NEET PhysicsNCERT Class 12Chapter 1

Electric Charges and Fields: NEET notes

This chapter is electrostatics: charges at rest and the forces and fields they produce. It starts with what charge is and how bodies get it, sets out its three properties (additivity, conservation, quantisation), states Coulomb's law and the superposition principle, and then builds the electric field, field lines, electric flux, the electric dipole and Gauss's law, which it uses to find the field of a long wire, a flat sheet and a spherical shell.

What NEET asks

NEET asks for Coulomb-force ratios when charge or distance changes, the net force or field from several charges by vector addition, the dipole field on the axis and the equator, torque on a dipole, flux through a closed surface from Gauss's law, and the fields of a line, sheet and shell. Marks are lost by forgetting that force and field are vectors, by using 1/r² for a dipole, and by counting charges outside a Gaussian surface in q_enclosed.

1. Electric charge, conductors and insulators

NCERT §1.2 and §1.3

Around 600 BC Thales of Miletus noticed that amber rubbed with wool or silk attracts light objects; the word electricity comes from the Greek elektron, meaning amber.
Rubbing experiments show only two kinds of charge: like charges repel and unlike charges attract. The property that tells the two kinds apart is called the polarity of charge.
Benjamin Franklin named the two kinds positive and negative. By convention a glass rod rubbed with silk is positive (the silk is negative), and a plastic rod rubbed with fur is negative (the fur is positive).
Charging means an excess or a deficit of electrons: when glass is rubbed with silk, some electrons pass from the rod to the silk, so the rod is left positive and the silk negative. No charge is created, and only a tiny fraction of the body's electrons move.
A gold-leaf electroscope detects charge: charge given to its metal knob spreads to two thin leaves, which repel and spread apart; the more the divergence, the more the charge.
Conductors (metals, the human and animal body, the earth) contain charges that can move freely; insulators (glass, porcelain, plastic, nylon, wood) do not. Semiconductors lie in between.
Charge given to a conductor spreads over its whole surface, while charge placed on an insulator stays where it was put.
A metal spoon held in the hand shows no charge after rubbing because the charge leaks through the body to the earth; given an insulating handle, it can be charged.

2. Basic properties of electric charge

NCERT §1.4

A charged body whose size is very small compared with the distances involved is treated as a point charge, with all its charge placed at one point.
Additivity: charge is a scalar, so a system's net charge is found by adding its charges as signed numbers. Unlike mass, charge can be negative.
Conservation: the total charge of an isolated system never changes. Charge only moves from one body to another; charged particles may be created or destroyed, but always in equal and opposite amounts, as when a neutron turns into a proton and an electron.
Quantisation: every free charge is a whole-number multiple of the basic charge e, q = ne with n = 0, ±1, ±2, …. An electron carries −e and a proton +e.
Quantisation was first suggested by Faraday's laws of electrolysis and shown experimentally by Millikan in 1912.
The SI unit of charge is the coulomb (C), the charge carried by a current of 1 A in 1 s. The basic charge is e = 1.602192 × 10⁻¹⁹ C, so −1 C is carried by about 6 × 10¹⁸ electrons.
Because 1 μC already holds about 10¹³ electronic charges, quantisation has no practical effect on large-scale charges and they can be treated as continuous; it matters only for charges of a few tens or hundreds of e.

