L-15: Electric Charge and Electric Field

Electric Charge and Electric Field — Electrostatics

Electrostatics is the branch of physics dealing with electric charges at rest — their properties, forces, fields, and the behaviour of surrounding space. Electrical energy powers nearly every modern device; this lesson lays the foundation for electricity and magnetism.

NIOS objectives: properties of charge; quantisation and conservation; Coulomb's law; electric field and field lines; dipole and dipole moment; Gauss's theorem; fields of point charge, line, shell, and plane sheet; Van de Graaff generator.

15.1 Frictional Electricity

Amber rubbed with fur attracted light objects — the word electric comes from Greek electron (amber). Rubbing a hard rubber rod with fur or a glass rod with silk produces charges that interact at a distance.

• Like charges repel; unlike charges attract.
• Franklin's convention: glass → positive; rubber → negative.
• Charging by conduction (touching) or induction (near + earth).

15.1.1 Conservation of Charge

When glass is rubbed with silk, the rod gains +q and silk gains −q of equal magnitude. Total charge of the isolated system is unchanged — charge is neither created nor destroyed, only transferred. Electrons move from glass to silk (or from fur to rubber).

15.1.2 Quantisation of Charge

Millikan (1909) showed charge always appears as an integral multiple of the fundamental charge e on an electron:

• Only two kinds of charge exist: positive and negative.
• Charge is conserved and quantised.
• Atom is neutral: equal electrons (−e) and protons (+e); neutron has no charge.

15.2 Coulomb's Law

Force between two stationary point charges q₁ and q₂ separated by distance r:

• ∝ product q₁q₂; ∝ 1/r² (inverse-square law, like gravitation).
• Along the line joining charges; repulsive for like, attractive for unlike.
• Valid for point charges only; acts at a distance.

One coulomb: two 1 C charges 1 m apart experience ~10¹⁰ N — enormous compared to everyday forces. Action-reaction: F₁₂ = −F₂₁.

15.2.2 Principle of Superposition

For multiple charges, net force on q₁ is the vector sum of individual Coulomb forces:

Example 15.2: Two 6.0×10⁻¹⁰ C charges 2.0 m apart → F = 81×10⁻¹¹ N.

15.3 Electric Field

Faraday introduced the electric field to explain action at a distance. Field at a point is force per unit positive test charge:

Superposition: E = E₁ + E₂ + E₃ + … = Σ k qᵢ/rᵢ² r̂ᵢ. At centroid of equilateral triangle with equal +q at corners, E = 0 by symmetry.

15.3.1 Electric Dipole

Two equal and opposite charges ±q separated by small distance 2l form a dipole (e.g. H₂O).

15.3.2 Dipole in Uniform Field

In uniform E, forces on ±q form a couple tending to align the dipole with E:

15.3.3 Electric Lines of Force (Field Lines)

Fictitious lines depicting field direction and strength. Tangent at any point gives E direction; density ∝ field strength.

• Start from +q radially outward to infinity; end on −q from infinity.
• Dipole: lines from +q terminate on −q.
• Two field lines never cross.
• Two equal +q: field zero at midpoint P.

15.4 Electric Flux and Gauss's Law

Electric flux through small area element: Δφ = E · Δs. For a closed Gaussian surface enclosing charge q:

15.4.1 Point Charge

15.4.2 Long Line Charge

Uniform linear charge density λ (C·m⁻¹). Cylindrical Gaussian surface of radius r, length l:

15.4.3 Uniformly Charged Spherical Shell

15.4.4 Infinite Plane Sheet of Charge

Uniform surface charge density σ (C·m⁻²). Cylindrical Gaussian surface piercing the sheet:

         GAUSS'S LAW — FIELD SUMMARY
         ============================
    Point charge     :  E = q / (4πε₀ r²)
    Infinite line    :  E = λ / (2πε₀ r)
    Shell outside    :  E = Q / (4πε₀ r²)
    Shell inside     :  E = 0
    Plane sheet      :  E = σ / (2ε₀)

15.5 Van de Graaff Generator

Electrostatic machine producing potentials of a few million volts (up to ~20 MV). Named after Robert J. van de Graaff.

• Large hollow metallic sphere S on insulating stand; rubber/silk belt on pulleys P₁ (motor-driven) and P₂ (at sphere centre).
• Comb C₁ at ~10⁴ V positive; comb C₂ connected to inner surface of sphere.
• Corona discharge near sharp needle points ionises air; belt carries + charge to C₂.
• Charge accumulates on outer surface of S; earthed chamber T at high pressure reduces leakage.
• Used to accelerate ion beams for nuclear reaction studies.

Quick Revision

• Q = Ne; charge conserved and quantised; e = 1.6×10⁻¹⁹ C.
• Coulomb: F = kq₁q₂/r²; k = 9×10⁹ N·m²·C⁻²; superposition for many charges.
• Field: E = F/q₀ = kq/r²; F = qE.
• Dipole: p = 2ql; end-on E = 2p/(4πε₀r³); broad-on E = p/(4πε₀r³); τ = pE sin θ.
• Gauss: Φ = q/ε₀; shell inside E = 0; sheet E = σ/(2ε₀).
• Van de Graaff: corona discharge + belt → megavolt sphere.