Coulomb's Law

Introduction

Key Concepts

Definition of Coulomb's Law

Coulomb's Law describes the force between two point charges. Formulated by Charles-Augustin de Coulomb in the 18th century, it establishes that the electric force ($F$) between two charges is directly proportional to the product of the magnitudes of the charges ($|q_1|$ and $|q_2|$) and inversely proportional to the square of the distance ($r$) between them. The law is mathematically expressed as: $$F = k_e \frac{|q_1 q_2|}{r^2}$$ where $k_e$ is the Coulomb constant, approximately equal to $8.988 \times 10^9 \, \text{N.m}²/\text{C}²$.

Mathematical Expression of Coulomb's Law

Coulomb's Law can be represented both in magnitude and vector form. The scalar form provides the magnitude of the force, while the vector form accounts for the direction of the force. **Scalar Form:** $$F = k_e \frac{|q_1 q_2|}{r^2}$$ **Vector Form:** $$\vec{F}_{12} = k_e \frac{q_1 q_2}{r^2} \hat{r}_{12}$$ where $\hat{r}_{12}$ is the unit vector pointing from charge $q_1$ to charge $q_2$.

Variables and Constants

Understanding the variables and constants in Coulomb's Law is crucial for applying the law correctly: - **$F$ (Electric Force):** Measured in newtons (N), it represents the magnitude of the force between two charges. - **$q_1$ and $q_2$ (Charges):** Measured in coulombs (C), these represent the magnitudes of the electric charges. - **$r$ (Distance):** Measured in meters (m), it is the separation between the two charges. - **$k_e$ (Coulomb Constant):** A proportionality constant given by: $$k_e = \frac{1}{4\pi\epsilon_0} \approx 8.988 \times 10^9 \, \text{N.m}²/\text{C}²$$ - **$\epsilon_0$ (Permittivity of Free Space):** Approximately $8.854 \times 10^{-12} \, \text{C}²/\text{N.m}²$.

Electric Force Between Charges

Coulomb's Law quantifies the electric force between two point charges. The force is: - **Attractive** if the charges are of opposite signs. - **Repulsive** if the charges are of the same sign. The direction of the force is along the line joining the two charges. **Example Calculation:** 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 $0.05 \, \text{m}$. $$F = k_e \frac{|q_1 q_2|}{r^2} = (8.988 \times 10^9) \frac{|3 \times 10^{-6} \times -2 \times 10^{-6}|}{(0.05)^2}$$ $$F = (8.988 \times 10^9) \frac{6 \times 10^{-12}}{0.0025}$$ $$F = (8.988 \times 10^9) \times 2.4 \times 10^{-9}$$ $$F \approx 21.57 \, \text{N}$$ Since $q_1$ and $q_2$ have opposite signs, the force is attractive.

Superposition Principle

Coulomb's Law adheres to the superposition principle, which states that the total electric force acting on a charge due to multiple other charges is the vector sum of the individual forces exerted by each charge. **Mathematically:** $$\vec{F}_{\text{total}} = \sum_{i=1}^{n} \vec{F}_i$$ **Example:** If a charge $q$ experiences forces $\vec{F}_1$ and $\vec{F}_2$ from two different charges, the total force is: $$\vec{F}_{\text{total}} = \vec{F}_1 + \vec{F}_2$$

Units and Dimensional Analysis

Ensuring consistency in units is vital when applying Coulomb's Law. The International System of Units (SI) standardizes this: - **Force ($F$):** Newtons (N) - **Charge ($q$):** Coulombs (C) - **Distance ($r$):** Meters (m) - **Coulomb Constant ($k_e$):** $8.988 \times 10^9 \, \text{N.m}²/\text{C}²$ Performing dimensional analysis helps verify the correctness of equations and calculations.

Applications of Coulomb's Law

Coulomb's Law is foundational in various applications within physics and engineering: - **Electric Fields:** Determines the electric field created by point charges. - **Chemical Bonds:** Explains the forces between ions in ionic compounds. - **Electrostatic Precipitators:** Utilizes electric forces to remove particles from exhaust streams. - **Capacitors:** Calculates the force between charges on capacitor plates.

Limitations of Coulomb’s Law

While Coulomb's Law is powerful, it has limitations: - **Point Charges:** It strictly applies to point charges or spherically symmetric charge distributions. - **Medium:** The law is valid in a vacuum or air; in other media, the permittivity changes. - **Non-Relativistic:** It doesn't account for relativistic effects at high velocities. - **Quantum Effects:** Doesn't incorporate quantum mechanical phenomena.

Relation to Electric Field and Potential

Coulomb's Law is intrinsically related to the concepts of electric field ($\vec{E}$) and electric potential ($V$): - **Electric Field:** Defined as the force per unit charge: $$\vec{E} = \frac{\vec{F}}{q} = k_e \frac{q}{r^2} \hat{r}$$ - **Electric Potential:** The work done per unit charge in bringing a charge from infinity to a point: $$V = k_e \frac{q}{r}$$

Comparison with Newton's Law of Universal Gravitation

Both Coulomb's Law and Newton's Law of Universal Gravitation describe forces that follow an inverse-square law, but they differ in nature and interaction: - **Coulomb's Law:** Deals with electric forces between charges. - **Newton's Law:** Describes gravitational forces between masses. These similarities and differences are further explored in the comparison table below.

Comparison Table

Aspect	Coulomb's Law	Newton's Law of Universal Gravitation
Type of Force	Electric force between charges	Gravitational force between masses
Mathematical Expression	$F = k_e \frac{|q_1 q_2|}{r^2}$	$F = G \frac{m_1 m_2}{r^2}$
Direction of Force	Repulsive or attractive depending on charges	Always attractive
Constant	Coulomb constant ($k_e = 8.988 \times 10^9 \, \text{N.m}²/\text{C}²$)	Gravitational constant ($G = 6.674 \times 10^{-11} \, \text{N.m}²/\text{kg}²$)
Strength of Force	Much stronger compared to gravity	Significantly weaker than electric force
Nature	Vector force	Vector force

Summary and Key Takeaways