example, a plastic rod can retain localized excess charge even though it does not conduct current like a metal wire.
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Capacitor
A capacitor stores energy by separating charge between conducting regions. In the ideal two-plate model, opposite plate charges create a potential difference related to charge by the capacitance. For example, disconnecting a charged capacitor from a battery leaves energy stored until a discharge path is provided.
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Quantization
Quantization means a quantity can take only certain allowed values in the stated quantum system. Bound-state boundary conditions can restrict energy to discrete levels rather than a continuous range. For example, an electron bound in a hydrogen atom cannot have an arbitrary energy between its allowed levels.