8.1 Electric Charge and Electric Force

Electric charge is a basic property of matter that comes in positive and negative types, and charged objects push or pull on each other through the electrostatic force. Coulomb's law gives the size of that force: it grows with the product of the charges and shrinks with the square of the distance between them.

Why This Matters for the AP Physics C: E&M Exam

This topic is the foundation for everything in AP Physics C: E&M. The forces you study here lead directly into electric fields, potential, and eventually circuits and magnetism.

Unit 8 carries one of the heaviest weightings on the multiple-choice section, so you will see Coulomb's law and charge reasoning often. On the free-response side, the first question rewards students who can derive a symbolic expression, build a clear representation of a setup, and justify claims with physical reasoning. Coulomb's law problems with two to four charges are a natural fit for that kind of multi-step work.

You should be ready to:

• Calculate the force between point charges using a logical computational pathway.
• Compare electric force across different scenarios when charges or distances change.
• Predict how the force changes when you scale a charge or the separation distance.
• Apply Coulomb's law and the definition of charge to support a claim.

Key Takeaways

• Charge is a scalar that is positive or negative, and the elementary charge $e$ is the smallest indivisible amount; protons carry $+e$, electrons carry $-e$, and neutrons are neutral.
• Coulomb's law: $\left|\vec{F}_{E}\right|=\dfrac{1}{4 \pi \varepsilon_{0}} \dfrac{\left|q_{1} q_{2}\right|}{r^{2}}=k \dfrac{\left|q_{1} q_{2}\right|}{r^{2}}$, an inverse-square law along the line connecting the charges.
• Like charges repel, opposite charges attract, and force direction always lies along the line of separation.
• For several charges, use superposition: the net force is the vector sum of each pairwise force.
• Between particles the electrostatic force is far stronger than gravity, but gravity controls large scales because big systems are nearly neutral.
• Permittivity measures how much a material polarizes in a field; free space has the constant $\varepsilon_0$, and matter differs based on how easily its electrons rearrange.

Electric Charge Fundamentals

Electric charge is a fundamental property of matter that exists as either positive or negative. This property determines how objects interact electromagnetically.

The elementary charge $e$ is the smallest indivisible unit of electric charge:

• Electrons carry a charge of $-e$ (negative)
• Protons carry a charge of $+e$ (positive)
• Neutrons have no electric charge (neutral)

When analyzing electric interactions, you often model charged objects as point charges, treating all the charge as if it were concentrated at a single point. This is a model that works when the object's physical size does not matter for the situation.

Charge is a scalar quantity, meaning it has magnitude and sign but no direction. Unlike force or velocity, charge is fully described by its numerical value and whether it is positive or negative.

Coulomb's Law and the Electric Force

Coulomb's law describes the electrostatic force between two charged objects. The force is directly proportional to the product of the charge magnitudes and inversely proportional to the square of the distance between them.

$\left|\vec{F}_{E}\right|=\frac{1}{4 \pi \varepsilon_{0}} \frac{\left|q_{1} q_{2}\right|}{r^{2}}=k \frac{\left|q_{1} q_{2}\right|}{r^{2}}$

Where:

• $\left|\vec{F}_{E}\right|$ is the magnitude of the electrostatic force
• $q_1$ and $q_2$ are the charges of the two objects
• $r$ is the distance between the charges
• $\varepsilon_0$ is the permittivity of free space
• $k$ is Coulomb's constant (approximately $9.0 \times 10^9$ N·m²/C²)

When more than two charges are present, the net force on any one charge is the vector sum of the individual forces acting on it. This is the principle of superposition.

Direction of the Electrostatic Force

The direction depends on the signs of the interacting charges:

• Like charges (both positive or both negative) repel each other.
• Unlike charges (one positive, one negative) attract each other.
• The force acts along the line connecting the two charges.

A quick way to remember it: opposites attract, likes repel. This is fundamentally different from gravity, which is always attractive.

Macroscopic Properties from Electric Forces

Electric forces between atoms and molecules are behind many everyday effects:

• The normal force when objects touch comes from electromagnetic repulsion between atoms.
• Friction arises from electromagnetic interactions between surface molecules.
• Tension in ropes and strings results from electromagnetic bonds between atoms.
• Material strength and elasticity come from electric forces between molecules.

Even though these trace back to electric interactions, it is more convenient to treat them as nonfundamental contact forces like normal force, friction, and tension.

Electrostatic vs Gravitational Forces

Relative Magnitudes

Between particles, the electrostatic force is vastly stronger than the gravitational force:

• The electric force between an electron and proton is roughly $10^{39}$ times stronger than their gravitational attraction.
• For two protons, the electric repulsion is about $10^{36}$ times stronger than their gravitational attraction.

This is why electric forces shape atomic and molecular behavior while gravity is negligible at those scales.

Why Gravity Controls Large Scales

Even though gravity is far weaker at the particle level, it has more influence at astronomical scales:

• Most large objects are electrically neutral, with nearly equal positive and negative charge.
• Electric forces between neutral objects largely cancel out.
• Gravitational forces are always attractive and add up.

So planets orbit stars because of gravity, not electric forces, even though electric forces are intrinsically stronger.

Electric Permittivity

Electric permittivity measures how readily a material or medium becomes polarized when placed in an electric field. It helps describe how electric fields behave inside different materials.

Polarization Model

When an electric field is applied to a material, it can rearrange charges within the material, which is called polarization:

• In atoms, the electron cloud shifts slightly relative to the nucleus.
• A separation of positive and negative charge develops within the material.
• This induced separation affects the overall field in the material.

