Lesson 13 of 13 · 17 min
Chapter review
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Must-know facts
18 facts
- 01Glass rubbed with silk is positive; plastic rubbed with fur is negative. Like charges repel, unlike attract.
- 02Charging by rubbing moves electrons; no charge is created.
- 03Charge is additive (scalar), conserved, and quantised: q = ne, e = 1.6 × 10⁻¹⁹ C.
- 04About 6 × 10¹⁸ electrons make up 1 C; 1 μC holds about 10¹³ electronic charges.
- 05Coulomb's law F = kq₁q₂/r², k = 1/(4πε₀) ≈ 9 × 10⁹ N m² C⁻², ε₀ = 8.854 × 10⁻¹² C² N⁻¹ m⁻².
- 06Electric to gravitational force, electron and proton: about 2.4 × 10³⁹.
- 07Superposition: forces and fields from several charges add as vectors.
- 08E = F/q, unit N C⁻¹ = V m⁻¹; point charge E = kQ/r², outward for +Q.
- 09Field 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
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.