3. Coulomb's law

NCERT §1.5

Coulomb's law: the force between two point charges at rest is proportional to the product of the charges, inversely proportional to the square of the distance between them, and acts along the line joining them: F = k|q₁q₂|/r².
Coulomb measured the force with a torsion balance (arriving at the law in 1785). He did not know the charges; he halved a sphere's charge by touching it to an identical uncharged sphere and compared forces.
In SI units k = 1/(4πε₀) ≈ 9 × 10⁹ N m² C⁻², where ε₀ = 8.854 × 10⁻¹² C² N⁻¹ m⁻² is the permittivity of free space. Two 1 C charges 1 m apart would repel with 9 × 10⁹ N, so the coulomb is a very large unit; μC and nC are used in practice.
In vector form F₂₁ = kq₁q₂ r̂₂₁/r₂₁², with r̂₂₁ pointing from q₁ to q₂. The same formula covers like charges (repulsion) and unlike charges (attraction), through the sign of q₁q₂.
F₁₂ = −F₂₁: the two charges pull or push each other equally and oppositely, as Newton's third law requires.
The law has been checked from laboratory distances down to about 10⁻¹⁰ m. It is the force in vacuum; matter between the charges changes the situation.
Electric forces dwarf gravity: between an electron and a proton the ratio of electric to gravitational force is about 2.4 × 10³⁹, and between two protons about 1.3 × 10³⁶.

4. Forces between multiple charges

NCERT §1.6

Principle of superposition: the force on a charge due to several others is the vector sum of the forces each would exert alone.
The force between any two charges is unaffected by the presence of other charges; each pair obeys Coulomb's law as if it were alone.
F₁ = F₁₂ + F₁₃ + … + F₁ₙ, added by the parallelogram (or component) method, never by adding magnitudes.
All of electrostatics follows from two ideas: Coulomb's law and the superposition principle.
Symmetry often gives the answer: equal charges at the corners of an equilateral triangle exert zero net force on a charge at its centroid.
For any set of charges held only by their mutual forces, the forces on all of them add to zero, because Coulomb forces come in equal and opposite pairs.

5. Electric field

NCERT §1.7

A charge Q sets up an electric field in the space around it; a charge q placed at a point feels a force because of the field there: F = qE.
The electric field at a point is the force per unit positive test charge, E = lim (q→0) F/q. The test charge is made very small so that it does not disturb the source charges. The SI unit is N C⁻¹ (equivalently V m⁻¹).
The field of a point charge Q is E = kQ/r² along r̂: radially outward for positive Q, inward for negative Q, and the same in magnitude at all points at the same distance (spherical symmetry).
E depends on the source charges and the position, not on the test charge, since F is proportional to q.
For several source charges, add the field each one makes on its own, as vectors: E = E₁ + E₂ + … + Eₙ.
A field is more than a bookkeeping device: when charges accelerate, their influence spreads at the speed of light c, and the field (strictly the electromagnetic field) carries the effect and energy across the delay. The idea of the field was introduced by Faraday.
In a uniform field a charged particle has constant acceleration qE/m. The same field gives an electron an acceleration about 1800 times that of a proton (the ratio of their masses), so unlike free fall under gravity, the time to cover a distance depends on the mass; gravity is negligible beside such forces.

6. Electric field lines

NCERT §1.8

A field line is a curve whose tangent at every point gives the direction of the net field there; an arrow on it shows which way. Field lines are curves in three dimensions.
The relative closeness of lines shows the relative strength of the field: lines crowd where E is strong and spread out where it is weak. In a uniform field they are equally spaced parallel straight lines.
For a point charge the number of lines through a sphere around it is the same at every radius, while the area grows as r², which is the picture behind E ∝ 1/r². (Solid angle ΔΩ = ΔS/r².)
Lines begin on + charges and finish on − charges; around a lone charge they run out to, or in from, infinity.
In a charge-free region field lines are continuous curves with no breaks.
Two field lines never cross, because at the crossing point the field would have two directions.
Electrostatic field lines never form closed loops, because the electrostatic field is conservative.
Faraday invented the picture and called them lines of force; the number drawn is a choice, only their relative density means anything.

7. Electric flux

NCERT §1.9

Electric flux through a small area element is Δφ = E·ΔS = E ΔS cos θ, where θ is the angle between E and the area vector ΔS. It is proportional to the number of field lines crossing the element.
An area is a vector: its magnitude is the area and its direction is along the normal to the surface. For a closed surface the outward normal is always used.
Flux is largest when the surface faces the field squarely (θ = 0), and zero when the field lines run along the surface (θ = 90°).
Flux is a scalar. Its SI unit is N m² C⁻¹ (equivalently V m).
The total flux through a surface is the sum of E·ΔS over all its small elements, exact in the limit of vanishing elements (an integral).
Unlike water through a ring, nothing physically flows in electric flux; the name comes from the analogy.

