Electric Charges and Fields

2.1 Core concepts

Electrostatics is the branch of physics that studies forces, fields and potentials arising from charges at rest. Almost every electrical phenomenon we encounter — a comb attracting bits of paper after running through dry hair, the spark when removing a sweater on a winter day, lightning during a thunderstorm — is electrostatic in origin (NCERT §1.1, p. 1–2). Two thousand years ago Greek philosophers noticed that amber rubbed with silk attracted small objects; the modern word "electricity" comes from elektron, the Greek for amber. The systematic study began with William Gilbert in 1600 and culminated in the precision experiments of Charles-Augustin de Coulomb in 1785.

Experiment shows that there are two kinds of electric charge — positive and negative (NCERT §1.2, p. 2–3). Conventionally, the charge developed on a glass rod rubbed with silk is positive, and that on a plastic rod rubbed with cat's fur is negative. Like charges repel (positive repels positive, negative repels negative) and unlike charges attract (positive attracts negative). The two kinds neutralise one another when combined in equal amounts, leaving a body electrically neutral overall — which is what most macroscopic matter is, because every proton's positive charge is balanced by an electron's negative charge.

Materials are broadly classified by how easily charge moves through them (NCERT §1.3, p. 3–4). Conductors (most metals, the human body, the Earth) allow charge to move freely; touching a charged body to a conductor disperses the charge throughout it. Insulators (glass, plastic, nylon, wood) do not allow charge to move easily; a charge deposited on one part of an insulator stays there. Semiconductors (silicon, germanium) have intermediate behaviour and are the basis of all modern electronics.

Three basic properties of electric charge (NCERT §1.4, p. 4–6). First, additivity: charges add algebraically like real numbers, with their signs included; if a body has charges +q₁, +q₂, −q₃, the net charge is q₁ + q₂ − q₃. Second, conservation: the total charge of an isolated system is constant in time. Charge cannot be created or destroyed, only transferred between bodies; even nuclear reactions and particle decays conserve total charge. Third, quantisation: electric charge always appears as integral multiples of the elementary charge e = 1.602192 × 10⁻¹⁹ C. Robert Millikan demonstrated this beautifully in his 1909–1912 oil-drop experiment by measuring the charge on isolated oil droplets and finding all values to be small integer multiples of e. At macroscopic scales the quantum is so small that charge appears continuous, but at the level of single electrons and ions it is sharply discrete.

Coulomb's law (NCERT §1.5, p. 6–9). Two point charges q₁ and q₂ separated by distance r in vacuum exert on each other a force of magnitude

F = k q₁ q₂ / r² = (1/4πε₀)(q₁ q₂ / r²) (NCERT Eq. 1.1, p. 6),

where k = 1/(4πε₀) ≈ 9 × 10⁹ N m² C⁻² and ε₀ = 8.854 × 10⁻¹² C² N⁻¹ m⁻² is the permittivity of free space. The force is along the line joining the two charges, repulsive if q₁q₂ > 0 and attractive if q₁q₂ < 0. In full vector form,

F₂₁ = (1/4πε₀)(q₁q₂/r²₂₁) r̂₂₁ (NCERT Eq. 1.3, p. 9),

where r̂₂₁ is the unit vector from charge 1 to charge 2. Coulomb's law manifestly obeys Newton's third law: F₁₂ = −F₂₁. It is valid down to subatomic distances (~10⁻¹⁰ m), where it underlies the structure of atoms and molecules.

The superposition principle (NCERT §1.6, p. 11–12) says that the force on a given charge due to a collection of other charges is the vector sum of the pairwise Coulomb forces, unaffected by the presence of the other charges. The deep simplicity of electrostatics is that two ingredients — Coulomb's law and superposition — suffice to predict every other result in the subject.

Electric field (NCERT §1.7, p. 14). Rather than thinking about forces between pairs of charges, it is often easier to think of each source charge as setting up a field in the space around it, and then asking what force this field exerts on any test charge placed at a given point. The electric field E at a point is defined as the force per unit positive test charge in the limit that the test charge goes to zero:

E(r) = lim_{q→0} F/q.

