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Electric Charges and Fields — Notes

5 sections 4 min readBeta

1. Electric charge

  • Frictional electricity: rubbing transfers electrons; the gainer becomes negative, the loser equally positive (glass–silk: glass +; ebonite–fur: ebonite −).
  • Two kinds of charge: like charges repel, unlike attract. Conductors let charge move freely; insulators do not.
  • Quantisation: q=neq = n e, e=1.6×10−19e = 1.6 \times 10^{-19} C, n an integer. Conservation: the total charge of an isolated system is constant. Charge is also additive and the same in all frames.

Formulas

  • Quantisation
    q=neq = n e

Key terms

Quantisation of charge

Every charge is an integral multiple of the electronic charge e.

Conservation of charge

The net charge of an isolated system remains constant.

2. Coulomb’s law and superposition

  • F=14πε0q1q2r2F = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1q_2}{r^2}, 14πε0=9×109\dfrac{1}{4\pi\varepsilon_0} = 9 \times 10^{9} N m² C⁻², ε0=8.85×10−12\varepsilon_0 = 8.85 \times 10^{-12} C² N⁻¹ m⁻²; along the line joining the charges.
  • Vector form: F⃗12=14πε0q1q2r2r^12\vec F_{12} = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1q_2}{r^2}\hat r_{12} (force on q2q_2, r^12\hat r_{12} from q1q_1 to q2q_2); F⃗21=−F⃗12\vec F_{21} = -\vec F_{12}.
  • In a medium: Fm=F/KF_m = F/K, with K=ε/ε0K = \varepsilon/\varepsilon_0 (dielectric constant, relative permittivity).
  • Coulomb: the charge that exerts 9×1099 \times 10^{9} N on an equal charge 1 m away in vacuum.
  • Compared with gravitation: both inverse-square and central; electric force can attract or repel, depends on the medium, and is about 103610^{36} times stronger between two protons.
  • Superposition: the net force on a charge is the vector sum of the forces due to each other charge, each found as if the others were absent.
Drawn to scale (90 px per newton); the resultant direction is checked.

Formulas

  • Coulomb’s law
    F=14πε0q1q2r2F = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1q_2}{r^2}
  • Superposition
    F⃗0=∑iF⃗0i\vec F_0 = \sum_i \vec F_{0i}

Key terms

Dielectric constant

K = ε/ε₀, the factor by which a medium reduces the force between charges.

3. Electric field and charge densities

  • E⃗=lim⁡q0→0F⃗/q0\vec E = \lim_{q_0 \to 0}\vec F/q_0; unit N C⁻¹ = V m⁻¹. A charge in a field feels F⃗=qE⃗\vec F = q\vec E.
  • Point charge: E=14πε0qr2E = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q}{r^2}, radially outward for q>0q > 0, inward for q<0q < 0. Several charges: vector sum.
  • Between two like charges q1q_1, q2q_2 distance dd apart, E=0E = 0 at x=dq1q1+q2x = \dfrac{d\sqrt{q_1}}{\sqrt{q_1} + \sqrt{q_2}} from q1q_1.
  • Charge densities: linear λ=dq/dl\lambda = dq/dl (C m⁻¹), surface σ=dq/dA\sigma = dq/dA (C m⁻²), volume ρ=dq/dV\rho = dq/dV (C m⁻³).
  • point charge: E ∝ 1/r²
Doubling r divides E by 4; tripling r divides it by 9.

Formulas

  • Field of a point charge
    E=14πε0qr2E = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q}{r^2}

Key terms

Electric field

Force per unit positive test charge at a point.

4. Electric lines of force

  • Tangent gives the direction of E⃗\vec E; lines start on + charges and end on − charges (or at infinity).
  • Two lines never intersect (unique direction of E⃗\vec E at a point); they are denser where E is larger; electrostatic lines never form closed loops; they meet a conductor normally.
  • Patterns: radial for a point charge; curved from + to − for a dipole; bending apart with a neutral point for two like charges; straight, parallel and equally spaced (uniform field) between oppositely charged parallel plates.
Traced numerically; arrowheads checked against the computed field.

Key terms

Neutral point

A point where the resultant electric field is zero.

5. Electric dipole

  • Dipole: +q+q and −q-q separated by 2a2a; moment p⃗=q(2a)\vec p = q(2a) from −q-q to +q+q (C m).
  • Axial point (distance r from centre): E=14πε02pr(r2−a2)2≈14πε02pr3E = \dfrac{1}{4\pi\varepsilon_0}\dfrac{2pr}{(r^2 - a^2)^2} \approx \dfrac{1}{4\pi\varepsilon_0}\dfrac{2p}{r^3}, along p⃗\vec p.
  • Equatorial point: E=14πε0p(r2+a2)3/2≈14πε0pr3E = \dfrac{1}{4\pi\varepsilon_0}\dfrac{p}{(r^2 + a^2)^{3/2}} \approx \dfrac{1}{4\pi\varepsilon_0}\dfrac{p}{r^3}, opposite to p⃗\vec p. For a short dipole, Eaxial=2EeqE_\text{axial} = 2E_\text{eq} at the same r.
  • Uniform field: net force zero; torque τ⃗=p⃗×E⃗\vec\tau = \vec p \times \vec E, τ=pEsin⁡θ\tau = pE\sin\theta (max pEpE at 90°, zero at 0° and 180°); the dipole rotates to align with E but does not translate. In a non-uniform field there is also a net force.
  • axial point: exact ÷ approximate
The short-dipole result 2kp/r³ is reliable only when r ≫ a.

Formulas

  • Axial field (short dipole)
    Eaxial≈14πε02pr3E_\text{axial} \approx \dfrac{1}{4\pi\varepsilon_0}\dfrac{2p}{r^3}
  • Equatorial field (short dipole)
    Eeq≈14πε0pr3E_\text{eq} \approx \dfrac{1}{4\pi\varepsilon_0}\dfrac{p}{r^3}
  • Torque on a dipole
    τ⃗=p⃗×E⃗\vec\tau = \vec p \times \vec E

Key terms

Dipole moment

Product of either charge and the separation, directed from −q to +q.

Exam tips

  • Convert μC, nC, cm, mm to SI first; give directions for forces and fields.
  • Superposition: diagram, components, then magnitude and angle.
  • Dipole derivations: labelled figure, exact result, then the r ≫ a approximation.
  • Torque: zero net force in a uniform field; τ = pE sin θ and τ = p × E.

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