8. Electric dipole

NCERT §1.10

An electric dipole is two charges +q and −q, equal in size, held a distance 2a apart. Its dipole moment is p = q × 2a, a vector pointing from −q to +q. The SI unit is C m.
Its total charge is zero, but its field is not: the fields of +q and −q do not cancel exactly, and far away the dipole field falls off as 1/r³, faster than the 1/r² of a single charge.
On the axis at distance r from the centre: E = 2kp/r³ (r >> a), directed along p. The exact expression is E = k·4qar/(r² − a²)².
On the equatorial plane at distance r: E = kp/r³ (r >> a), directed opposite to p. The exact expression is E = k·2qa/(r² + a²)^(3/2).
At the same large distance, the axial field is twice the equatorial field.
Far fields depend on q and a only through the product p. A point dipole is the limit 2a → 0 with p kept finite; for it the 1/r³ formulas are exact.
Polar molecules such as H₂O have a permanent dipole moment because their centres of positive and negative charge do not coincide; CO₂ and CH₄ have zero dipole moment but acquire an induced one in an applied field.

9. Dipole in a uniform external field

NCERT §1.11

In a uniform field the forces qE and −qE on the two charges are equal and opposite, so the net force on a dipole is zero.
The two forces act at different points and form a couple: torque τ = p × E, of magnitude τ = pE sin θ, where θ is the angle between p and E.
The torque turns the dipole towards alignment with E. It is maximum (pE) at θ = 90° and zero when p is parallel or antiparallel to E.
In a non-uniform field there is also a net force. With p parallel to E the dipole is pulled towards the stronger field; with p antiparallel it is pushed towards the weaker field.
A comb run through dry hair attracts uncharged bits of paper: the comb's field polarises the paper, inducing a dipole along the field, and because the comb's field is non-uniform the paper is pulled towards the comb.

10. Continuous charge distribution

NCERT §1.12

When charge is spread over a wire, a surface or a volume, it is described by a density rather than by listing charges.
Linear charge density λ = ΔQ/Δl (C m⁻¹), surface charge density σ = ΔQ/ΔS (C m⁻²) and volume charge density ρ = ΔQ/ΔV (C m⁻³).
Each element Δl, ΔS or ΔV is small on the everyday scale but still contains a very large number of charged particles, so the density is a smoothed average that ignores quantisation, like the density of a liquid.
The field of a continuous distribution is found the same way as for point charges: divide it into elements, find each element's field from Coulomb's law, and add them as vectors (an integral in the limit).
Coulomb's law with superposition gives the field of any distribution, discrete, continuous or mixed.

11. Gauss's law

NCERT §1.13

For a point charge q at the centre of a sphere of radius r, E is the same everywhere on the sphere and along the normal, so the flux is E × 4πr² = q/ε₀, whatever the radius.
Gauss's law: the total electric flux through any closed surface equals 1/ε₀ times the net charge enclosed, φ = ∮E·dS = q_enc/ε₀.
It holds for a closed surface of any shape or size; the enclosed charges may lie anywhere inside it.
q_enc counts only charges inside the surface, but the field E on the surface is produced by all charges, inside and outside.
If a closed surface encloses no net charge, the total flux through it is zero, as for a closed cylinder in a uniform field: the flux in through one end equals the flux out through the other, and none crosses the curved side.
The chosen surface is the Gaussian surface. It must not pass through a point charge (the field is undefined there), though it may pass through a continuous distribution.
Gauss's law rests on the inverse-square form of Coulomb's law; it is most useful for finding E when the charge has symmetry.