The degree of polarization depends on the material's composition and arrangement.

Permittivity of Free Space

The permittivity of free space $\varepsilon_0$ is a fundamental physical constant:

• It appears in Coulomb's law and many other electromagnetic equations.
• Its value is approximately $8.85 \times 10^{-12}$ F/m (farads per meter).
• It serves as the baseline against which materials are compared.

Permittivity in Matter

Different materials have different permittivities based on how their internal structure responds to electric fields. In a given material, permittivity depends on how easily electrons can change configurations.

• Conductors are made of materials in which charge carriers move easily.
• Insulators are made of materials in which charge carriers cannot move easily.
• The permittivity of matter differs from that of free space because the electrons rearrange in response to an external field.

How to Use This on the AP Physics C: E&M Exam

Problem Solving

Use a consistent process for Coulomb's law problems:

1. Sketch the charges and label positions and signs. A clear diagram earns representation points and prevents sign errors.
2. Use the magnitude form $F = k\dfrac{|q_1 q_2|}{r^2}$ to find the size of each pairwise force.
3. Decide direction separately using the rule that like charges repel and opposite charges attract, along the line connecting them.
4. Add forces as vectors. For charges along a line, assign positive and negative directions; for two-dimensional setups, break each force into components.

Free Response

When deriving an expression, keep your work symbolic as long as possible and plug in numbers only at the end. State which law or definition you are applying before you use it, and justify direction claims with the sign rule rather than just asserting an answer.

Common Trap

Watch the inverse-square dependence. If the separation distance doubles, the force drops to one quarter, not one half. If you triple a charge, the force triples. Practice predicting these factor-of-change results, since the exam likes to test scaling.

Practice Problem 1: Coulomb's Law

Solution

Use Coulomb's law:

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

Known values:

• $q_1 = 4.0 \times 10^{-6}$ C (positive)
• $q_2 = -2.0 \times 10^{-6}$ C (negative)
• $r = 0.3$ m
• $k = 9.0 \times 10^9$ N·m²/C²

Substituting:

$F = (9.0 \times 10^9)\frac{|4.0 \times 10^{-6} \times (-2.0 \times 10^{-6})|}{(0.3)^2}$

$F = (9.0 \times 10^9)\frac{8.0 \times 10^{-12}}{0.09}$

$F = (9.0 \times 10^9) \times 8.89 \times 10^{-11}$

$F = 0.80 \text{ N}$

For direction, one charge is positive and one is negative, so the force is attractive. The charges are pulled toward each other along the line connecting them.

Practice Problem 2: Multiple Charges

Solution

Find the forces on $q_2$ from both $q_1$ and $q_3$, then add them as vectors.

Force from $q_1$ on $q_2$:

• $q_1 = 3.0 \times 10^{-6}$ C
• $q_2 = -5.0 \times 10^{-6}$ C
• $r_{12} = 4.0$ m

$F_{12} = k\frac{|q_1 q_2|}{r_{12}^2} = (9.0 \times 10^9)\frac{|3.0 \times 10^{-6} \times (-5.0 \times 10^{-6})|}{(4.0)^2}$

$F_{12} = (9.0 \times 10^9)\frac{15.0 \times 10^{-12}}{16.0} = 8.44 \times 10^{-3} \text{ N}$

Since $q_1$ is positive and $q_2$ is negative, this force is attractive, pulling $q_2$ toward $q_1$ (negative x-direction).

Force from $q_3$ on $q_2$:

• $q_2 = -5.0 \times 10^{-6}$ C
• $q_3 = 2.0 \times 10^{-6}$ C
• $r_{23} = 3.0$ m

$F_{23} = k\frac{|q_2 q_3|}{r_{23}^2} = (9.0 \times 10^9)\frac{|-5.0 \times 10^{-6} \times (2.0 \times 10^{-6})|}{(3.0)^2}$

$F_{23} = (9.0 \times 10^9)\frac{10.0 \times 10^{-12}}{9.0} = 10.0 \times 10^{-3} \text{ N}$

Since $q_2$ is negative and $q_3$ is positive, this force is attractive, pulling $q_2$ toward $q_3$ (positive x-direction).

The net force is the vector sum:

$F_{net} = F_{23} - F_{12} = 10.0 \times 10^{-3} \text{ N} - 8.44 \times 10^{-3} \text{ N} = 1.56 \times 10^{-3} \text{ N}$

The net force on $q_2$ is $1.56 \times 10^{-3}$ N in the positive x-direction.

Common Misconceptions

• Charge is not a vector. It is a scalar with a sign. Direction belongs to the force, not the charge.
• The negative sign on a charge does not go into the magnitude form of Coulomb's law. Use absolute values for magnitude, then decide direction from the signs.
• Doubling the distance does not halve the force. Because of the inverse-square law, the force drops to one quarter.
• Electric force is not always weaker than gravity. Between charged particles it is enormously stronger; gravity only wins at large scales because big objects are nearly neutral.
• "Opposites attract" describes direction, not magnitude. The size of the force is the same on both charges by Newton's third law, regardless of which charge is larger.
• Permittivity is not the same as conductivity. It measures how much a material polarizes in a field, which is tied to how easily electrons rearrange, not simply whether current flows.

Related AP Physics C: E&M Guides

• 8.2 Electric Charge and the Process of Charging
• 8.3 Electric Fields
• 8.4 Electric Fields of Charge Distributions
• 8.6 Gauss's Law
• 8.5 Electric Flux