Electric Charges and Fields — NEET Notes

Electric charge and its three basic properties

Long before there was an SI unit, Thales of Miletus noticed (around 600 BC) that amber rubbed with wool attracted bits of straw — and the Greek word for amber, elektron, gave us the word "electricity." Two kinds of charge exist: glass rubbed with silk acquires what Benjamin Franklin labelled positive; plastic rubbed with cat's fur acquires negative. Like charges repel, unlike charges attract. The mechanism is electron transfer — no new charge is ever created during rubbing; electrons simply migrate from the rod to the silk (or vice versa). NCERT crystallises three foundational properties of charge that NEET tests in almost every form.

At macroscopic scales the granularity is invisible — a 1 μC charge holds about 6 × 10¹² electrons, so adding or removing a few is undetectable and the charge appears continuous. At the microscopic scale (a few tens of electronic charges), quantisation matters and can be directly counted.

Conductors and insulators

Whether a substance can carry charge depends on the mobility of its electrons. In conductors — metals, the human body, the Earth itself — the outer electrons are only loosely bound and migrate freely through the lattice. Any charge placed on a conductor spreads rapidly over its outer surface (you will see why in the next chapter on potential). In insulators — glass, porcelain, plastic, nylon, dry wood — electrons are tightly tied to their parent atoms; charge deposited on the surface stays where it was placed. This single difference explains a familiar puzzle: a plastic comb rubbed against dry hair becomes electrified and lifts paper shreds, but a metal spoon rubbed identically does not — the metal's charge leaks through your hand to the ground the moment it appears. Hold the metal rod by an insulating handle and the same rod will, in fact, retain its charge.

A third intermediate class — semiconductors (silicon, germanium) — sits between the two, with resistance higher than metals but lower than glass. NEET rarely tests semiconductors at this chapter level; they appear in the Semiconductor Electronics chapter at the end of Class 12.

Charging by induction

You can charge a body without ever touching it. Hold a positively charged rod near (but not in contact with) an isolated metal sphere mounted on an insulating stand. The free electrons inside the sphere migrate towards the rod, leaving the far side electron-deficient. Now ground the far side momentarily — electrons flow up from the earth to neutralise the positive face, while the rod still holds the near-side electrons in place. Disconnect the ground first, then remove the rod. The sphere is left with a net negative charge — the opposite sign of the inducing rod. The rod itself has lost nothing. This is charging by induction, and it works because the conductor's electrons can move freely while the charging rod's charges cannot.

Coulomb's law

Coulomb's law is the quantitative heart of electrostatics. Two point charges q₁ and q₂ separated by a distance r in vacuum exert a force on each other along the line joining them. The magnitude is

F = k q₁q₂ / r²     k = 1/(4πε₀)

Coulomb's law in vacuum — the inverse-square heart of electrostatics

The force is directly proportional to the product of the magnitudes of the two charges and inversely proportional to the square of their separation. The proportionality constant k = 1/(4πε₀) has the value 8.99 × 10⁹ N·m²/C² in vacuum, conventionally rounded in NEET working to 9 × 10⁹. The constant ε₀ is the permittivity of free space, equal to 8.854 × 10⁻¹² C²·N⁻¹·m⁻². In vector form, the force on q₂ due to q₁ is F₂₁ = (1/4πε₀)(q₁q₂/r²) r̂₂₁ — automatically capturing attraction (opposite signs give a force along the inward-pointing unit vector) and repulsion (like signs along the outward unit vector). The law obeys Newton's third law: F₁₂ = −F₂₁.

Coulomb's law was first established at the macroscopic scale; it has since been verified all the way down to atomic-nuclear separations of about 10⁻¹⁰ m. The strength is staggering: between two protons at any distance, the electrostatic repulsion is about 10³⁶ times stronger than their gravitational attraction. The reason gravity dominates astrophysics is simply that on large scales positive and negative charges cancel almost perfectly, while mass — being always positive — never does.

