AP Physics 2 Unit 2 Foundations: Charge, Conservation, and the Electric Force

Electric Charge

What electric charge is

Electric charge is a property of matter that determines how objects participate in electric interactions. If an object has a net charge, it will exert forces on other charged objects—and experience forces from them—even without touching. This “action at a distance” is one of the big ideas that makes electrostatics feel different from contact forces like friction or tension.

In everyday materials, charge comes from the particles that make up atoms:

  • Protons have charge +e
  • Electrons have charge -e
  • Neutrons have charge 0

Here e is the elementary charge, the smallest magnitude of free charge you will typically deal with in basic physics:

e = 1.60 \times 10^{-19} \text{ C}

The unit of charge is the coulomb (C). A coulomb is a large unit compared to the charges of individual particles, which is why typical lab charges are often in microcoulombs \left(\mu\text{C} = 10^{-6}\text{ C}\right) or nanocoulombs.

Why charge matters

Charge is the “source” of electric forces and (in the next parts of Unit 2) electric fields and electric potential. If you understand what charge is and how it moves between objects, then Coulomb’s law, fields, and circuits become much more intuitive—because they all depend on where net charge exists and how it is distributed.

Key properties of charge

Two types: positive and negative

There are two kinds of charge. Like charges repel and opposite charges attract.

A useful way to think about net charge is as an imbalance between protons and electrons:

  • An object is positively charged if it has fewer electrons than protons.
  • An object is negatively charged if it has more electrons than protons.

A common misconception is that “positive charge moves.” In most solid materials, electrons are the mobile charges; protons are bound in atomic nuclei and do not move through the material.

Quantization of charge

Charge is quantized, meaning net charge comes in integer multiples of e:

q = n e

Here q is the net charge on an object and n is an integer (positive, negative, or zero). In many AP problems, you will treat charge as a continuous variable because the numbers involve enormous counts of electrons, but the underlying idea remains: you cannot have “half an electron’s worth” of net charge.

Charge can be transferred, not created (in ordinary situations)

In electrostatics, charging an object means transferring electrons to or from it. You are not creating new charge; you are rearranging existing charge.

Conductors, insulators, and polarization

To predict how charge behaves, you need to know how easily charges can move within a material.

  • Conductors allow charges (usually electrons) to move freely through the material. Metals are the typical example.
  • Insulators do not allow charges to move freely. Charge can remain “stuck” where it was placed (like on a plastic rod).

Even in an insulator, charges within atoms and molecules can shift slightly when an external charge is nearby. This is called polarization: the material develops small regions of slightly positive and slightly negative charge, even if its net charge is zero.

Polarization matters because it explains real observations like “a charged balloon sticks to a wall.” The wall is neutral overall, but it polarizes, creating an attractive effect.

How objects become charged (mechanisms)

When AP Physics asks about charging, it’s usually one of these processes.

Charging by friction (triboelectric effect)

Rubbing two different materials together can transfer electrons from one to the other. Afterward:

  • One object has gained electrons and becomes negatively charged.
  • The other has lost electrons and becomes positively charged.

A crucial point: the total charge of the two-object system is unchanged (this connects directly to conservation of charge).

Charging by conduction (contact)

If a charged object touches a conductor, electrons can flow between them. After contact, charge redistributes, and both objects can end up with net charge.

In many AP problems with two identical conducting spheres that touch and then separate, the charge ends up shared equally (because identical objects at equilibrium tend to the same potential; at this level you can treat it as equal sharing).

Charging by induction (no contact)

Induction charges an object without touching it, using the fact that charges in a conductor can move.

A standard induction sequence for charging a conductor (like a metal sphere) is:

  1. Bring a charged rod near (but not touching) the conductor. Charges in the conductor separate (polarize): opposite charges gather nearer the rod.
  2. While the rod remains nearby, connect the conductor to ground. Electrons flow to or from the conductor depending on the rod’s sign.
  3. Remove the ground connection first.
  4. Then remove the rod.

The conductor is left with a net charge opposite the rod’s charge.

Students often mix up the order of steps 3 and 4. If you remove the rod before disconnecting the ground, the induced separation disappears and the object may not end up charged.

Worked example: counting electrons from a net charge

An object has net charge q = -3.2 \times 10^{-9}\text{ C}. How many excess electrons does it have?