12. Applications of Gauss's law

NCERT §1.14

Infinitely long straight wire with uniform λ: by symmetry the field is radial and depends only on r. A coaxial cylinder of length l gives E × 2πrl = λl/ε₀, so E = λ/(2πε₀r), falling as 1/r.
Infinite plane sheet with uniform σ: the field is normal to the sheet on both sides. A box straddling the sheet gives 2EA = σA/ε₀, so E = σ/(2ε₀), the same at every distance from the sheet.
For a large finite sheet, E = σ/(2ε₀) holds well in the middle region away from the edges; for a long finite wire, λ/(2πε₀r) holds near its middle.
Uniformly charged thin spherical shell, outside (r > R): E = q/(4πε₀r²), as if the whole charge q = 4πR²σ were at the centre.
Inside the shell (r < R): the Gaussian sphere encloses no charge, so E = 0 everywhere inside. Experiments confirming this confirm the 1/r² law.
A solid sphere with uniform volume charge also has, outside it, the field of a point charge at its centre.
In each case the field on the Gaussian surface comes from the whole distribution, even though only the enclosed part appears in the law; the answers rely on the symmetry of an infinite wire or sheet.

Must-know facts

Glass rubbed with silk is positive; plastic rubbed with fur is negative. Like charges repel, unlike attract.
Charging by rubbing moves electrons; no charge is created.
Charge is additive (scalar), conserved, and quantised: q = ne, e = 1.6 × 10⁻¹⁹ C.
About 6 × 10¹⁸ electrons make up 1 C; 1 μC holds about 10¹³ electronic charges.
Coulomb's law F = kq₁q₂/r², k = 1/(4πε₀) ≈ 9 × 10⁹ N m² C⁻², ε₀ = 8.854 × 10⁻¹² C² N⁻¹ m⁻².
Electric to gravitational force, electron and proton: about 2.4 × 10³⁹.
Superposition: forces and fields from several charges add as vectors.
E = F/q, unit N C⁻¹ = V m⁻¹; point charge E = kQ/r², outward for +Q.
Field lines: start on +, end on −, never cross, never form closed loops, crowd where E is strong.
Flux φ = E·S = ES cos θ, unit N m² C⁻¹; outward normal for closed surfaces.
Dipole moment p = q × 2a, from −q to +q, unit C m.
Dipole far field: axial 2kp/r³ along p; equatorial kp/r³ opposite to p; falls as 1/r³.
Dipole in uniform field: net force zero, torque τ = p × E, τ = pE sin θ.
Non-uniform field: dipole parallel to E moves towards stronger field.
λ in C m⁻¹, σ in C m⁻², ρ in C m⁻³.
Gauss's law: φ = q_enc/ε₀ for any closed surface.
Infinite line: E = λ/(2πε₀r); infinite sheet: E = σ/(2ε₀), independent of distance.
Thin shell: E = q/(4πε₀r²) outside, E = 0 inside.

Common traps

Adding the magnitudes of forces from several charges.

Forces and fields are vectors. Resolve into components or use the parallelogram law, and check the direction of each force from the signs first.

Thinking that a rubbed glass rod has gained positive charge (protons).

Only electrons move in rubbing. The rod is positive because it lost electrons to the silk.

Using a 1/r² law for the field of a dipole.

A dipole's far field falls as 1/r³: doubling the distance cuts it to one-eighth. Axial field is twice the equatorial field at the same distance.

Pointing the dipole moment from + to −, or the equatorial field along p.

p points from −q to +q. On the axis E is along p; on the equator E is opposite to p.

Concluding that a dipole in a uniform field has no effect on it at all.

The net force is zero, but there is a torque pE sin θ that turns it towards E. Only in a non-uniform field is there also a net force.

Counting charges outside a Gaussian surface in q_enc, or saying they make the field on the surface zero.

Outside charges add nothing to the total flux, but they do change E at points on the surface.

Believing the flux through a closed surface depends on its size or shape.