For a point charge Q the field at distance r is

E = (1/4πε₀)(Q/r²) r̂ (NCERT Eq. 1.6, p. 14),

directed radially outward for +Q and radially inward for −Q. SI unit of E is N C⁻¹ (equivalently V m⁻¹). For a system of charges, the total field at a point is the vector sum of the fields produced by each charge separately (NCERT §1.7.1, Eq. 1.10, p. 15).

Electric field lines (NCERT §1.8, p. 19–21). Field lines are a pictorial device introduced by Michael Faraday. They have four important properties: (i) the tangent to a field line at any point gives the direction of E; (ii) the density of lines through unit area perpendicular to them is proportional to |E|; (iii) lines start on positive charges and end on negative charges, or extend to infinity; and (iv) lines never cross, because E must have a unique direction at every point. Electrostatic field lines do not form closed loops — that property is reserved for magnetic field lines.

Electric flux (NCERT §1.9, p. 21–23). For a small area element ΔS (treated as a vector along the outward normal), the electric flux of E through it is

ΔΦ = E · ΔS = E ΔS cos θ.

For a finite surface S, Φ = ∫_S E · dS. Unit: N C⁻¹ m² (or V m). Geometrically, flux measures the number of field lines piercing the surface, with directionality.

Electric dipole (NCERT §1.10, p. 23–25). A pair of equal and opposite charges +q and −q separated by 2a is an electric dipole. The dipole moment

p = q × 2a, directed from −q to +q (NCERT Eq. 1.19, p. 24), unit C m.

The field of a short dipole (r ≫ a) falls off as 1/r³, much faster than a point charge's 1/r². On the axial line at distance r from the centre, E = 2p/(4πε₀ r³) along p̂. On the equatorial plane, E = −p/(4πε₀ r³), anti-parallel to p̂. The axial field is exactly twice the equatorial field in magnitude.

Dipole in a uniform external field (NCERT §1.11, p. 27). A dipole feels equal and opposite forces from a uniform E, so the net force is zero. But the two forces form a couple producing a torque

τ = p × E, magnitude pE sin θ,

which tries to align p with E. The torque vanishes when p is parallel (or anti-parallel) to E.

Continuous charge distributions (NCERT §1.12, p. 28). Real charges occupy macroscopic regions, so we work with charge densities: linear (λ = ΔQ/Δl, C m⁻¹) along thin wires; surface (σ = ΔQ/ΔS, C m⁻²) on thin sheets and conductors' surfaces; and volume (ρ = ΔQ/ΔV, C m⁻³) inside extended bodies. The field of a continuous distribution is found by integrating the contributions of infinitesimal charge elements, with superposition replacing the discrete sum by an integral.

Gauss's law (NCERT §1.13, p. 29–31). The net electric flux through any closed surface (a Gaussian surface) equals the net enclosed charge divided by ε₀:

∮ E · dS = q_enc / ε₀ (NCERT Eq. 1.31, p. 30).

This is one of Maxwell's four equations and a direct consequence of the inverse-square Coulomb law plus superposition. Two crucial features: the flux depends only on the enclosed charge, not on its location inside the surface or on the shape of the surface; and the E on the left-hand side is due to all charges (inside and outside), although the contribution of outside charges to the net flux is exactly zero. When a charge configuration has enough symmetry (spherical, cylindrical or planar), Gauss's law lets us compute E with almost no integration.

Three classic applications (NCERT §1.14, p. 33–36).

