Moving Charges
and Magnetism

This chapter is unusually unforgiving of weak foundations. Fix these first or it will feel far harder than it is. What follows is not a summary — it is enough to actually learn or refresh each prerequisite in about ten minutes each.

1 · Cross product and the right-hand rule

The cross product A × B produces a new vector — perpendicular to both A and B. Two things matter about that vector: its magnitude, and its direction.

• Magnitude | A × B | = A B sin θ, where θ is the smaller angle between the two vectors. Maximum when they are perpendicular (sin 90° = 1), zero when they are parallel or antiparallel (sin 0° = sin 180° = 0). This is why a charge moving along a magnetic field feels no force.
• Direction — the right-hand rule. Point the fingers of your right hand along the first vector (A), curl them toward the second (B) through the smaller angle. The thumb now points along A × B.
• Anti-commutative: A × B = − B × A. Swap the order and the direction flips. This is the source of most sign errors in the chapter.
• Unit-vector shortcut: î × ĵ = k̂ · ĵ × k̂ = î · k̂ × î = ĵ. Cyclic. Any other order gives the negative.

2 · Circular motion

A charged particle in a uniform magnetic field, moving perpendicular to it, traces a perfect circle. Half the "charged particle" results in this chapter are direct consequences of one idea: a force always perpendicular to velocity changes only direction, never speed.

• Centripetal acceleration a = v² / r, always pointing toward the centre of the circle.
• Centripetal force required F_c = m v² / r. This is not a new force — it is whatever real force plays the role.
• For a charged particle in B, that real force is the magnetic Lorentz force q v B. Setting q v B = m v² / r and cancelling v gives directly r = m v / q B — the radius formula.
• Period T = 2π r / v = 2π m / q B — notice that v and r cancel out, so the period is independent of the particle's speed and radius. That single fact is the reason a cyclotron works.
• Work done by the magnetic force = 0. Force ⟂ velocity means F · v = 0, so kinetic energy is conserved. A particle in a pure B field never speeds up or slows down.

3 · Vector addition of fields

Magnetic fields obey superposition: the total field at a point is the vector sum of the fields from every source. Two wires, three wires, a wire and a loop — all superposition. Fields are not scalars; they cannot be added arithmetically unless they happen to be collinear.

• Same direction — magnitudes add: B_total = B₁ + B₂.
• Opposite direction — magnitudes subtract, direction follows the larger: B_total = |B₁ − B₂|.
• Perpendicular — use Pythagoras: B_total = √(B₁² + B₂²); direction from tan θ = B₂ / B₁.
• General angle — resolve each field into x and y components, sum the components separately, then recombine. This is the only method that always works.

4 · Symmetry arguments (Ampère's law is unusable without them)

Ampère's law ∮ B · dl = μ₀ I_enc is always true — but it only becomes usable when B is constant in magnitude and simple in direction along the chosen loop. That happens only in a few high-symmetry configurations.

• Long straight wire. Cylindrical symmetry about the wire axis. Amperian loop: a circle centred on the wire. B is tangent to the circle everywhere and has the same magnitude at every point. Answer falls out in two lines.
• Long solenoid. Translational symmetry along the axis. Amperian loop: a rectangle with two sides parallel to the axis, one inside and one outside. Only the inside side contributes. Yields B = μ₀ n I.
• Toroid. Rotational symmetry about the toroid's central axis. Amperian loop: a circle coaxial with that central axis, running inside the toroid. Yields B = μ₀ N I / (2 π r).
• Everything else — a finite wire, a circular loop off-centre, a square loop — has too little symmetry to reduce the line integral. For these, fall back to Biot–Savart. Recognising this in under five seconds is the judgement-call skill of Pillar 1.

5 · The small extras — trigonometry, algebra, units

• Angles matter — never drop the sinθ. Force on a wire: F = B I L sin θ. Torque on a loop: τ = N I A B sin θ. When the current is perpendicular to B, sinθ = 1 and the angle vanishes — but half of NEET's harder problems set θ ≠ 90° deliberately to see who dropped it.
• Standard-angle values — sin 30° = 1/2 · sin 45° = 1/√2 · sin 60° = √3/2 · sin 90° = 1. Memorise cold; NEET expects instant recall.
• The 1/r vs 1/r² distinction — magnetic field of an infinite wire falls as 1/r (single power), but the on-axis field of a loop or the field of a magnetic dipole falls as 1/r³ at large distance. Getting the exponent wrong makes ratio questions unanswerable.
• Unit awareness — magnetic field in Tesla (T), current in Ampere (A), μ₀ = 4π × 10⁻⁷ T·m/A. If a NEET answer has an implausible power of ten (say 10⁺⁷), the unit or the constant was fumbled.
• Components resolve everything — any awkward geometry (wire at an angle, loop tilted, force at an angle) reduces to resolving each vector along two perpendicular axes and summing separately. This is the one tool that never fails.

