Lecture-3-Coulomb's Law

Key Concepts

• Coulomb's Law: Describes the electrostatic force between two point charges.
• Electrostatic Force (F): The force of attraction or repulsion between charged objects.
• Permittivity of Free Space (ε₀): A physical constant representing the ability of a vacuum to permit electric fields.
• Superposition: The principle that the total force on a charge due to multiple charges is the vector sum of the individual forces.
• Charge Density (ρ): The amount of electric charge per unit volume.
• Vector: A quantity with both magnitude and direction.
• Unit Vector: A vector with a magnitude of 1, indicating direction.
• SI Units: The International System of Units, a standard system of measurement.

Coulomb's Law: Introduction and Formulation

The lecture introduces Coulomb's Law as the first application of vectors in electromagnetics.

• Basic Principle: The force between two charges (Q1 and Q2) acts along the line joining them. The force on Q2 due to Q1 is equal and opposite to the force on Q1 due to Q2.
• Magnitude: The force magnitude is proportional to the product of the charges (Q1Q2) and inversely proportional to the square of the distance (r12²) between them.
• Direction: The force direction is given by the unit vector along the line joining the charges (r12̂).
• Equation: F = (1 / 4πε₀) * (Q1Q2 / r12²) * r12̂
• SI Units: The lecture emphasizes the use of SI units, where the proportionality constant is 1 / 4πε₀.
• Permittivity of Free Space (ε₀): ε₀ = 8.8 x 10⁻¹² Coulombs² per Newton meter².

Strength of Electrostatic Force: Feynman's Example

The lecture uses an example from Feynman's Lectures on Physics to illustrate the immense strength of the electrostatic force.

• Scenario: Two humans, each with a 1% excess of electrons.
• Assumptions: Human weight = 100 kg, density = 1 g/cm³, distance between humans = 1 meter.
• Calculation:
  • 1% excess electrons ≈ 10²⁶ electrons.
  • Force magnitude ≈ (1 / 4πε₀) * (10²⁶ * e)² / 1² Newtons, where e is the charge of an electron (approximately 1.6 x 10⁻¹⁹ Coulombs).
  • Result: Force ≈ 4 x 10²⁴ Newtons.
• Comparison: This force is strong enough to lift the Earth (mass ≈ 10²⁴ kg), highlighting the extraordinary strength of the electric force.

Comparison with Gravitational Force

The lecture compares the electrostatic and gravitational forces between two electrons.

• Gravitational Force (Fg): Fg = G * me² / d², where G is the gravitational constant (6.7 x 10⁻¹¹ Nm²/kg²) and me is the mass of the electron (9.1 x 10⁻³¹ kg).
• Electrostatic Force (Fe): Fe = (1 / 4πε₀) * e² / d².
• Ratio: Fe / Fg ≈ 10⁴³. The electric force is 43 orders of magnitude stronger than the gravitational force between two electrons.
• Protons: Even when considering protons (which are much heavier), the electric force is still significantly stronger (Fe / Fg ≈ 10³⁷).

Why Gravitation Dominates at Large Scales

The lecture addresses the apparent paradox of why the much weaker gravitational force dominates at large scales.

• Initial Argument and its Flaw: The initial argument is that equal numbers of electrons and protons cancel out electric forces. However, even with slight charge separation (d), the electric force only reduces by a factor of approximately d/r, where r is the distance of observation. This reduction is insufficient to explain gravitational dominance.
• Explanation based on Force Hierarchy: The true explanation lies in the hierarchy of forces: nuclear > electromagnetic > gravitational.
• Cancellation Mechanism: In systems with charged particles (e.g., nebulae), particles adjust their positions to minimize electric fields. This adjustment leads to an exponential decay of the Coulomb force (e^(-r/λ)), where λ is a characteristic length scale.
• Comparison of Decay Rates: Comparing the exponential decay of the electric force with the inverse square law of gravity explains why gravity dominates at large distances.
• Estimation of Crossover Point: The electric field becomes as weak as the gravitational field at around r/λ ≈ 80-90. Since λ is small for most materials (angstroms to micrometers), gravity takes over within centimeters.
• Key Difference: Gravitational force is always attractive, leading to cumulative effects at large scales, while electric forces can be both attractive and repulsive, leading to cancellation.

Force Scale Diagram

The lecture presents a diagram illustrating the scale lengths at which different forces dominate:

• Nuclear Forces: Dominate at 10⁻¹⁵ meters (nucleus formation).
• Electric Forces: Dominate at angstrom distances (chemistry, molecule formation, metals).
• Magnetic Forces: Dominate at galactic scale lengths (weaker than electric forces, e.g., Earth's ionosphere, solar system).
• Gravitational Forces: Dominate at the largest scales (universe formation).

Superposition Principle

The lecture introduces the superposition principle for calculating the force on a charge due to multiple charges.

• Principle: The total force on a charge is the vector sum of the individual forces due to each other charge, as if each charge were acting alone.
• Equation: F = (q / 4πε₀) * Σ(Qi * rî / ri²), where the summation is over all charges Qi.
• Validity: Superposition holds true when the presence of other charges does not affect the interaction between the charge in question and any individual charge.
• Accuracy: Coulomb's Law and the superposition principle have been verified to high precision (approximately 20 digits) in vacuum.

Continuous Charge Distributions

The lecture discusses how to calculate the force due to a continuous charge distribution.

• Problem: Summing over a large number of individual charges (e.g., 10²³) is impractical.
• Solution: Divide the charge distribution into small volume elements (boxes) of size Δ.
• Charge Density (ρ): Define charge density as the charge per unit volume (Q / Δ³).
• Volume Integral: Replace the summation over individual charges with a volume integral: F = (1 / 4πε₀) * ∫(ρ / r²) dV.

Conclusion

The lecture provides a detailed explanation of Coulomb's Law, its implications, and its limitations. It emphasizes the immense strength of the electrostatic force, explains why gravity dominates at large scales, introduces the superposition principle, and outlines the method for calculating forces due to continuous charge distributions. The lecture sets the stage for further exploration of electromagnetism by establishing a solid foundation in electrostatics.