10.1 Electric Charge and Electric Force

Electric charge is a basic property of matter that comes in positive and negative types, and the force between two charges follows Coulomb's law: $F = k\frac{|q_1 q_2|}{r^2}$. Like charges repel, opposite charges attract, and the force gets much weaker as distance grows because it depends on $1/r^2$.

Why This Matters for the AP Physics 2 Exam

This topic gives you the foundation for all of electrostatics. You will use Coulomb's law to calculate forces, reason about whether charges attract or repel, and explain why electric forces overwhelm gravity at small scales but lose out at astronomical scales. These ideas show up in both multiple-choice and free-response questions, and they connect to later representations like electric field maps and equipotential lines that appear on the Translation Between Representations free-response question.

You will also practice translating between words, diagrams, and equations, which is exactly the kind of thinking the exam rewards. Knowing how each variable in Coulomb's law affects the force lets you predict what happens when charge or distance changes without redoing the whole calculation.

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Key Takeaways

• Charge is quantized: every net charge is an integer multiple of the elementary charge $e = 1.60 \times 10^{-19}$ C, so $q = \pm ne$.
• Coulomb's law, $F = k\frac{|q_1 q_2|}{r^2}$, is directly proportional to each charge and inversely proportional to the square of the distance.
• Same-sign charges repel, opposite-sign charges attract, and the force always acts along the line connecting the charges.
• The electric force is far stronger than gravity for tiny particles, but gravity controls large scales because big objects are mostly neutral.
• Electric permittivity measures how much a material polarizes in a field; conductors let charge move freely, insulators do not.
• You only need to calculate force for four or fewer interacting charges unless the situation is highly symmetric.

The Electric Force Between Charged Objects

Electric charge is a fundamental property of all matter. Charge comes in two types: positive and negative. The smallest indivisible amount of charge is the elementary charge, $e = 1.60 \times 10^{-19}$ C. A proton has charge $+e$, an electron has charge $-e$, and a neutron has zero charge. Because electric charge is quantized, the net charge on an object occurs in integer multiples of the elementary charge: $q = \pm ne$, where $n$ is an integer.

Electric force governs how charged particles interact. The force follows specific patterns based on the charges involved:

• When objects have the same charge sign (both positive or both negative), they repel each other
• When objects have opposite charge signs (one positive, one negative), they attract each other
• The force acts along the line connecting the centers of the charged objects
• Electric forces follow Coulomb's law: $F = k\frac{|q_1q_2|}{r^2} = \frac{1}{4\pi\varepsilon_0}\frac{|q_1q_2|}{r^2}$ where $k$ is Coulomb's constant, $q_1$ and $q_2$ are the charges, and $r$ is the distance between them

A point charge is a model used when the size of a charged object is negligible compared with the distance between objects, so the object can be treated as if all its charge were concentrated at a single point.

Electric forces are responsible for some macroscopic properties of objects in everyday experience. However, because there are enormous numbers of particle interactions in real materials, we usually model these effects using nonfundamental contact forces such as normal force, friction, and tension instead of tracking every individual electric interaction. The small shock you feel after walking across a carpet and touching a doorknob is one everyday example of these forces, but it is an application, not required content for this topic.

Comparing Electric and Gravitational Forces

Both electric and gravitational forces are fundamental forces in nature, but they have important differences:

• Electric forces can be either attractive or repulsive, while gravitational forces are always attractive
• For objects with both mass and charge, the electric force is typically much stronger than the gravitational force
• For elementary particles, the electric force is enormously larger than the gravitational force. For example, the ratio $F_e/F_g$ is about $10^{36}$ for two protons and about $10^{39}$ for a proton and an electron.

Even though electric forces are stronger, gravity has more influence at astronomical scales because:

• Large objects tend to be electrically neutral (equal amounts of positive and negative charge)
• Electric forces between neutral objects largely cancel out
• Mass is always positive and never cancels, so gravity keeps adding up

This is why planets orbit stars due to gravity rather than electric forces, even though electric forces are inherently stronger.

Electric Permittivity

Electric permittivity measures the degree to which a material or medium becomes polarized in the presence of an electric field, and this affects how electric charges interact within that material.

When an electric field is applied to a material, it can cause polarization. Electric polarization can be modeled as an induced rearrangement of electrons by an external electric field, creating a separation of positive and negative charge within the material even if the material as a whole remains electrically neutral.

Key aspects of permittivity include:

• Free space (vacuum) has a constant permittivity value of $\varepsilon_0 = 8.85 \times 10^{-12}$ F/m, and this constant appears in physical relationships such as Coulomb's law in free space: $F = \frac{1}{4\pi\varepsilon_0}\frac{|q_1q_2|}{r^2}$.
• Materials have permittivity values different from free space, often expressed as a relative permittivity $\varepsilon_r$ (ratio to free space).
• The permittivity affects the strength of electric forces within the material: $F = \frac{1}{4\pi\varepsilon}\frac{q_1q_2}{r^2}$

The permittivity of a material depends on its composition and arrangement. In a given material, permittivity is determined by how easily electrons can change configurations in response to an external electric field; the more easily the electron distribution shifts, the more strongly the material can be polarized.

