Understanding Coulomb's Law in Physics

Coulomb's law is a fundamental principle in electrostatics that quantifies the force between two stationary electric charges. It establishes the mathematical relationship between the magnitude of two point charges, their separation, and the force acting along the line joining them. This law is central to understanding forces in electric fields and is foundational for problems related to electric charges in physics.

Coulomb's Law Statement and Explanation

Coulomb's law states that the electrostatic force between two point charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between their centers. The force may be attractive or repulsive depending on the nature of the charges.

If two point charges $q_1$ and $q_2$ are separated by a distance $r$ in vacuum, the force $F$ between them is given by:

$F = k \dfrac{|q_1 q_2|}{r^2}$

Here, $k$ is the Coulomb constant, and the direction is along the straight line passing through both charges. The sign of the charges determines whether the force is attractive or repulsive.

Mathematical Form and Units in Coulomb's Law

The scalar form of Coulomb's law provides the magnitude of force without direction. In the SI system, the unit for charge is Coulomb (C), for distance is metre (m), and for force is Newton (N).

The vector form of Coulomb's law includes force direction:

$\vec{F} = k \dfrac{q_1 q_2}{r^2} \hat{r}$

Where $\hat{r}$ is the unit vector from one charge to the other. The value of $k$ in vacuum or air is $9.0 \times 10^9\ \text{N·m}^2/\text{C}^2$.

For further clarity on electric field properties, refer to Electric Field Lines Properties.

Derivation of Coulomb's Law and Proportionality

Coulomb's law is derived from experimental results using a torsion balance, as demonstrated by Charles-Augustin de Coulomb. The force varies directly with each charge and inversely with the square of their separation.

If either charge doubles, the force also doubles. If the distance is doubled, the force becomes one-fourth. Therefore, proportionality can be represented as:

$F \propto \dfrac{|q_1 q_2|}{r^2}$

The constant of proportionality $k$ depends on the medium between charges. In vacuum, $k = \dfrac{1}{4\pi \varepsilon_0}$, where $\varepsilon_0$ is the permittivity of free space.

Effect of Medium and Permittivity

The electrostatic force is affected by the nature of the intervening medium. In other materials, force is reduced by the material’s relative permittivity, also called the dielectric constant ($\varepsilon_r$).

The formula for force in a medium becomes:

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

In water or glass, the permittivity is much higher, resulting in a weaker force between the same charges compared to a vacuum or air.

For an overview of electrostatics and its key concepts, visit Electrostatics Overview.

Solved Example Using Coulomb's Law

Consider two point charges, $q_1 = +2 \times 10^{-6}$ C and $q_2 = -3 \times 10^{-6}$ C, separated by $r = 0.2$ m in air. The Coulomb constant $k = 9 \times 10^9$ N·m$^2$/C$^2$.

Using $F = k \dfrac{|q_1 q_2|}{r^2}$:

$F = \dfrac{9 \times 10^9 \times 2 \times 10^{-6} \times 3 \times 10^{-6}}{(0.2)^2}$

$F = \dfrac{9 \times 2 \times 3 \times 10^9 \times 10^{-12}}{0.04}$

$F = \dfrac{54 \times 10^{-3}}{0.04} = 1.35$ N$

The negative and positive signs show the force is attractive, acting along the line joining the charges.

Applications and Limits of Coulomb's Law

Coulomb's law applies accurately to stationary point charges and is essential for calculations in electrostatics. It explains interactions such as the bonding of ions, forces inside atoms, and the behaviors of static electricity in insulators.

For extended or continuous charge distributions, the law must be applied to each element of charge, and the resultant found by vector addition or integration.

For more details on electrostatic potential and capacitance, consult Electrostatic Potential and Capacitance.

Common Errors and Important Points in Exams

• Always use the square of distance, not just distance.
• Check the sign of charges for direction of force.
• Use vector form for multiple charges and directions.
• Apply the formula only to point or small charges.
• Convert all values to SI units before calculations.
• Adjust for the medium using the correct permittivity.

To review major formulas for JEE Main, refer to Revision Notes on Coulomb's Law.

Comparison with Newton's Law of Gravitation

Both Coulomb's law and Newton's law of gravitation are inverse square laws. However, Coulomb’s law describes the force between electric charges, which can be attractive or repulsive, while gravitation only describes attraction between masses.

Key Features and Summary of Coulomb’s Law

• Quantifies force between two point charges
• Depends on product of charges and inverse square of distance
• Force acts along the line joining charges
• Medium affects magnitude through permittivity
• Base for solving electrostatics problems

For an understanding of types of capacitors linked to Coulomb's law, review Capacitor Types and Formula.

Coulomb’s law is essential for analyzing electrostatic interactions, calculating forces in JEE Main problems, and provides a foundation for advanced concepts in physics.