Coulomb's Law is a fundamental principle in physics that quantifies the electrostatic force between electrically charged particles. Essential for understanding electric interactions, Coulomb's Law is a cornerstone topic in the Collegeboard AP Physics 2: Algebra-Based curriculum. Mastery of this law enables students to analyze and predict the behavior of charges in various physical scenarios, laying the groundwork for more advanced studies in electromagnetism and electrical engineering.

Coulomb's Law is named after Charles-Augustin de Coulomb, a French physicist who, in the late 18th century, conducted experiments using a torsion balance to measure the forces between electric charges. Published in 1785, Coulomb's findings established the quantitative relationship between electric charges, laying the foundation for the study of electrostatics.

Electric charge is a fundamental property of matter responsible for electric and magnetic interactions. There are two types of charges: positive and negative. Like charges repel each other, while opposite charges attract. The unit of electric charge in the International System of Units (SI) is the coulomb (C).

Electric force is the force exerted between charged particles. It can either be attractive or repulsive, depending on the nature of the charges involved. This force is a key concept in understanding how charged particles interact within atoms and molecules.

Coulomb's Law mathematically expresses the electric force ($F$) between two point charges. The formula is given by: $$ F = k_e \cdot \frac{|q_1 \cdot q_2|}{r^2} $$ where:

• $F$ is the magnitude of the electric force between the charges.
• $k_e$ is Coulomb's constant, approximately $8.988 \times 10^9 \, \text{N.m}^2/\text{C}^2$.
• $q_1$ and $q_2$ are the amounts of the charges.
• $r$ is the distance between the centers of the two charges.

This equation highlights that the electric force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

Coulomb's constant ($k_e$) plays a crucial role in Coulomb's Law. Its value in SI units is: $$ k_e = \frac{1}{4\pi \varepsilon_0} \approx 8.988 \times 10^9 \, \text{N.m}^2/\text{C}^2 $$ where $\varepsilon_0$ is the vacuum permittivity. This constant ensures that the units of force come out correctly when calculating using the law.

While Coulomb's Law provides the magnitude of the electric force, the force itself is a vector quantity, meaning it has both magnitude and direction. The direction of the force depends on the nature of the charges:

• If both charges are of the same type (both positive or both negative), the force is repulsive, pushing the charges apart.
• If the charges are of opposite types (one positive and one negative), the force is attractive, pulling the charges together.

The direction is along the line joining the centers of the two charges.

The Superposition Principle states that when multiple charges are present, the total electric force acting on a particular charge is the vector sum of the individual forces exerted by each of the other charges. This principle allows for the analysis of complex systems by breaking them down into simpler interactions.

An electric field ($E$) is defined as the electric force per unit charge experienced by a small positive test charge placed in the field. Coulomb's Law can be used to derive the expression for the electric field created by a point charge: $$ E = k_e \cdot \frac{|q|}{r^2} $$ where $q$ is the charge creating the field, and $r$ is the distance from the charge. The electric field is a vector pointing away from a positive charge and toward a negative charge.

Coulomb's Law has numerous applications across various fields of physics and engineering:

• Atomic Structure: It explains the electrostatic forces between electrons and the nucleus, contributing to the stability of atoms.
• Electrostatic Precipitators: Used in industries to remove particles from exhaust gases by charging them and using electric fields to attract and remove them.
• Biology: Understanding interactions between molecules and ions in biological systems.
• Electric Force Calculations: Essential in solving problems related to electric potential energy and field configurations.

While Coulomb's Law is fundamental, it has limitations:

• Point Charges: The law applies strictly to point charges or spherically symmetric charge distributions.
• Static Charges: It is valid only for stationary charges and does not account for magnetic effects or moving charges.
• Medium Considerations: The presence of a medium with permittivity other than vacuum ($\varepsilon_0$) affects the force, requiring adjustments in calculations.

Electric potential energy ($U$) between two point charges can be expressed as: $$ U = k_e \cdot \frac{q_1 \cdot q_2}{r} $$ The force can be derived by taking the negative gradient of the potential energy with respect to distance: $$ F = -\frac{dU}{dr} = k_e \cdot \frac{q_1 \cdot q_2}{r^2} $$ This derivation reinforces the inverse-square relationship between force and distance inherent in Coulomb's Law.

Applying Coulomb's Law involves identifying the charges involved, determining their magnitudes, measuring the distance between them, and then substituting these values into the law's equation. Example 1: Calculate the electric force between two charges, $q_1 = 3 \times 10^{-6} \, \text{C}$ and $q_2 = -2 \times 10^{-6} \, \text{C}$, separated by a distance of $0.05 \, \text{meters}$. $$ F = k_e \cdot \frac{|q_1 \cdot q_2|}{r^2} = 8.988 \times 10^9 \cdot \frac{(3 \times 10^{-6}) \cdot (2 \times 10^{-6})}{(0.05)^2} $$ $$ F = 8.988 \times 10^9 \cdot \frac{6 \times 10^{-12}}{0.0025} = 8.988 \times 10^9 \cdot 2.4 \times 10^{-9} = 21.57 \, \text{N} $$ The negative sign indicates that the force is attractive. Example 2: Two identical positive charges, each of $1 \times 10^{-6} \, \text{C}$, are placed $0.1 \, \text{meters}$ apart. Determine the magnitude of the force between them. $$ F = k_e \cdot \frac{q_1 \cdot q_2}{r^2} = 8.988 \times 10^9 \cdot \frac{(1 \times 10^{-6}) \cdot (1 \times 10^{-6})}{(0.1)^2} $$ $$ F = 8.988 \times 10^9 \cdot \frac{1 \times 10^{-12}}{0.01} = 8.988 \times 10^9 \cdot 1 \times 10^{-10} = 0.8988 \, \text{N} $$ Since both charges are positive, the force is repulsive.

• Coulomb's Law quantifies the electric force between two charged particles.
• The electric force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
• Electric force is a vector quantity, exhibiting both magnitude and direction.
• The Superposition Principle allows for the calculation of forces in systems with multiple charges.
• Coulomb's Law is foundational for understanding electric fields, atomic structure, and various technological applications.