JEE Physics Power Batch Lecture 03: Coulomb's Law and Electrostatic Force Properties

Introduction: Initial overview of electrostatics.
Electric Charges: Fundamental concepts and definitions.
Basic Properties of Electric Charge: Quantization, conservation, and additivity.
Charging Methods: Friction, conduction, and induction.

Coulomb's Law: Definition, mathematical derivation, and limitations.
Properties of Coulomb's Law: Essential characteristics and physical implications.

Proton vs. Electron Charge Comparison [HCV]: - Question: The charge on a proton is $+1.6 \times 10^{-19}\,C$ and that on an electron is $-1.6 \times 10^{-19}\,C$ . Does it mean that the electron has a charge $3.2 \times 10^{-19}\,C$ less than the charge of a proton? - Conceptual Insight: The positive and negative signs represent the nature (polarity) of the charge, not algebraic value. Magnitude-wise, they are equal ( $1.6 \times 10^{-19}\,C$ ).
Charge on an $\alpha$ -particle [NCERT]: - An $\alpha$ -particle is a Helium nucleus expressed as ${}_{2}^{4}He^{2+}$ . - The charge of an $\alpha$ -particle is equivalent to $2$ protons: $q = +2e$ . - Calculation: $\text{Charge} = 2 \times 1.6 \times 10^{-19}\,C = 3.2 \times 10^{-19}\,C$ . - Options Provided: - A: $4.8 \times 10^{-19}\,C$ - B: $1.6 \times 10^{-19}\,C$ - C: $3.2 \times 10^{-19}\,C$ (Correct Answer) - D: $6.4 \times 10^{-19}\,C$

Definition: The magnitude of the electrostatic 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.
Mathematical Representation: - Consider two point charges $q_1$ and $q_2$ separated by a distance $r$ . - $F \propto q_1 q_2$ - $F \propto \frac{1}{r^2}$ - Combining these: $F = \frac{k q_1 q_2}{r^2}$
Scope of Application: This law is strictly applicable only to point charges.
Interaction Direction: - The force between charges acts along the line joining the centers of the charges. - $F_{12}$ is the force on charge $q_1$ due to $q_2$ . - $F_{21}$ is the force on charge $q_2$ due to $q_1$ . - Magnitude: $|F_{12}| = |F_{21}| = F$ . - Direction: They are always in opposite directions ( $F_{12} = -F_{21}$ ).

Constant of Proportionality ( $k$ ): - $k = \frac{1}{4 \pi \epsilon}$ , where $\epsilon$ is the permittivity of the medium.
In Vacuum (Free Space): - Permittivity of free space $\epsilon_0 = 8.85 \times 10^{-12}\,C^2 N^{-1} m^{-2}$ . - $k = \frac{1}{4 \pi \epsilon_0} = 9 \times 10^9\,N m^2 C^{-2}$ .
In a Medium: - Permittivity is defined as $\epsilon = \epsilon_0 \epsilon_r$ . - $\epsilon_r$ is the Relative Permittivity (also known as the Dielectric Constant).

Ratio Definition: $\epsilon_r = \frac{\epsilon}{\epsilon_0}$ .
Dimension: Since it is a ratio of similar quantities, relative permittivity is dimensionless.
Values for Common Media: - Vacuum: $\epsilon_r = 1$ - Air: $\epsilon_r \approx 1$ - Water: $\epsilon_r = 81$ - Metals/Conductors: $\epsilon_r = \infty$

Dimension of $k$ : - From $F = \frac{k q_1 q_2}{r^2}$ , we get $k = \frac{F r^2}{q^2}$ . - $[k] = \frac{[M L T^{-2}] [L^2]}{[A T]^2} = [M L^3 T^{-4} A^{-2}]$ .
Dimension of $\epsilon$ : - Since $k = \frac{1}{4 \pi \epsilon}$ , the dimensions of $\epsilon$ are the inverse of $k$ . - $[\epsilon] = [M^{-1} L^{-3} T^4 A^2]$ .

Force Reduction Principle: The electrostatic force between two charges decreases when they are placed in a medium other than vacuum.
Formula for Force in Medium ( $F_m$ ): - $F_m = \frac{1}{4 \pi \epsilon_0 \epsilon_r} \frac{q_1 q_2}{r^2}$ - $F_m = \frac{F_{vacuum}}{\epsilon_r}$
Key Observation: Since $\epsilon_r \ge 1$ , it follows that $F_m \le F_{vacuum}$ .
Example 1: If force in vacuum $F_v = 100\,N$ and medium relative permittivity $\epsilon_r = 20$ , then force in medium $F_m = \frac{100}{20} = 5\,N$ .
Example 2: - Given: Force in vacuum $F_v = 48\,N$ . - Question A: Find force in medium with $\epsilon_r = 3$ . - Calculation: $F_m = \frac{48}{3} = 16\,N$ . - Question B: If force in medium is found to be $12\,N$ , find the relative permittivity $\epsilon_r$ . - Calculation: $\epsilon_r = \frac{F_v}{F_m} = \frac{48}{12} = 4$ .

Independence Principle: The electrostatic force of attraction or repulsion between two specific charges is independent of the presence or absence of other surrounding charges. (Note: While individual pair forces remain constant, the net force on a charge may change).
Central Force Nature: - The force acting on one point charge due to another is always along the line joining these two charges. - It is equal in magnitude and opposite in direction regardless of the magnitudes of the charges (Newton's Third Law). - Electrostatic force is characterized as a central force.
Conservative Nature: - Electrostatic force is a conservative force. - The Work Done ( $W D$ ) by the electrostatic force in moving a point charge along any closed loop shape is zero.

The electrostatic force ( $\mathbf{F}$ ) is related to the Potential Energy ( $U$ ) via the negative gradient: - $\mathbf{F} = -\nabla U = - \left[ \frac{\partial U}{\partial x} \hat{i} + \frac{\partial U}{\partial y} \hat{j} + \frac{\partial U}{\partial z} \hat{k} \right]$
Mathematical Example: - Let $U = x^2 y^3 z$ . - Partial derivative with respect to $x$ : $\frac{\partial U}{\partial x} = 2 x y^3 z$ - Partial derivative with respect to $y$ : $\frac{\partial U}{\partial y} = 3 x^2 y^2 z$ - Partial derivative with respect to $z$ : $\frac{\partial U}{\partial z} = x^2 y^3$ - Thus, $\mathbf{F} = -(2 x y^3 z \hat{i} + 3 x^2 y^2 z \hat{j} + x^2 y^3 \hat{k})$ .