Class 12 · Chapter 1

Electric Charges & Fields

Overview, notes, short notes, formula sheet, daily practice problems, previous year questions, and videos for this chapter — all in one place.

Electric Charges & Fields Short Notes

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Condensed revision points for Electric Charges & Fields — for quick recall before exams, not a substitute for the full notes.

q = ne; e = 1.6×10⁻¹⁹ C. Charge is quantised, conserved, invariant, and always associated with mass.
Positive = electron deficiency; Negative = electron excess. Charge is a scalar (adds algebraically).
True test of electrification is repulsion, not attraction.
Charging methods: friction (transfer), conduction (contact, sharing), induction (redistribution, no contact, no net charge change on inducing body).
In induction: induced charge is opposite in sign and never greater in magnitude than the inducing charge.

F = kq₁q₂/r²; k = 1/(4πε₀) ≈ 9×10⁹ N·m²/C².
Vector form: F⃗₁₂ = (kq₁q₂/r²) r̂₂₁. Superposition: F⃗ = ΣF⃗ᵢ (two-body interactions add vectorially).
Conservative force; stronger than gravity; can attract or repel (gravity only attracts); depends on the medium (gravity doesn't).
Equilibrium of charges is never stable under electrostatic forces alone.

3 collinear charges: outer two same sign, middle one opposite sign.
Equilateral triangle (charge q at each vertex): centre charge for equilibrium = −q/√3.
Square (charge q at each vertex): centre charge for equilibrium = −q(2√2+1)/4.
Suspended charges: tanθ = F_electric/mg, where θ is the angle the thread makes with the vertical.

E⃗ = F⃗/q₀ (test charge, q₀→0). E = kq/r²; unit N/C; vector quantity.
Points away from +q, toward −q. Obeys superposition.
Force on a charge: F⃗ = qE⃗ — along E⃗ for +q, opposite for −q.
Continuous distributions: E = ∫kλdl/r² (linear), ∫kσdA/r² (surface), ∫kρdV/r² (volume).

Field lines: start on +, end on −; never cross; never form closed loops; perpendicular to conductor surface.
Φ = ∫E⃗·dA⃗; scalar; unit N·m²/C or V·m.
Φ = 0 if: no charge enclosed, equal +/− charge enclosed (e.g. a dipole), or incoming flux = outgoing flux.
Φ = 0 does NOT mean E = 0 on the surface; but E = 0 everywhere does mean Φ = 0.

∮E⃗·dA⃗ = q_enc/ε₀ — depends only on enclosed charge, not on surface shape/size or exact charge position inside.
Field at the Gaussian surface is due to ALL charges (inside + outside), even though flux depends only on enclosed charge.
Symmetric solid, charge q at centre, n identical faces: flux per face = q/(nε₀) (e.g. hemisphere split through centre: q/(2ε₀) each half).
Charge in the plane of one face (hemisphere/cylinder/cube standing on that plane): flux through rest of solid = q/(2ε₀).
Charge at a cube's corner: total flux through whole cube = q/(8ε₀) (8 cubes share a corner). On an edge: q/(4ε₀) (4 cubes share an edge).
Isolated charged conductor surface: E = σ/ε₀ (all flux pushed outward through one side).

Conducting sphere/shell: E_in = 0 (r<R); E_surface = kQ/R²; E_out = kQ/r² (r>R) — acts like a point charge outside.
Non-conducting uniform sphere: E_in = kQr/R³ = ρr/3ε₀ (grows linearly, r<R); same surface and outside formulas as conductor.
No discontinuity at r=R for either case — inside and surface formulas match exactly at r=R.

Infinite line charge: E = λ/(2πε₀r) = 2kλ/r — falls off as 1/r, field is radial.
Infinite plane sheet: E = σ/(2ε₀) — uniform, independent of distance from the sheet.
Conductor surface (σ): E = σ/ε₀ — exactly twice the isolated-sheet value, since all flux exits one side only.

p⃗ = qd⃗, directed from −q to +q. Unit: C·m.
Uniform field: F_net = 0, but τ = p⃗×E⃗ = pE sinθ (max at θ=90°, zero at θ=0° or 180°).
Work rotating θ₁→θ₂: W = pE(cosθ₁−cosθ₂). U = −p⃗·E⃗ = −pEcosθ (zero reference at θ=90°).
U minimum (−pE, most stable) at θ=0°; U maximum (+pE, least stable) at θ=180°. W(0°→180°) = 2pE.

Axial (r≫d): E = 2kp/r³, along p⃗.
Equatorial (r≫d): E = kp/r³, opposite to p⃗. Axial field = 2 × equatorial field at same r.
Both fall off as 1/r³ — faster than a point charge's 1/r², since the dipole is net-neutral.
General point: E = (kp/r³)√(1+3cos²θ); tanα = ½tanθ (α = angle of E from line OP, ≠ θ in general).
Non-uniform field: dipole feels both a net force, F = p(dE/dr), and a torque.

Trajectory: y = (qE/2mv²)x² — a parabola, exactly analogous to projectile motion (qE/m plays the role of g).
Velocity along original direction (v) stays constant; field-direction velocity builds up over time.
Time to cross field region of length l: T = l/v. Total deflection: y = ½(qE/m)(l/v)².

Field inside a conductor is always zero; field inside a uniformly charged non-conducting sphere is NOT zero — it grows linearly with r.
Φ=0 across a closed surface doesn't mean no charge is present anywhere nearby — only that net enclosed charge is zero (e.g. an enclosed dipole).
Don't confuse the isolated-sheet field (σ/2ε₀) with the conductor-surface field (σ/ε₀) — easy to drop the factor of 2 either way.
Axial dipole field is twice the equatorial field at equal distance — a frequently tested ratio, easy to misremember as equal or inverted.
Coulomb's law gives a force; treating extended charged spheres as point charges is valid only for points outside the sphere, never for points inside.
A dipole in a uniform field has zero net force but a non-zero torque (unless aligned/anti-aligned) — don't assume zero force means zero effect.

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