## **Statement**
*Coulomb's Law* states:
> *"The magnitude of the force 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."*
---
## **Introduction**
Coulomb's Law quantifies the force between two electric [[charge]]s. First established by **Charles Augustin de Coulomb** in **1785**, this law was derived through experiments using a **torsion balance**. It provides a mathematical description of the interaction between electric charges, forming a cornerstone of **electrostatics**.
---
## **Mathematical Form**
### **Diagram**:
![[Pasted image 20241130091128.png]]
- **Force $\vec{F}_{21}$**: Acts on charge $q_2$ due to $q_1$, directed along the unit vector $\hat{\boldsymbol{r}}_{12}$.
- **Force $\vec{F}_{12}$**: Acts on charge $q_1$ due to $q_2$, directed along $\hat{\boldsymbol{r}}_{21}$.
### **a) Magnitude**
The force $F$ between two charges $q_1$ and $q_2$ separated by a distance $r$ is expressed as:
$
F \propto q_1 q_2 \quad \text{(A)}
$
$
F \propto \frac{1}{r^2} \quad \text{(B)}
$
Combining (A) and (B):
$
F \propto \frac{q_1 q_2}{r^2}
$
or
$
F = k \frac{q_1 q_2}{r^2}
$
Where:
- **$k$** is the proportionality constant whose value depends on the units and medium.
- **$q_1$** and **$q_2$** are localized or point charges.
---
### **b) Direction**
The force between two charges acts along the line joining them:
1. For charge $q_2$ due to $q_1$, the force is:
$
\vec{F}_{21} = k \frac{q_1 q_2}{r^2} \hat{\boldsymbol{r}}_{12}
$
2. For charge $q_1$ due to $q_2$, the force is:
$
\vec{F}_{12} = k \frac{q_1 q_2}{r^2} \hat{\boldsymbol{r}}_{21}
$
Since $\hat{\boldsymbol{r}}_{12} = -\hat{\boldsymbol{r}}_{21}$:
$
\vec{F}_{21} = -\vec{F}_{12}
$
This implies the forces are equal in magnitude but opposite in direction.
---
### **c) Nature of Electrostatic Force**
- **Like charges ($q_1 q_2 > 0$)**: Experience a **repulsive** force.
- **Unlike charges ($q_1 q_2 < 0$)**: Experience an **attractive** force.
---
## **Coulomb's Law in Vacuum**
In a **vacuum**, the electrostatic force is strongest. The proportionality constant **$k$** is expressed in terms of the **permittivity of free space** ($\varepsilon_0$):
$
k = \frac{1}{4 \pi \varepsilon_0}
$
Thus:
$
F = \frac{1}{4 \pi \varepsilon_0} \frac{q_1 q_2}{r^2}
$
Where:
- **$\varepsilon_0 = 8.85 \times 10^{-12} \, \mathrm{C^2 \, N^{-1} \, m^{-2}}$**
- **$k = 9.0 \times 10^9 \, \mathrm{N \, m^2 \, C^{-2}}$**
---
## **Coulomb's Law in Material Media**
When a **medium** (insulator) is placed between charges, the force reduces. This is described as:
$
F_{\text{med}} = \frac{1}{4 \pi \varepsilon} \frac{q_1 q_2}{r^2}
$
Where **$\varepsilon$** is the permittivity of the medium.
---
## **Relative Permittivity or Dielectric Constant**
The **relative permittivity** (or **dielectric constant**) **$\varepsilon_r$** compares the permittivity of a medium with that of a vacuum:
$
\varepsilon_r = \frac{\varepsilon}{\varepsilon_0}
$
### **Key characteristics**:
- **Dimensionless**.
- Always **greater than 1** for dielectric materials.
---
## **Summary**
### **Key Points**
| **Concept** | **Explanation** |
|----------------------------|---------------------------------------------------------------------------------|
| **Coulomb's Law** | Describes the force between two charges based on their magnitudes and separation. |
| **Force** | Directly proportional to product of charges; inversely proportional to distance squared. |
| **Permittivity** | Affects the force; vacuum has the highest force magnitude. |
| **Nature of Force** | Attractive for opposite charges; repulsive for like charges. |
### **Real-World Application**:
Coulomb's Law is fundamental to understanding **electrostatic** interactions in physics and engineering. It underpins technologies like **capacitors**, **electrostatic sensors**, etc.
---
## **References**