Everything in it belongs to one of these. Knowing which pillar a question sits in is most of the work of solving it.

Every derivation is done by her on paper, not read off the page. Reading a derivation and reproducing one are different skills, and only the second is tested.

Both take about forty minutes each and are used at every revisit for the rest of the year.

Asset 1 — the geometry-to-formula sheet

• One page. For each configuration, a small sketch on the left and its field expression on the right. Never a formula without its picture.
• Cover: infinite wire, finite wire, loop at centre, loop on axis, arc of angle θ, semicircle, quarter circle, solenoid (inside and at the end), toroid, square loop at centre.
• Add a column for direction — an arrow on the sketch, not words.
• The reason for the sketches: a plain formula list is exactly why students apply the on-axis expression at the centre. Attaching the picture makes that error visible before it happens.

Asset 2 — the direction drill

• Twenty quick items that ask only for a direction — which way does B point, which way does F act. No numbers, no calculation.
• Five minutes a day for a week makes the right-hand rule reflexive.
• This is disproportionately valuable: in this chapter, a candidate who computes the correct magnitude and picks the wrong direction scores the same as one who knew nothing.
• Include the awkward cases: current out of the page, antiparallel wires, a negative charge, and a loop viewed edge-on.

Derivation-heavy chapters decay faster than recall chapters. Plan for that.

1. Day 10. Reproduce the geometry-to-formula sheet from memory on blank paper. Correct in red. Then re-derive any two formulas that were wrong.
2. Day 25. Same reproduction, plus twenty mixed problems chosen so that no two consecutive questions come from the same pillar.
3. Day 50. Re-sit the 45-question test cold. Compare error types against the log, not just the score.
4. Final month. Formula sheet and direction drill only. By this stage new problems add little; speed and direction accuracy add a lot.

Each revisit should take less time than the one before. If it doesn't, the first pass was passive rather than active — the fix is more re-derivation, not more reading.

Almost none of these are conceptual failures. They are pattern errors, which means they are fixable by drill.

• Applying the on-axis field expression of a circular loop at its centre, or the reverse.
• Using total turns where the solenoid formula requires turns per unit length, n = N / L.
• Forgetting that parallel currents attract and antiparallel currents repel — the opposite of the intuition carried over from charges.
• Missing that the magnetic force does zero work, so the speed and the kinetic energy of a charged particle never change in a pure magnetic field.
• Forgetting that the cyclotron period is independent of speed and radius — which is the whole reason the device works.
• Assuming a cyclotron can accelerate any particle; it cannot usefully accelerate neutral particles, and electrons are impractical.
• Confusing the shunt (parallel, for an ammeter) with the series resistance (for a voltmeter), or mixing up their formulas.
• Treating the field from two wires as an arithmetic sum when the two contributions are not collinear.
• Dropping the angle in the force on a wire when the current is not perpendicular to the field.
• Reading a figure too quickly — currents drawn into or out of the page are the most common source of sign errors in this chapter.

• 60–75 seconds per numerical is normal here, not slow. Budget for it rather than panicking mid-paper — this is a different pace from an inorganic chemistry chapter, where 35–40 seconds is the target.
• Always draw the figure, even when one is given. Redrawing it with the direction arrows marked takes eight seconds and prevents the most expensive class of error.
• If the geometry is not clear within twenty seconds, mark and move. Geometry confusion rarely resolves itself by staring.

• The standard approach — read the chapter, memorise the formula list, then solve problems — fails here because the formulas are nearly identical in form and differ only by geometry. A list without pictures gives no way to tell them apart under time pressure.
• This is not a recall chapter, so the chemistry-style method of read → tabulate → recall is the wrong tool. Derivation is the study; the formula is only its residue.
• The chapter pays twice. Magnetism & Matter and Electromagnetic Induction both assume it, so time invested here reduces the cost of two later chapters.
• Its errors are unusually systematic — direction, geometry, wrong law. Systematic errors respond to drill, which is why the direction drill and error log matter more here than in most chapters.