Step 1: Use quantization idea

q = n e

Here n = \frac{q}{e}.

Step 2: Compute

n = \frac{-3.2 \times 10^{-9}}{1.60 \times 10^{-19}} = -2.0 \times 10^{10}

Interpretation: The negative sign means excess electrons. The object has 2.0 \times 10^{10} extra electrons.

Exam Focus

  • Typical question patterns
    • Determine whether an object becomes positive or negative after friction, conduction, or induction (often with a grounding step).
    • Convert between net charge and number of electrons transferred using q = n e.
    • Conceptual questions about conductors vs insulators and why neutral objects can be attracted to charged ones (polarization).
  • Common mistakes
    • Saying “protons move” during charging in solids; in AP contexts it’s almost always electrons moving.
    • Confusing polarization with net charging: polarization can cause attraction even when net charge remains zero.
    • Messing up induction order (ground removal must happen before removing the external charged object).

Conservation of Electric Charge

What conservation of charge means

Conservation of electric charge means that the total charge in an isolated system remains constant. Charge can move from one object to another, and it can spread out differently, but the algebraic sum of charge stays the same.

If you define a system of objects and no charge enters or leaves that system, then:

q_{\text{total, initial}} = q_{\text{total, final}}

This is not just a bookkeeping rule—it is the principle that lets you solve many AP problems with simple algebra before you ever use Coulomb’s law.

Why it matters

In electrostatics, you often cannot “see” charge directly, but conservation gives you a reliable constraint. For example, if two objects start neutral and end up with charges +Q and -Q after rubbing, conservation tells you immediately those magnitudes must match.

Later in the unit, conservation of charge also helps connect electrostatics to circuits: charge doesn’t vanish at a resistor; it flows through.

How to use conservation in typical scenarios

1) Friction: equal and opposite charges in a two-object system

Suppose two neutral objects are rubbed together and separated. The system initially has:

q_{\text{total}} = 0

After rubbing, if one object has +Q, the other must have -Q so that the total remains zero.

A common misconception is that both objects can end up with the same sign from friction. In a two-object isolated system, they must be opposite signs because electrons are transferred between them.

2) Conduction with identical conductors: sharing charge

When two identical conducting spheres touch, charge redistributes until they reach electrostatic equilibrium. For intro AP problems, the result is equal sharing:

If initial charges are q_1 and q_2, then after touching and separating:

q_{\text{final on each}} = \frac{q_1 + q_2}{2}

If the spheres are not identical (different radii), the final sharing is not simply half-and-half; AP Physics 2 may address that qualitatively (larger conductors can hold more charge at the same potential), but many algebra-based problems stick to identical spheres.

3) Induction with grounding: charge enters or leaves through Earth

Induction with grounding is the main case where students get confused about “where the charge comes from.” The key is that the Earth acts as a massive reservoir of charge.

  • If a negatively charged rod is brought near a conductor, it repels electrons in the conductor.
  • If the conductor is grounded, those electrons can flow into Earth.
  • When the ground is removed, the conductor has a deficit of electrons, so it is positively charged.

Total charge is still conserved if you include the Earth in your system. If you treat only the sphere as the system, then charge is not conserved within that smaller system because charge crossed its boundary.

Worked example: two identical spheres touch

Sphere A has q_A = +6.0\,\mu\text{C} and sphere B has q_B = -2.0\,\mu\text{C}. They are identical conductors. They touch and are separated. Find the final charge on each.

Step 1: Conserve total charge

q_{\text{total}} = q_A + q_B = +6.0\,\mu\text{C} + (-2.0\,\mu\text{C}) = +4.0\,\mu\text{C}

Step 2: Share equally (identical spheres)

q_{\text{final}} = \frac{q_{\text{total}}}{2} = \frac{+4.0\,\mu\text{C}}{2} = +2.0\,\mu\text{C}

Each sphere ends with +2.0\,\mu\text{C}.

Worked example: friction charging in an isolated system

Two objects start neutral. After rubbing, object 1 has q_1 = -3.0\,\mu\text{C}. What is the charge on object 2?

