Electrostatics
Easy Overview
Electrostatics is the study of electric charges that are not moving — charges at rest. It might sound static, but the principles of electrostatics govern everything from the spark you get when you touch a metal doorknob after walking on a carpet, to how a photocopier works, to why lightning strikes. Understanding electrostatics is the foundation for everything else in electricity and magnetism. It all starts with electric charge. Charge is a fundamental property of matter. There are two types: positive and negative. Like charges repel, opposite charges attract. The unit of charge is the coulomb (C). One coulomb is enormous — it takes about 6.24 × 10¹⁸ electrons to make one coulomb. The charge on a single electron is e = 1.6 × 10⁻¹⁹ C. Charge is quantised — it always comes in multiples of e. Charge is also conserved. Coulomb's law gives the force between two point charges: F = k|q₁q₂|/r², where k = 1/(4πε₀) ≈ 9 × 10⁹ N·m²/C². The force is inversely proportional to the square of the distance — double the distance, force drops to one-quarter. The electric force between two protons is about 10³⁶ times stronger than the gravitational force between them. Electric field is the concept that makes it easier to think about forces without dealing with individual charges. An electric field exists in the space around a charge. For a point charge Q, E = kQ/r². Electric fields obey superposition — total field = vector sum of fields from individual charges. Gauss's law relates flux through a closed surface to enclosed charge, making it a powerful tool for symmetric charge distributions. Electric potential is the scalar counterpart — work done per unit charge to bring a charge from infinity. Potential is easier to work with because it is a scalar. Capacitors store charge and energy, and dielectrics increase their capacitance.
Electric Charge and Its Properties
Electric charge is a fundamental property. Like charges repel, unlike attract. Charge is quantised (Q = ±ne, e = 1.6 × 10⁻¹⁹ C) and conserved. Conductors allow free charge motion; insulators do not. Semiconductors have intermediate conductivity. Electrostatic induction: charges redistribute in a conductor near a charged object without charge transfer.
Coulomb's Law
Coulomb's law: F = k|q₁q₂|/r², where k = 1/(4πε₀) = 9 × 10⁹ N·m²/C². ε₀ = 8.85 × 10⁻¹² C²/N·m² is the permittivity of free space. Force acts along the line joining charges. The force is attractive for unlike charges, repulsive for like. It is an inverse-square law, like gravity. Valid only for point charges.
Electric Field and Electric Field Lines
Electric field E = F/q₀. For a point charge Q, E = kQ/r² r̂. Electric fields obey superposition. Field lines start at positive charges and end at negative. Tangent gives field direction; density gives field strength. Field lines never cross. They are perpendicular to equipotential surfaces and to conductor surfaces.
Electric Dipole and Dipole Moment
An electric dipole consists of +q and -q separated by 2a. Dipole moment p = q × 2a (from −q to +q). Axial field: E = 2kp/r³. Equatorial field: E = kp/r³. Dipole in uniform field experiences torque τ = p × E and has potential energy U = −p·E. In non-uniform field, it also experiences net force.
Gauss's Law
Gauss's law: ∮E·dA = Q_enclosed/ε₀. Electric flux Φ_E = ∮E·dA = ∮E cosθ dA. Useful for symmetric distributions: (1) Spherical: E = Q/(4πε₀r²) outside sphere. (2) Cylindrical: E = λ/(2πε₀r) for infinite line. (3) Planar: E = σ/(2ε₀) for infinite sheet. One of Maxwell's four equations.
Applications of Gauss's Law
Outside a charged spherical shell: E = Q/(4πε₀r²) — like point charge. Inside: E = 0. Inside a uniformly charged non-conducting sphere: E = ρr/(3ε₀). Infinite line charge: E = λ/(2πε₀r). Infinite sheet: E = σ/(2ε₀). Near a conductor surface: E = σ/ε₀.
Electric Potential and Potential Difference
Electric potential V = W/q₀ (work per unit charge from infinity). For a point charge: V = kQ/r. SI unit: volt (1 V = 1 J/C). Potential is scalar — easier than field. For multiple charges, V = V₁ + V₂ + V₃ + ... E = −dV/dr — field points in direction of steepest potential decrease. Equipotential surfaces: constant V, no work moving along them.
Potential Due to Different Charge Distributions
Point charge: V = kQ/r. Electric dipole: V = kp cosθ/r². Charged spherical shell: outside V = kQ/r, inside V = kQ/R (constant). Non-conducting sphere: inside V = kQ(3R² − r²)/(2R³). Work to assemble a system: U = ½Σ qᵢVᵢ.