Forces between multiple charges — the superposition principle

Coulomb's law fixes the force between just two charges. What if there are three, ten, a hundred? Experimentally, the answer is the simplest possible: the principle of superposition. The total electrostatic force on any one charge is the vector sum of the Coulomb forces exerted on it by every other charge taken one at a time. Crucially, each pair-wise Coulomb interaction is unaffected by the presence of the other charges — there is no "screening" of one pair by a third charge sitting nearby. The force F₁ on charge q₁ in a system of n charges is therefore

F₁ = (q₁/4πε₀) Σᵢ (qᵢ / r₁ᵢ²) r̂₁ᵢ

Superposition — Coulomb's law summed over all other charges

Two consequences worth keeping in mind. First, a charge placed at the centroid of an equilateral triangle of three identical charges experiences zero net force — the three vectors cancel by symmetry. Second, all of electrostatics — every field calculation, every dipole derivation, even Gauss's law in its derivation from first principles — is just Coulomb's law plus superposition, applied to a continuum of charges.

The electric field and its lines

It is awkward to talk about forces between charges that may be far apart, or to keep recalculating the force whenever the "test charge" we drop into the region changes. The electric field E at a point is defined as the force that a unit positive test charge would experience at that point:

E = F / qₜ     (test charge qₜ → 0)

Definition of the electric field — force per unit positive test charge

The field is a vector. Its magnitude is N/C (or, equivalently, V/m), and its direction is the direction of force on a positive test charge. For a single point charge Q at the origin, the field at distance r is E = kQ/r² along r̂. For a system of charges, the field at any point is the vector sum of the fields each charge would produce there alone — superposition again. The test charge is taken in the limit qₜ → 0 so its own field does not disturb the source charges; in practice, we treat it as vanishingly small.

Electric field lines are an old visualisation due to Faraday: imaginary curves drawn so that the tangent at every point gives the direction of E there, and the density of lines indicates the magnitude. Lines start from positive charges, end on negative charges, and never cross. NEET tests the rules constantly through diagram-based MCQs.

Electric flux

Imagine the electric field as the velocity field of a fictitious fluid streaming through space. The electric flux through a surface measures how much of this "field-fluid" pierces the surface. For a tiny patch of area dA, with outward unit normal n̂, the elementary flux is dΦ = E · dA, where dA = n̂ dA. Summing — or rather integrating — over the whole surface gives

Φₑ = ∫ₛ E · dA = ∫ₛ E cosθ dA

Electric flux through a surface — scalar, sign depends on orientation

Flux is a scalar; its sign depends on whether E points along or against the outward normal n̂. Its SI unit is N·m²/C (or V·m). The angle θ in the dot product is the angle between E and the outward normal — when E is parallel to the surface (θ = 90°), the flux is zero; when E is perpendicular and outward (θ = 0°), the flux is maximum and positive. For a closed surface, the convention is that n̂ always points outward, so positive flux means net outflow.

The electric dipole and its field

An electric dipole is the simplest non-trivial charge configuration: two equal and opposite charges +q and −q separated by a fixed distance 2a. Despite having zero net charge, a dipole sets up a distinctive non-zero field — and a great deal of chemistry (polar molecules like HCl, H₂O) and biology (membrane potentials) is dipole physics in disguise. The dipole moment is

p = q × 2a    (vector, directed from −q to +q)

Electric dipole moment — magnitude q · separation, units C·m

The dipole moment is a vector quantity. In physics convention (NCERT, NEET), p points from the negative charge to the positive charge. Its SI unit is the coulomb-metre (C·m). The dipole's field at a far-away point depends on where you measure it relative to the dipole's axis. Two lines have particularly clean answers — the axial line (the line through both charges) and the equatorial line (the perpendicular bisector of the dipole). For a short dipole (r ≫ a):

The general formula for a short dipole at a point making angle θ with the axis is E = (kp/r³) √(1 + 3cos²θ), which collapses to the two formulae above at θ = 0° (axial) and θ = 90° (equatorial). The 1/r³ dependence is the dipole's signature — a single point charge falls as 1/r², a dipole falls one power faster because the two charges' contributions partially cancel.

Dipole in a uniform external field

Place a dipole in a uniform external electric field E. Each charge feels equal and opposite forces — +qE on the positive charge, −qE on the negative — so the net force is zero. The dipole does not translate. But the two forces form a couple, and the couple produces a torque that tries to align p with E.