Use conservation: initial total is zero.

q_1 + q_2 = 0

So:

q_2 = -q_1 = +3.0\,\mu\text{C}

Exam Focus

  • Typical question patterns
    • “Two spheres touch and separate” problems where conservation determines final charges.
    • Friction and separation scenarios that require recognizing equal and opposite charges in an isolated two-object system.
    • Induction-with-ground questions asking for the final sign of charge (and where electrons moved).
  • Common mistakes
    • Forgetting to define the system: if grounding occurs, charge can leave/enter the object, so the Earth must be included for conservation.
    • Treating charge magnitudes as always equal after contact even when objects are not identical (unless stated identical).
    • Losing track of signs: charge is algebraic, so -2\,\mu\text{C} plus +6\,\mu\text{C} is +4\,\mu\text{C}, not 8\,\mu\text{C}.

Coulomb's Law and Electric Force

What the electric force is

The electric force is the interaction between charges. In electrostatics (charges at rest), the force between two point charges is described by Coulomb’s law. This law is the electric analog of Newton’s law of gravitation in two important ways:

  • It is an inverse-square law (depends on \frac{1}{r^2}).
  • It acts along the line connecting the two objects.

But unlike gravity, which is always attractive, the electric force can attract or repel depending on the signs of the charges.

Why Coulomb’s law matters

Coulomb’s law is the starting point for nearly everything else in electrostatics:

  • The electric field is defined from the electric force per unit charge.
  • The electric potential relates to the work done by electric forces.

So learning Coulomb’s law isn’t just learning one formula—it’s learning the fundamental interaction that later concepts repackage in more convenient forms.

Coulomb’s law (magnitude) and what each symbol means

For two point charges q_1 and q_2 separated by distance r, the magnitude of the force is:

F = k \frac{|q_1 q_2|}{r^2}

Where:

  • F is the magnitude of the electric force (newtons)
  • q_1 and q_2 are charges (coulombs)
  • r is the separation distance between charge centers (meters)
  • k is Coulomb’s constant

A commonly used value is:

k = 8.99 \times 10^9 \text{ N m}^2/\text{C}^2

You may also see Coulomb’s constant written using the permittivity of free space \epsilon_0:

k = \frac{1}{4\pi\epsilon_0}

with:

\epsilon_0 = 8.85 \times 10^{-12} \text{ C}^2/(\text{N m}^2)

AP problems often provide k, so focus on meaning and use rather than memorizing decimals.

Direction of the force (attraction vs repulsion)

Coulomb’s law gives the magnitude; direction comes from charge signs:

  • If q_1 q_2 > 0 (same sign), the force is repulsive: each charge pushes the other away.
  • If q_1 q_2 < 0 (opposite signs), the force is attractive: each charge pulls the other closer.

A reliable way to avoid sign mistakes is to:

  1. Use F = k \frac{|q_1 q_2|}{r^2} for magnitude.
  2. Decide direction separately with a sketch.

Newton’s third law and electric forces

Electric forces come in pairs. The force that charge 1 exerts on charge 2 has the same magnitude and opposite direction as the force that charge 2 exerts on charge 1:

  • \vec{F}_{12} = -\vec{F}_{21}

Students sometimes think “the bigger charge exerts a bigger force.” That is false for the interaction pair: both feel equal magnitude forces. What can differ is the acceleration, because acceleration depends on mass:

a = \frac{F}{m}

Superposition: forces from multiple charges add as vectors

If more than two charges are present, the net force on a given charge is the vector sum of the forces from every other charge.

This is the principle of superposition:

\vec{F}_{\text{net}} = \vec{F}_1 + \vec{F}_2 + \cdots

Each individual force is found with Coulomb’s law using the distance to that particular source charge. Then you add them as vectors (often in 1D along a line, or in 2D with components).

A classic mistake is to add force magnitudes without considering direction. Always decide whether each force contribution points left/right (or up/down) before combining.

When Coulomb’s law applies (modeling choices)

Coulomb’s law is exact for point charges. Many objects are not literally points, but you can often treat them as point charges when:

  • The object’s size is much smaller than the separation distance, so the charge distribution details don’t matter much.
  • The object is spherical and charged, and you are calculating the force outside the sphere (a uniformly charged sphere behaves like a point charge located at its center for outside points).

AP questions will usually tell you to model objects as point charges or small spheres.

Also note: the constant k above is for charges in vacuum (and is a good approximation for air). In other materials, the force can be reduced due to polarization effects in the medium, but AP Physics 2 typically keeps you in vacuum/air unless explicitly stating a dielectric context.

