Learning Objectives
- Understand electric charge, its properties, and conservation
- State and apply Coulomb's law for electrostatic force
- Define electric field and calculate it for various charge distributions
- Understand electric flux and Gauss's law and its applications
- Learn about electric dipoles and their behaviour in uniform fields
Key Concepts
Electric Charge
Electric charge is a fundamental property of matter. It is quantized: q = ne (e = 1.6 × 10⁻¹⁹ C, n is integer). Charge is conserved -- the net charge of an isolated system remains constant. Charge is additive and invariant (does not depend on the frame of reference).
Methods of charging: Friction, conduction, and induction.
Coulomb's Law
F = kq₁q₂/r², where k = 1/(4πε₀) = 9 × 10⁹ N·m²/C²
ε₀ = 8.854 × 10⁻¹² C²/(N·m²) is the permittivity of free space.
In a medium: F = kq₁q₂/(κr²), where κ (or εᵣ) is the dielectric constant.
Coulomb's law obeys the superposition principle: the net force on a charge is the vector sum of forces due to all other charges.
Electric Field
E = F/q₀ (force per unit positive test charge). SI unit: N/C or V/m.
Due to a point charge: E = kq/r² (directed away from +q, toward -q).
Electric field lines: Start from positive charges, end on negative charges. They never cross. The density of lines indicates field strength. They are perpendicular to equipotential surfaces.
Electric Dipole
A pair of equal and opposite charges (+q and -q) separated by distance 2a.
Dipole moment: p = q × 2a (directed from -q to +q). SI unit: C·m.
Field on axial line: E = 2kp/r³ (for r >> a)
Field on equatorial line: E = kp/r³ (for r >> a), opposite to p
Torque on dipole in uniform field: τ = p × E, |τ| = pE sin θ
Potential energy: U = -p · E = -pE cos θ. Stable equilibrium at θ = 0°, unstable at θ = 180°.
Electric Flux and Gauss's Law
Electric flux: Φ = ∫E · dA = ∫E cos θ dA. SI unit: N·m²/C or V·m.
Gauss's Law: The total electric flux through a closed surface equals q_enclosed/ε₀.
Φ = q_enclosed/ε₀
Applications of Gauss's Law:
- Infinite line charge: E = λ/(2πε₀r), where λ is linear charge density
- Infinite plane sheet: E = σ/(2ε₀), where σ is surface charge density
- Uniformly charged sphere (outside): E = kQ/r² (like a point charge)
- Uniformly charged sphere (inside): E = kQr/R³ (increases linearly with r)
- Charged conducting sphere: E = 0 inside; E = kQ/r² outside
Summary
Electric charge is quantized, conserved, and additive. Coulomb's law gives the force between point charges following an inverse square law. The electric field is force per unit charge. Electric dipoles experience torque in uniform fields. Gauss's law relates flux through a closed surface to enclosed charge and simplifies field calculations for symmetric charge distributions.
Important Terms
- Coulomb (C): SI unit of charge; 1 C = 6.25 × 10¹⁸ electrons
- Electric Field: Region around a charge where another charge experiences a force
- Electric Flux: Number of electric field lines passing through a surface
- Dipole Moment: Product of charge and separation distance
- Dielectric Constant: Factor by which electric field is reduced in a medium
- Gauss's Law: Relates total flux to enclosed charge
Quick Revision
- F = kq₁q₂/r²; k = 9 × 10⁹ N·m²/C²
- E = kq/r² (point charge); E = 2kp/r³ (axial); E = kp/r³ (equatorial)
- τ = pE sin θ; U = -pE cos θ
- Gauss's law: Φ = q/ε₀
- Line charge: E = λ/2πε₀r; Plane sheet: E = σ/2ε₀
- Inside conductor: E = 0; charge resides on the surface