τ = p × E    (magnitude pE sinθ)

Torque on a dipole in a uniform field

The torque is maximum (τ = pE) when p is perpendicular to E and zero when p is parallel (stable equilibrium) or anti-parallel (unstable equilibrium) to E. Rotating the dipole against this torque stores energy as electrostatic potential energy:

U = − p · E = − pE cosθ

Dipole potential energy in a uniform field

U is minimum (U = −pE) when θ = 0 — the aligned position is the most stable. U is maximum (U = +pE) when θ = 180° — anti-aligned is unstable. NEET 2021 used exactly this: a dipole anti-parallel to E (θ = 180°, U = +pE) in a non-uniform field experiences a net force pushing it towards the weaker-field region, reducing its potential energy as it moves.

If the field is non-uniform, the forces on the two charges no longer cancel exactly — there is a net translational force in addition to the torque. This is why a charged comb attracts (uncharged) paper bits: the comb induces a dipole in the paper, and its non-uniform field then pulls that dipole towards itself.

Continuous charge distribution

Real charged bodies — a wire, a sheet, a sphere — don't consist of a handful of point charges. They have charge spread continuously. To handle them, we replace the discrete sum from Coulomb's law with an integral, and define three densities depending on geometry. Linear charge density λ = dq/dl (units C/m) is used for charged wires; surface charge density σ = dq/dA (units C/m²) for sheets and conductor surfaces; volume charge density ρ = dq/dV (units C/m³) for charged volumes. The field at a point P due to a continuous distribution is the vector integral E(P) = (1/4πε₀) ∫ (dq/r²) r̂, where the integration runs over the entire distribution. In principle this works for any geometry; in practice, doing the integral is often punishing — which is precisely the motivation for Gauss's law.

Gauss's law and its applications

Gauss's law is the second great organising principle of electrostatics, after Coulomb's law itself. It states that the total electric flux through any closed surface in vacuum equals the net charge enclosed divided by ε₀:

Φₑ = ∮ₛ E · dA = q₄ₐₐ / ε₀

Gauss's law — flux through a closed surface depends only on enclosed charge

The closed surface is called a Gaussian surface; it is purely a mathematical construction, not a physical boundary. Three features make Gauss's law astonishingly useful. First, the flux depends only on the net charge inside — the positions of those charges, and the entire universe of charges outside the surface, do not matter. Second, when the charge distribution has high symmetry (line, plane, sphere), you can choose a Gaussian surface that lets you pull E outside the integral and reduce the law to a one-line algebraic equation. Third, Gauss's law and Coulomb's law are logically equivalent in electrostatics — each can be derived from the other.

Application 1 — Infinitely long straight wire

A wire with uniform linear charge density λ. Choose a coaxial cylindrical Gaussian surface of radius r and length L. By symmetry, E is radial, with the same magnitude everywhere on the curved face and zero flux through the flat end-caps (E parallel to those surfaces). Flux through the cylinder is E · (2πrL). Enclosed charge is λL. Equating: E (2πrL) = λL / ε₀, giving

E = λ / (2πε₀ r)

Infinite line charge — field falls as 1/r, not 1/r²

Note the 1/r dependence — different from a point charge's 1/r². The reason: for a line, the symmetry distributes the flux over a cylindrical surface (area ∝ r), not a spherical surface (area ∝ r²).

Application 2 — Infinite uniformly charged plane sheet

A sheet with uniform surface charge density σ. Choose a Gaussian pillbox — a short cylinder with its axis perpendicular to the sheet, end-caps of area A on either side. By symmetry, E is perpendicular to the sheet and equal in magnitude on both faces; flux through the curved side is zero. Total flux: 2EA. Enclosed charge: σA. So E (2A) = σA / ε₀:

E = σ / (2ε₀)

Infinite plane sheet — field is uniform and independent of distance

The remarkable feature: E does not depend on the distance from the sheet at all. Move closer, move further — the field stays the same. (For a conducting plate with surface density σ on each face, the result becomes E = σ/ε₀ because all the field is pushed to one side.)

Application 3 — Uniformly charged thin spherical shell

A spherical shell of radius R carrying total charge q distributed uniformly. By spherical symmetry, the field outside is radial and depends only on the distance from the centre. Outside the shell (r > R): pick a Gaussian sphere of radius r; flux is E (4πr²), enclosed charge is q, giving E = q / (4πε₀ r²) — exactly the field of a point charge q sitting at the centre. Inside the shell (r < R): the Gaussian sphere encloses zero charge, so E = 0 everywhere inside.