Worked example 1: force between two charges

Two point charges are separated by r = 0.30\text{ m}. Charge 1 is q_1 = +2.0\,\mu\text{C} and charge 2 is q_2 = -5.0\,\mu\text{C}. Find the magnitude of the force and describe its direction.

Step 1: Convert to coulombs

q_1 = 2.0 \times 10^{-6}\text{ C}

q_2 = -5.0 \times 10^{-6}\text{ C}

Step 2: Use Coulomb’s law for magnitude

F = k \frac{|q_1 q_2|}{r^2}

Compute the product:

|q_1 q_2| = |(2.0 \times 10^{-6})(-5.0 \times 10^{-6})| = 1.0 \times 10^{-11}

Compute:

F = (8.99 \times 10^9) \frac{1.0 \times 10^{-11}}{(0.30)^2}

F = (8.99 \times 10^9) \frac{1.0 \times 10^{-11}}{0.090}

F \approx 1.0 \text{ N}

Step 3: Direction
The charges are opposite signs, so they attract. Each force vector points toward the other charge along the line connecting them.

Worked example 2: net force on a charge in 1D (superposition)

Three charges lie on the x-axis:

  • q_A = +2.0\,\mu\text{C} at x = 0.00\text{ m}
  • q_B = +1.0\,\mu\text{C} at x = 0.40\text{ m}
  • q_C = -3.0\,\mu\text{C} at x = 0.70\text{ m}

Find the net force on q_B.

Step 1: Identify forces on q_B

  • Force on q_B due to q_A: like charges (both positive) so **repulsion**. Since q_A is left of q_B, repulsion pushes q_B to the right.
  • Force on q_B due to q_C: opposite charges, so **attraction**. Since q_C is right of q_B, attraction pulls q_B to the right.

Both contributions point right, so magnitudes will add.

Step 2: Distances

r_{AB} = 0.40\text{ m}

r_{BC} = 0.70 - 0.40 = 0.30\text{ m}

Step 3: Magnitudes by Coulomb’s law

F_{AB} = k \frac{|q_A q_B|}{r_{AB}^2}

F_{AB} = (8.99 \times 10^9) \frac{(2.0 \times 10^{-6})(1.0 \times 10^{-6})}{(0.40)^2}

F_{AB} = (8.99 \times 10^9) \frac{2.0 \times 10^{-12}}{0.16} \approx 0.112\text{ N}

Now for q_C:

F_{CB} = k \frac{|q_C q_B|}{r_{BC}^2}

F_{CB} = (8.99 \times 10^9) \frac{(3.0 \times 10^{-6})(1.0 \times 10^{-6})}{(0.30)^2}

F_{CB} = (8.99 \times 10^9) \frac{3.0 \times 10^{-12}}{0.090} \approx 0.300\text{ N}

Step 4: Add with direction
Both are to the right, so:

F_{\text{net on }B} = 0.112\text{ N} + 0.300\text{ N} = 0.412\text{ N}

Direction: to the right.

What to notice: The closer charge q_C produced the larger force even though the charge magnitudes were comparable—distance matters strongly because of the \frac{1}{r^2} dependence.

Common conceptual pitfalls (and how to self-correct)

  • Mixing up force and field: In this section you are dealing with forces between charges. Later you will use electric field to avoid repeatedly computing pairwise forces. For now, remember force depends on both charges; field depends only on the source charges.
  • Forgetting vector direction: Always decide whether each interaction is attraction or repulsion and draw the direction along the line connecting charges.
  • Using the wrong distance: r is the center-to-center separation between charges (or between the modeled point locations).
  • Assuming neutral means “no force”: Neutral objects can be attracted to charged objects due to polarization, even if their net charge is zero.

Exam Focus

  • Typical question patterns
    • Straight Coulomb’s law calculations: given q_1, q_2, and r, find F and describe attraction vs repulsion.
    • Superposition problems: compute the net force on one charge due to two or more others, often in a line (1D) or with right-angle geometry (2D components).
    • Conceptual comparisons: how the force changes if distance doubles, or if one charge is tripled (testing the inverse-square and proportionality to charge).
  • Common mistakes
    • Treating force as scalar: adding magnitudes without assigning directions.
    • Thinking the larger charge “pushes harder” in a third-law pair; the forces are equal in magnitude on each charge.
    • Not squaring distance or misapplying units (mixing \mu\text{C} with C without converting).