Class 12 · Chapter 4

Moving Charges & Magnetism

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

Moving Charges & Magnetism Short Notes

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

dB = (μ₀/4π)(I dl sinθ)/r²; vector form dB = (μ₀/4π)(I dl × r̂)/r². μ₀/4π = 10⁻⁷ T·m/A.
Field on the axis of a straight wire (θ = 0°/180°) is always zero.
Right-hand thumb rule: thumb along current → curl gives field (straight wire); curl along current → thumb gives field (circular wire/loop).

B = (μ₀I)/(4πa)(sinφ₁ + sinφ₂); infinite wire: B = μ₀I/(2πa); semi-infinite: B = μ₀I/(4πa).
If the point lies outside the wire's span (off to the side along its length), the sines subtract instead of adding.

Centre of loop: B₀ = μ₀I/(2R); N turns: B₀ = μ₀NI/(2R); arc of angle α (rad): B = μ₀Iα/(4πR) = (α/2π)B₀.
Two arcs of a uniform-wire loop, fed at the same two points, always give equal and opposite fields at the centre — net zero, regardless of terminal placement.
Regular polygon, side a: triangle B = 9μ₀I/(2πa); square B = 2√2μ₀I/(πa); hexagon B = √3μ₀I/(πa).
Axial field: B(x) = μ₀NIR²/[2(R²+x²)^1.5] = B₀sin³θ. Points of inflection at x = ±R/2, separated by R — basis of Helmholtz coils.
Far axial field (x ≫ R): B ≈ μ₀M/(2πx³), exactly like a dipole.

∮B·dl = μ₀I_enclosed — only current inside the loop counts; best suited to infinite, symmetric distributions.
Solid cylinder: inside B = μ₀Ir/(2πR²) ∝ r; surface B = μ₀I/(2πR) (max); outside B = μ₀I/(2πr) ∝ 1/r.
Hollow cylinder (radii a, b): B = 0 for r < a; B = μ₀I(r²−a²)/[2πr(b²−a²)] for a<r<b; B = μ₀I/(2πr) for r > b.
Outside any cylindrical conductor, B depends only on total current and distance — never on shape, solid/hollow, or radius.

Long solenoid, inside: B = μ₀nI (n = turns/length); independent of cross-sectional radius.
Finite solenoid, axial point: B = (μ₀nI/2)(cosθ₁−cosθ₂); at an open end of a long solenoid: B = μ₀nI/2 (half the central value).
Toroid: B = μ₀nI inside the core, n = N/(2πR_m), R_m = (R₁+R₂)/2. B = 0 outside the toroid and in the central empty space.

F = q(v×B); zero if v ∥ B or v = 0; max (qvB) when v ⊥ B. Always ⊥ to v ⟹ does no work ⟹ speed (KE) is constant, only direction changes.
Circular motion (v ⊥ B): r = mv/(qB) = √(2mE_k)/(qB); T = 2πm/(qB) (independent of v, r); ω = qB/m.
Helical motion (angle θ to B): r = mv sinθ/(qB); T unchanged = 2πm/(qB); pitch p = (v cosθ)T.
Field of finite width d: completes semicircle if r < d, grazes boundary if r = d, exits early with deviation θ = sin⁻¹(dqB/mv) if r > d.

Semicircle time t = πm/(qB) is independent of speed — this is what makes resonant acceleration possible.
Resonance: oscillator T = 2πm/(qB); cyclotron frequency ν = qB/(2πm).
(E_K)_max = q²B²r²_max/(2m); after N revolutions, 2NqV = q²B²r²_max/(2m).
Can't accelerate neutral particles; impractical for electrons (too light — go relativistic quickly, breaking resonance).

dF = I(dl×B); zero if I ∥/antiparallel to B, max (BIdl) if I ⊥ B. Always ⊥ to both dl and B.
Uniform field: F = I(L×B), where L is the net displacement vector from start to end — shape/path length doesn't matter.
Net force on any closed loop in a uniform field is always zero (torque may not be); in a non-uniform field, both force and torque can be nonzero.
Parallel wires: f = μ₀I₁I₂/(2πd) per unit length. Like currents attract, unlike currents repel — opposite of the rule for charges.
Definition of ampere: current giving f = 2×10⁻⁷ N/m between two such wires 1 m apart.
Free wire above a fixed wire needs unlike currents (stable, T = 2π√(h/g)); free wire below needs like currents (unstable).

M = NIA (vector along the area vector via right-hand rule); SI unit A·m².
τ = M×B = NIAB sinθ; max at θ = 90° (plane ∥ B); zero at θ = 0°/180° (plane ⊥ B).
Net force on a loop in uniform B is always zero; torque is what a galvanometer measures.
Galvanometer (radial field, sinα fixed at 90°): NIAB = Cθ ⟹ I ∝ θ (linear scale).
Current sensitivity = NAB/C; voltage sensitivity = NAB/(CR).
Sensitivity raised by ↑N, ↑B, ↑A, or ↓C; fixed stray-field immunity from the strong magnet; eddy currents damp the coil quickly.

Orbital current I = ef = e/T; orbital magnetic moment M = evr/2.
M = −eL/(2m) — M and orbital angular momentum L are antiparallel (electron's charge is negative).
Bohr quantization L = nh/(2π) ⟹ M = neh/(4πm); for n = 1, M = Bohr magneton μ_B ≈ 9.27×10⁻²⁴ A·m².
For any uniformly charged spinning body, M/L = q/(2m) — a universal ratio, independent of shape (only the moment of inertia changes between ring, disc, sphere, shell).

Don't confuse 'like currents attract' with the electrostatic rule — for currents it's the reverse of like-charges-repel.
Cyclotron period/frequency depend only on q, m, B — never on v or r; only the radius grows with speed, not the timing.
Net magnetic force on a closed current loop in a uniform field is zero even though the torque on it generally isn't — don't assume zero force means zero torque.
Outside a cylindrical conductor, B depends only on total enclosed current and r — solid vs hollow, thick vs thin makes no difference.
Magnetic force never changes a charged particle's speed (work done is always zero) — only direction changes, so KE is conserved in any purely magnetic field region.

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