For a given enclosed charge the total flux is q_enc/ε₀ for any closed surface; a bigger sphere has weaker E over a larger area.

Making the field of an infinite sheet fall off with distance.

E = σ/(2ε₀) at every distance from an infinite sheet; for a line charge it falls as 1/r, for a point charge as 1/r².

Taking the field inside a charged shell to be the field of a point charge.

Inside a uniformly charged thin shell, E = 0. Outside, it acts as a point charge at the centre.

Formulas

Quantisation of charge

q = ne

n an integer; e = 1.602192 × 10⁻¹⁹ C.

Coulomb's law

F = (1/4πε₀) q₁q₂/r²

1/(4πε₀) ≈ 9 × 10⁹ N m² C⁻²; ε₀ = 8.854 × 10⁻¹² C² N⁻¹ m⁻².

Coulomb's law, vector form

F₂₁ = (1/4πε₀) q₁q₂ r̂₂₁/r₂₁²

Force on q₂ due to q₁; r̂₂₁ points from q₁ to q₂; F₁₂ = −F₂₁.

Superposition

F₁ = F₁₂ + F₁₃ + … + F₁ₙ

Vector sum; the same holds for fields.

Electric field

E = F/q

Test charge q → 0; unit N C⁻¹.

Field of a point charge

E = (1/4πε₀) Q/r²

Radially outward for Q > 0.

Electric flux

Δφ = E·ΔS = E ΔS cos θ

Unit N m² C⁻¹; outward normal for a closed surface.

Dipole moment

p = q × 2a

Directed from −q to +q; unit C m.

Dipole field on the axis

E = 2p/(4πε₀r³)

r >> a; along p. Exact: E = 4qar/[4πε₀(r² − a²)²].

Dipole field on the equatorial plane

E = −p/(4πε₀r³)

r >> a; opposite to p. Exact magnitude: 2qa/[4πε₀(r² + a²)^(3/2)].

Torque on a dipole

τ = p × E; τ = pE sin θ

Uniform field; net force zero.

Charge densities

λ = ΔQ/Δl; σ = ΔQ/ΔS; ρ = ΔQ/ΔV

Units C m⁻¹, C m⁻², C m⁻³.

Gauss's law

φ = ∮E·dS = q_enc/ε₀

Any closed surface; q_enc is the net charge inside.

Infinite line charge

E = λ/(2πε₀r)

Radial; r is the perpendicular distance from the wire.

Infinite plane sheet

E = σ/(2ε₀)

Normal to the sheet, independent of distance.

Thin spherical shell

E = q/(4πε₀r²) for r ≥ R; E = 0 for r < R

q = 4πR²σ.

Key terms

Electric charge: A property of matter that comes in two kinds and makes bodies exert electric forces on each other.
Polarity of charge: The property that distinguishes positive charge from negative charge.
Conductor: A material in which charge can move freely, such as a metal.
Insulator: A material in which charge cannot move freely, so charge stays where it is placed.
Point charge: A charged body small enough compared with the distances involved to be treated as a point.
Quantisation of charge: The fact that every free charge is a whole-number multiple of e.
Permittivity of free space: The constant ε₀ = 8.854 × 10⁻¹² C² N⁻¹ m⁻² that fixes the size of Coulomb's force in vacuum.
Superposition principle: The net force or field due to many charges is the vector sum of those due to each one alone.
Test charge: A very small charge used to probe a field without disturbing the source charges.
Electric field line: A curve whose tangent at each point gives the direction of the electric field there.
Electric flux: The dot product of the field with the area vector, summed over a surface.
Electric dipole: Two equal and opposite charges separated by a small distance.
Dipole moment: The product of either charge of a dipole and their separation, pointing from − to +.
Polar molecule: A molecule with a permanent dipole moment, such as water.
Gaussian surface: Any closed surface chosen to apply Gauss's law.
Surface charge density: Charge per unit area of a surface, in C m⁻².

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