Lesson 13 of 13 · 17 min

Chapter review

Watch a class

The whole chapter on YouTube

Full chapter with derivations and practice
Arvind Academy · Hinglish · Whole chapter · Open on YouTube

Coulomb to Gauss in one sitting
NCERT Wallah · Hinglish · Whole chapter · Open on YouTube

Must-know facts

18 facts

1Glass rubbed with silk is positive; plastic rubbed with fur is negative. Like charges repel, unlike attract.
2Charging by rubbing moves electrons; no charge is created.
3Charge is additive (scalar), conserved, and quantised: q = ne, e = 1.6 × 10⁻¹⁹ C.
4About 6 × 10¹⁸ electrons make up 1 C; 1 μC holds about 10¹³ electronic charges.
5Coulomb's law F = kq₁q₂/r², k = 1/(4πε₀) ≈ 9 × 10⁹ N m² C⁻², ε₀ = 8.854 × 10⁻¹² C² N⁻¹ m⁻².
6Electric to gravitational force, electron and proton: about 2.4 × 10³⁹.
7Superposition: forces and fields from several charges add as vectors.
8E = F/q, unit N C⁻¹ = V m⁻¹; point charge E = kQ/r², outward for +Q.
9Field lines: start on +, end on −, never cross, never form closed loops, crowd where E is strong.
10Flux φ = E·S = ES cos θ, unit N m² C⁻¹; outward normal for closed surfaces.
11Dipole moment p = q × 2a, from −q to +q, unit C m.
12Dipole far field: axial 2kp/r³ along p; equatorial kp/r³ opposite to p; falls as 1/r³.
13Dipole in uniform field: net force zero, torque τ = p × E, τ = pE sin θ.
14Non-uniform field: dipole parallel to E moves towards stronger field.
15λ in C m⁻¹, σ in C m⁻², ρ in C m⁻³.
16Gauss's law: φ = q_enc/ε₀ for any closed surface.
17Infinite line: E = λ/(2πε₀r); infinite sheet: E = σ/(2ε₀), independent of distance.
18Thin shell: E = q/(4πε₀r²) outside, E = 0 inside.

Common traps

Where marks are lost

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

16 to know

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

16 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.

Test yourself: 10 questionsExam-style questions on Electric Charges and Fields, with full solutions.Start