1. Concept Overview
Coulomb’s law tells us how strongly two point charges push or pull on each other. It is the electrostatic equivalent of Newton’s law of gravitation, but the electrical force can be much stronger. The law shows how the force depends on the size of the charges and how far apart they are.
2. Definition
Coulomb’s law states that the electric force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
In mathematical form:
\( F = k \dfrac{|q_1 q_2|}{r^2} \)
Here, \( k = 9 \times 10^9 \ \text{N m}^2 \text{C}^{-2} \).
3. Nature of the Electric Force
- The force acts along the line joining the two charges.
- It is repulsive when charges have the same sign.
- It is attractive when charges have opposite signs.
- The force is a vector quantity, meaning it has both magnitude and direction.
4. Vector Form of Coulomb’s Law
The force on charge \( q_1 \) due to charge \( q_2 \) is:
\( \vec{F}_{12} = k \dfrac{q_1 q_2}{r^2} \hat{r}_{12} \)
4.1. Meaning of Symbols
- \( \hat{r}_{12} \) is a unit vector from \( q_1 \) towards \( q_2 \).
- The direction automatically gives attraction or repulsion.
5. Superposition Principle and Coulomb’s Law
When more than two charges are present, the total force on any one charge is the vector sum of forces from all other charges. Coulomb’s law works perfectly with the superposition principle.
5.1. Expression
If charges \( q_1, q_2, q_3, ... \) exert forces \( \vec{F}_{21}, \vec{F}_{31}, ... \) on \( q_1 \), then:
\( \vec{F}_{\text{net}} = \vec{F}_{21} + \vec{F}_{31} + \cdots \)
6. Permittivity and the Constant k
The medium around the charges affects the force. The constant \( k \) depends on the permittivity of the medium, \( \varepsilon \).
6.1. Formula for k
\( k = \dfrac{1}{4 \pi \varepsilon} \)
In vacuum, permittivity is \( \varepsilon_0 = 8.85 \times 10^{-12} \ \text{C}^2 \text{N}^{-1} \text{m}^{-2} \).
6.2. Effect of Medium
A medium with higher permittivity reduces the electric force between charges.
7. Worked Examples
7.1. Example 1: Finding the Force
Two charges \( q_1 = 4 \times 10^{-6} \text{ C} \) and \( q_2 = 3 \times 10^{-6} \text{ C} \) are placed \( 0.2 \text{ m} \) apart. The force between them is:
\( F = k \dfrac{q_1 q_2}{r^2} = 9 \times 10^9 \times \dfrac{(4 \times 10^{-6})(3 \times 10^{-6})}{(0.2)^2} = 2.7 \text{ N} \)
Since both charges are positive, the force is repulsive.
7.2. Example 2: Opposite Charges
If one charge is negative, say \( q_2 = -3 \times 10^{-6} \text{ C} \), the magnitude stays the same, but the force becomes attractive.
8. Physical Interpretation
Coulomb’s law shows that electric force weakens very quickly with distance because of the \( 1/r^2 \) factor. This is why electrostatic effects are strong at short distances but fade at large distances. Charges act as if they create an invisible influence around them, pulling or pushing other charges depending on their sign and distance.