10.1 Coulomb's Law

Coulomb's Law is a fundamental principle in physics that describes the electrostatic force of interaction between two stationary, electrically charged particles. Formulated by Charles-Augustin de Coulomb in the 18th century, this law is the cornerstone of electrostatics.

Statement: The force of attraction or repulsion between two 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.

If two point charges, $q_1$ and $q_2$, are separated by a distance $r$, the magnitude of the electrostatic force ($F$) between them is given by: $F=k\frac{|q_1{q}_2|}{r^2}$

The Formula and Constants

The relationship can be broken down:

• Force is proportional to the product of the charges: $F\propto |q_1{q}_2|$
• Force is inversely proportional to the square of the distance: $F\propto \frac{1}{r^2}$

The constant of proportionality, $k$, is known as Coulomb's constant. $k=\frac{1}{4\pi {\epsilon }_0}\approx 9\times 10^9\mathrm{N}\cdot {\mathrm{m}}^2/{\mathrm{C}}^2$ Here, ${\epsilon }_0$ is the permittivity of free space (${\epsilon }_0\approx 8.85\times 10^{-12}{\mathrm{C}}^2/(\mathrm{N}\cdot {\mathrm{m}}^2)$), a fundamental constant that describes how an electric field permeates a vacuum.

Vector Form of Coulomb's Law

Force is a vector, having both magnitude and direction. The force exerted on charge $q_1$ by charge $q_2$ is: ${\vec{F}}_{12}=k\frac{q_1{q}_2}{r^2}{\hat{r}}_{21}$ Where ${\hat{r}}_{21}$ is a unit vector pointing from $q_2$ to $q_1$. Similarly, the force on $q_2$ by $q_1$ is: ${\vec{F}}_{21}=k\frac{q_1{q}_2}{r^2}{\hat{r}}_{12}$ Since the unit vectors are in opposite directions (${\hat{r}}_{12}=-{\hat{r}}_{21}$), this demonstrates Newton's Third Law: ${\vec{F}}_{12}=-{\vec{F}}_{21}$ The forces are equal in magnitude and opposite in direction. This confirms that Coulomb's force is a mutual force.

Scalar And Vector Quantities→

Direction of the Force

• If the charges have the same sign (both positive or both negative), the product $q_1{q}_2$ is positive, and the force is repulsive.
• If the charges have opposite signs (one positive, one negative), the product $q_1{q}_2$ is negative, and the force is attractive.

Comparison with Gravitational Force

Coulomb's Law has a mathematical form very similar to Newton's Law of Universal Gravitation ($F=G\frac{m_1{m}_2}{r^2}$).

The Superposition Principle

If more than two charges are present, the net force on any single charge is the vector sum of the individual forces exerted on it by all the other charges. For a charge $q_1$ in the presence of charges $q_2,q_3,\dots ,q_n$, the total force is: ${\vec{F}}_{1,\text{total}}={\vec{F}}_{12}+{\vec{F}}_{13}+\dots +{\vec{F}}_{1n}$

Effect of Medium

When an insulating medium (dielectric) is placed between the charges, the electrostatic force decreases. The relative permittivity ${\epsilon }_r$ (also called the dielectric constant) of the medium is defined such that the force in the medium $F_{\text{med}}$ is: $F_{\text{med}}=\frac{1}{4\pi {\epsilon }_0{\epsilon }_r}\frac{q_1{q}_2}{r^2}=\frac{F_{\text{vac}}}{{\epsilon }_r}$

Coulomb's Law and Spherical Conductors

For a point outside a uniformly charged spherical conductor, the entire charge of the sphere may be treated as a point charge concentrated at its center. This means Coulomb's Law applies exactly with $r$ measured from the center of the sphere to the external point.