Electric Charges & Fields— Notes, Formulas & Revision

Force between two point charges in vacuum is F = kq₁q₂/r² where k = 1/(4πε₀) = 8.99 × 10⁹ N·m²/C².

Force is along the line joining the charges. Like charges repel, unlike charges attract.

Coulomb force obeys Newton's third law — charges experience equal and opposite forces.

In a medium of relative permittivity εᵣ, force becomes 1/εᵣ times the vacuum value.

Inverse-square dependence: doubling distance reduces force by factor 4.

The net force on a charge from many charges is the vector sum of individual Coulomb forces.

Forces add as vectors — magnitudes can never just be added unless all forces are colinear.

Each pairwise force is independent of all other charges in the system.

Used to compute fields from charge distributions: dE = k dq / r² and integrate.

E-field around a +charge points radially outward; around −charge, radially inward.

Number of lines per unit area is proportional to field magnitude — denser near the charge.

Field lines never cross — direction would be ambiguous.

Lines start on positive and end on negative charges (or extend to infinity).

Dipole = +q and −q separated by small distance d. Dipole moment p⃗ = qd⃗ (from − to +).

Field lines emerge from + and curve to enter −. Density highest along the dipole axis.

Far away, |E| ∝ 1/r³ — falls faster than a single point charge (which goes as 1/r²).

Net charge of dipole = 0, but E ≠ 0 because charges aren't at the same point.

Field at any point = vector sum of fields from each charge.

Two equal +q create a 'saddle' point at the midpoint (E=0 there).

Quadrupole: alternating ± charges create a 4-leaf pattern with E=0 at center.

Lines bunch up where field is strong; spread out where weak.

E from a point charge falls as 1/r² — fast decay with distance.

Doubling r → E/4. Tripling r → E/9. Halving r → 4E.

Plot of E vs r is a hyperbola with asymptotes along both axes.

On the axis, E points along axis by symmetry. Components perpendicular to axis cancel.

E(z) = kQz/(z²+R²)^(3/2). Maximum at z = R/√2.

At z=0 (centre of ring): E=0. At z → ∞: E → kQ/z² (looks like point charge).

Field around an infinite straight charged line is radial (perpendicular to the line).

Magnitude depends only on perpendicular distance r — falls as 1/r (not 1/r²).

Use Gauss's law with a cylindrical Gaussian surface coaxial with the wire.

Field of an infinite charged sheet is uniform, perpendicular to the sheet, and independent of distance.

On both sides of an isolated sheet, E = σ/(2ε₀) outward.

Between two oppositely charged parallel sheets, fields add: E_inside = σ/ε₀.

Axial point: E_ax = 2kp/r³, parallel to p⃗.

Equatorial point: E_eq = kp/r³, antiparallel to p⃗.

Both fall as 1/r³ for short dipole — much faster than point charge's 1/r².

Uniform field exerts no net force on a dipole, but creates torque τ⃗ = p⃗ × E⃗.

|τ| = pE sinθ where θ = angle between p⃗ and E⃗.

Torque tries to align p⃗ with E⃗ (stable equilibrium at θ = 0°).

For small displacements about θ=0, equation of motion: I θ̈ = −pE sinθ ≈ −pE θ.

This is SHM with angular frequency ω = √(pE/I).

Period T = 2π√(I/(pE)).

Electric flux Φ = ∫ E⃗·dA⃗ measures the number of field lines crossing a surface.

For uniform E: Φ = EA cosθ, where θ is the angle between E⃗ and area normal n̂.

Maximum when n̂ ∥ E⃗ (θ=0), zero when perpendicular (θ=90°).

Total flux through a closed surface = (charge enclosed)/ε₀.

Charges OUTSIDE the closed surface contribute zero net flux.

Used to find E for symmetric distributions: spherical, cylindrical, planar.

Uniformly charged spherical shell: outside, E = kQ/r² (acts like point charge at center).

Inside the shell: E = 0 everywhere (a key Gauss's law result).

For solid uniform sphere of charge: E inside ∝ r, E outside same as point charge.

Use a cylindrical Gaussian surface coaxial with a long line charge.

Flux through curved side = E·2πrL; flux through end caps = 0 (E ⟂ caps).

Result: E = λ/(2πε₀r), perpendicular to wire.

In electrostatic equilibrium, E = 0 inside a conductor.

All excess charge resides on the OUTER surface.

Surface field is perpendicular to the surface (else electrons would move).

A closed conducting shell shields its interior from external static fields.

Charges on the shell rearrange to cancel the external field inside.

The cage works for static fields exactly; for AC fields, mesh size and frequency matter.

On a conductor, σ ∝ 1/R_curvature — sharper regions accumulate more charge.

Surface field E_surface = σ/ε₀ — also largest where curvature is highest.

This is why corona discharge starts near sharp points.

Two conducting spheres connected by a wire reach the same potential V.

V = kQ/R same on each → Q ∝ R; σ = Q/(4πR²) ∝ 1/R.

Smaller sphere has higher σ and stronger surface E.