Equipotential Surfaces and Relation Between E and V
Equipotential surfaces have constant potential. Properties: (1) No work moving charge along them. (2) E is perpendicular to them. (3) Closer spacing = stronger field. For a point charge, equipotentials are concentric spheres. Inside a conductor, the entire conductor is at constant potential — E = 0.
Conductors in Electrostatics
In electrostatic equilibrium: (1) E = 0 inside conductor. (2) Excess charge on surface. (3) E just outside = σ/ε₀, perpendicular to surface. (4) V constant throughout. (5) Charge accumulates at sharp points (lightning rods). Corona discharge at sharp points — principle of van de Graaff generator.
Capacitance and Capacitors
Capacitance C = Q/V. For an isolated sphere: C = 4πε₀R. Parallel-plate capacitor: C = ε₀A/d. Unit: farad (F). 1 F is enormous — typical capacitors are µF or pF. Series: 1/C_eq = 1/C₁ + 1/C₂ + ... Parallel: C_eq = C₁ + C₂ + ...
Energy Stored in a Capacitor
Energy stored U = Q²/(2C) = ½CV² = ½QV. Energy is stored in the electric field. Energy density u = ½ε₀E². For a parallel-plate capacitor, U = u × volume = (½ε₀E²)(Ad). Capacitors release energy much faster than batteries — used in camera flashes.
Dielectrics and Polarisation
Dielectrics increase capacitance by factor κ: C = κC₀ = κε₀A/d. κ is the dielectric constant. When placed in an electric field, molecules polarise — non-polar molecules develop induced dipoles, polar molecules align. Polarisation P = dipole moment per unit volume. D = ε₀E + P = κε₀E.
Dielectric Strength and Breakdown
Every dielectric has a maximum electric field it can withstand — dielectric strength. Air: 3 × 10⁶ V/m (3 kV/mm). Paper: κ ≈ 3.7, strength 16 kV/mm. Mica: κ ≈ 7, strength 150 kV/mm. Ceramic: κ ≈ 10-10000. Electrolytic capacitors use a thin oxide layer for high capacitance.
Van de Graaff Generator
A van de Graaff generator produces very high voltages (millions of volts). A motor-driven belt carries charge up inside a hollow metal sphere. Inside the sphere, a comb collects the charge and transfers it to the outer surface. The sphere's potential rises as charge accumulates. Used as particle accelerators for nuclear physics.
Key Points
- •Charge is quantised (Q = ±ne) and conserved. e = 1.6 × 10⁻¹⁹ C.
- •Coulomb's law: F = k|q₁q₂|/r², k = 9 × 10⁹ N·m²/C².
- •Electric field E = F/q₀. For point charge: E = kQ/r².
- •Electric dipole moment p = q × 2a. τ = p × E. U = −p·E.
- •Gauss's law: ∮E·dA = Q_enclosed/ε₀.
- •Electric potential V = kQ/r. E = −dV/dr.
- •Conductors: E = 0 inside, charge on surface, V constant.
- •Capacitance C = Q/V. Parallel-plate: C = ε₀A/d.
- •Energy in capacitor: U = ½CV². Energy density: u = ½ε₀E².
- •Dielectric: C = κC₀. κ = ε/ε₀ = 1 + χ_e.
Practice Questions
- State Gauss's law. Use it to derive E due to (a) infinite line charge, (b) infinite sheet, (c) spherical shell.
- Define electric potential. Derive potential due to a point charge. Four charges at corners of square of side 1 m: +2 µC, −3 µC, +4 µC, −5 µC. Find potential at centre.
- Explain electric dipole. Derive E on axial and equatorial lines. A dipole of moment 2 × 10⁻⁹ C·m at 60° to field of 10⁵ N/C. Find torque.
- What is a capacitor? Derive capacitance of parallel-plate capacitor with dielectric. A capacitor of 8 µF with air — capacitance when distance halved and κ = 4 introduced?
- Derive expression for energy stored in a capacitor. Show energy density = ½ε₀E². A 10 µF capacitor charged to 100 V. Find energy and energy density if plate separation = 1 mm.
- Explain behaviour of conductors in electrostatic equilibrium. Why is E = 0 inside? Why does charge reside on surface?
- What is an equipotential surface? State properties. Show that E is perpendicular to equipotential surfaces.
- Describe construction and working of van de Graaff generator.