Coulomb's law Knowledge-Based Study

Study notes

1. Electric Charges and Mutual Interaction

   Electrostatics deals with charges that are at rest. From fundamental observations, electric charges occur in two distinct types: positive and negative. The fundamental law of interaction dictates that:

   • Like charges repel one another (positive repels positive; negative repels negative).
   • Opposite charges attract one another (positive attracts negative).

   In 1785, French physicist Charles-Augustin de Coulomb quantified this interaction using a sensitive torsion balance, establishing the relationship between the magnitudes of charges, the distance separating them, and the resulting electrostatic force.

   The Force Formula

   Coulomb's law states that the electrostatic force of attraction or repulsion between two stationary point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them.

   Mathematically, the magnitude of the force $F$ is expressed as:

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

   Where:

   • $F$ is the magnitude of the electrostatic force, measured in newtons (N).
   • $q_1$ and $q_2$ are the electric charges, measured in coulombs (C).
   • $r$ is the straight-line distance between the centers of the charges, measured in meters (m).
   • $k_e$ is Coulomb's constant (electrostatic constant), equal to approximately $8.99 \times 10^9\text{ N}\cdot\text{m}^2/\text{C}^2$ in a vacuum.

   Direction of Electrostatic Forces

   Force is a vector quantity, possessing both magnitude and direction. The electrostatic force between two point charges always acts along the line joining their centers.

   • Repulsive force: If both charges carry the same sign ($q_1 q_2 > 0$), the forces push the charges away from each other.
   • Attractive force: If the charges carry opposite signs ($q_1 q_2 < 0$), the forces pull the charges toward each other.

   Regardless of whether the charge values are equal or different, the force exerted by $q_1$ on $q_2$ is equal in magnitude and opposite in direction to the force exerted by $q_2$ on $q_1$:

   $$\vec{F}_{12} = -\vec{F}_{21}$$

   This directly obeys Newton's third law of motion (action-reaction pair).

   Influence of the Medium and Multiple Charges

   Permittivity of the Medium

   Coulomb's constant $k_e$ is defined in terms of the vacuum permittivity $\varepsilon_0$:

   $$k_e = \frac{1}{4\pi \varepsilon_0}$$

   where $\varepsilon_0 \approx 8.85 \times 10^{-12}\text{ C}^2/(\text{N}\cdot\text{m}^2)$. When charges are placed in an insulating (dielectric) medium rather than a vacuum, the net electrostatic force decreases by a factor equal to the relative permittivity (dielectric constant $\kappa$ or $\varepsilon_r$) of that medium:

   $$F = \frac{1}{4\pi \varepsilon_r \varepsilon_0} \frac{|q_1 q_2|}{r^2}$$

   Principle of Superposition

   When more than two charges are present, the net electrostatic force acting on any single charge is the vector sum of the individual forces exerted by all other charges independently:

   $$\vec{F}_{\text{net}} = \vec{F}_1 + \vec{F}_2 + \dots + \vec{F}_n$$