1. Infinite straight line charge of linear density λ. Use a coaxial cylindrical Gaussian surface of radius r and length l. By symmetry E is radial; only the curved side contributes flux: E × 2πrl = λl/ε₀ ⇒ E = λ/(2πε₀ r) (NCERT Eq. 1.32, p. 34).
2. Infinite plane sheet of surface density σ. Use a pillbox Gaussian surface straddling the sheet, with end caps of area A on either side. Both end caps contribute flux EA in the same outward sense: 2EA = σA/ε₀ ⇒ E = σ/(2ε₀), independent of distance from the sheet (NCERT Eq. 1.33, p. 34).
3. Uniformly charged thin spherical shell of total charge q. Concentric Gaussian sphere of radius r. For r ≥ R (outside the shell) the entire charge is enclosed: E = q/(4πε₀ r²) — the shell behaves as if all its charge were at the centre. For r < R (inside) the Gaussian sphere encloses no charge: E = 0 (NCERT Eqs. 1.34–1.35, p. 35–36). These three results are the staple of CUET problems and should be at the fingertips.

2.2 Definitions to memorise

2.3 Diagrams / processes to remember

• Fig. 1.1 (p. 2) — like-charges repel / unlike-charges attract between glass-silk and plastic-fur rods.
• Fig. 1.2 (p. 4) — gold-leaf electroscope to detect charge; the divergence of the leaves measures the amount of charge.
• Fig. 1.3 (p. 8) — geometry and direction of vector Coulomb force showing F₁₂ = −F₂₁ (Newton's third law).
• Fig. 1.5 (p. 11) — system of three / many charges showing superposition of pairwise forces.
• Fig. 1.12 and 1.13 (p. 19–20) — radial field lines of a point charge; the line density falls as 1/r² because the area through which lines pass grows as r².
• Fig. 1.14 (p. 21) — field-line patterns for single +q, single −q, two equal positive charges, and a dipole (+q, −q).
• Fig. 1.17 (p. 24) — geometry for the dipole field at an axial point and at an equatorial point.
• Fig. 1.19 (p. 27) — dipole in a uniform external field: equal opposite forces produce a torque τ = p × E.
• Fig. 1.22 (p. 29) — sphere of radius r around a point charge q used to derive Gauss's law Φ = q/ε₀.
• Fig. 1.26 (p. 33) — cylindrical Gaussian surface for an infinite line charge: only the curved side contributes (E × 2πrl = λl/ε₀).
• Fig. 1.27 (p. 34) — pillbox Gaussian surface for an infinite plane sheet: 2EA = σA/ε₀.
• Fig. 1.28 (p. 35) — concentric spherical Gaussian surfaces for a charged thin shell: field outside is q/(4πε₀ r²), zero inside.

2.4 Common confusions / NTA trap points

• Confusing the Coulomb constant k = 1/(4πε₀) ≈ 9 × 10⁹ N m² C⁻² with ε₀ itself (8.854 × 10⁻¹² C² N⁻¹ m⁻²); NTA loves to put both numerical values among the options.
• For the infinite plane sheet the field is σ/(2ε₀) and is independent of distance; students often write σ/ε₀ (which is the field outside a conductor's surface) and expect a 1/r-type fall-off.
• The field of a uniformly charged thin shell is zero strictly inside (r < R) but q/(4πε₀ r²) outside (r ≥ R); a typical distractor treats the field as q/(4πε₀ R²) at every interior point.
• Dipole field falls as 1/r³ (not 1/r²), and the axial field is exactly twice the equatorial field in magnitude and opposite in direction.
• In Gauss's law, q on the right side is only the enclosed charge, but E on the left is due to all charges (inside and outside the surface).
• The torque τ = p × E has magnitude pE sin θ; the net force on a dipole in a uniform field is zero — students often mark a non-zero net force.
• Two equal point charges of opposite sign at the same distance produce field-vectors that add at the midpoint (not cancel), giving twice each single-charge contribution.
• Charge quantisation does not mean charges can be any integer multiple of some unit — they must be integer multiples of the elementary charge e specifically.
• A test charge q used to define E must be taken in the limit q → 0, otherwise it disturbs the source distribution it is measuring.
• Field lines start on +q, end on −q, never cross, and never form closed loops in electrostatics.
• For an infinite line charge, E falls off as 1/r (not 1/r² as for a point charge).
• Don't confuse the term "uniformly charged" (constant σ or ρ) with "uniform field" (constant E).

2.5 Key formulas table