• Conductors (like metals) are made from electrically conducting materials in which charge carriers move easily throughout the material
• Insulators (like rubber or plastic) are made from electrically nonconducting materials in which charge carriers cannot move easily

How to Use This on the AP Physics 2 Exam

Problem Solving

• Identify each charge with its sign and convert units (μC to C, nm to m) before plugging into Coulomb's law.
• Use magnitudes inside the equation to get the size of the force, then decide direction separately by checking the signs: same sign means repel, opposite means attract.
• For multiple charges, find each pairwise force as a vector, then add them using components. Remember the boundary: you will work with four or fewer charges unless there is high symmetry.
• When a question changes one variable, use the proportionality. Doubling one charge doubles the force; tripling the distance cuts the force to one ninth.

Free Response

• Connect words, diagrams, and equations clearly. If asked to explain why a force is attractive, name the charge signs and state that opposite charges attract.
• When comparing electric and gravitational forces, justify your claim with the idea that electric force can be attractive or repulsive while gravity is always attractive, and that large objects are mostly neutral.
• Show your reasoning, not just numbers. Stating the relationship between a variable and the force earns explanation credit.

Common Trap

• Forgetting to square the distance. The $r^2$ in the denominator is easy to miss and changes your answer by a lot.

Practice Problem 1: Electric Force Calculation

Solution

Use Coulomb's law:

$F = k\frac{|q_1q_2|}{r^2}$

Where:

• $k = 9.0 \times 10^9$ N·m²/C²
• $q_1 = +5.0 \times 10^{-6}$ C
• $q_2 = -2.0 \times 10^{-6}$ C
• $r = 0.3$ m

Substituting these values:

$F = (9.0 \times 10^9) \times \frac{|5.0 \times 10^{-6} \times (-2.0 \times 10^{-6})|}{(0.3)^2}$

$F = (9.0 \times 10^9) \times \frac{10.0 \times 10^{-12}}{0.09}$

$F = (9.0 \times 10^9) \times (1.11 \times 10^{-10})$

$F = 1.0 \times 10^0$ N = 1.0 N

Since one charge is positive and one is negative, the force is attractive, meaning the charges pull toward each other.

Practice Problem 2: Comparing Electric and Gravitational Forces

Solution

Calculate both forces and find their ratio.

Electric force using Coulomb's law: $F_e = k\frac{|q_1q_2|}{r^2}$

$F_e = k\frac{|q_p \times q_e|}{r^2} = k\frac{|(1.6 \times 10^{-19})(-1.6 \times 10^{-19})|}{r^2} = k\frac{2.56 \times 10^{-38}}{r^2}$

Gravitational force using Newton's law of gravitation: $F_g = G\frac{m_1m_2}{r^2}$

$F_g = G\frac{m_p \times m_e}{r^2} = G\frac{(1.67 \times 10^{-27})(9.11 \times 10^{-31})}{r^2} = G\frac{1.52 \times 10^{-57}}{r^2}$

The ratio of electric to gravitational force: $\frac{F_e}{F_g} = \frac{k(2.56 \times 10^{-38})/r^2}{G(1.52 \times 10^{-57})/r^2} = \frac{k}{G} \times \frac{2.56 \times 10^{-38}}{1.52 \times 10^{-57}}$

$\frac{F_e}{F_g} = \frac{9.0 \times 10^9}{6.67 \times 10^{-11}} \times \frac{2.56 \times 10^{-38}}{1.52 \times 10^{-57}}$

$\frac{F_e}{F_g} = 1.35 \times 10^{20} \times 1.68 \times 10^{19} = 2.27 \times 10^{39}$

So the electric force is approximately $2.3 \times 10^{39}$ times stronger than the gravitational force between a proton and an electron. Notice that the distance $r$ cancels out, so this ratio is the same no matter how far apart the particles are.

Common Misconceptions

• "Bigger charge always means bigger force regardless of distance." Both charge and distance matter. Because force depends on $1/r^2$, a small increase in distance can outweigh a large charge.
• "Coulomb's law gives the direction automatically." The equation with absolute values only gives the magnitude. You decide direction by looking at the signs of the charges.
• "Gravity is stronger than electric force because it controls planets." Gravity actually has more influence at large scales only because big objects are nearly neutral, so their electric forces cancel. Per particle, the electric force is vastly stronger.
• "A neutron has a small charge." A neutron has exactly zero net charge.
• "Charge can be any amount." Charge is quantized, so any net charge is a whole-number multiple of the elementary charge $e$.
• "Permittivity only matters in a vacuum." Every material has its own permittivity, which is why charges interact differently inside a material than in free space.

Related AP Physics 2 Guides

• 10.2 The Process of Charging
• 10.3 Electric Fields
• 10.5 Electric Potential
• 10.6 Capacitors
• 10.7 Conservation of Electric Energy
• 10.4 Electric Potential Energy