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'Moving Charges and Magnetism' sits at the heart of electromagnetic theory — establishing the connection between electric current and magnetic fields that powers all modern electrical technology. From motors to MRI scanners, this chapter is the foundation. It builds on Current Electricity and feeds directly into Electromagnetic Induction.
| Related Chapter | Connection to This Chapter |
|---|---|
| Electric Charges & Fields | Electric force generalised to magnetic force; field-line analogy |
| Electrostatic Potential | U = −MBcosθ parallels electric dipole potential energy concept |
| Current Electricity | Moving charges (current) are the source of all magnetic fields here |
| Magnetism & Matter | Bar magnet treated as magnetic dipole; M = m×2l analogy |
| Electromagnetic Induction | Magnetic flux Φ = BA; change in B derived here drives Faraday's Law |
| Alternating Current | AC motors/generators use F = BIL and τ = NIAB principles |
Electric field E exists due to charge (stationary or moving). Magnetic field B exists ONLY due to moving charges. A stationary charge in a magnetic field experiences NO magnetic force!
θ = angle between velocity v and field B
| Feature | Electric Force (FE) | Magnetic Force (FB) |
|---|---|---|
| Formula | F = qE | F = qvB sinθ |
| Acts on | Stationary OR moving charges | ONLY moving charges |
| Direction | Along or opposite to E | Perpendicular to BOTH v and B |
| Does work? | YES Can accelerate particle | NO Always ⊥ to v |
| Changes speed? | YES | NO (only direction changes) |
| Changes KE? | YES | NO (|v| stays constant) |
| Condition | Force | Result | Example |
|---|---|---|---|
| Only E, no B | F = qE | Force ∥ to E | CRT electron gun |
| Only B, θ = 0° | F = 0 | No force; straight line | Along field lines |
| Only B, θ = 90° | F = qvB | Max force; circular orbit | Cyclotron |
| Both E and B (balanced) | qE = qvB → v = E/B | Straight path | Velocity selector |
Point fingers of RIGHT hand along v (velocity). Curl them toward B (field). Thumb points in direction of v × B. For NEGATIVE charge, reverse the force direction.
Left hand: Forefinger → B (field), Middle finger → I (current), Thumb → F (force). FBI — Field, Current (Bundles together I), Motion.
The time period T = 2πm/qB and frequency f = qB/2πm are INDEPENDENT of particle speed v and orbit radius r. This is the fundamental principle behind the cyclotron!
| Formula | Depends On | Independent Of |
|---|---|---|
| r = mv/qB | mass m, velocity v, charge q, field B | — |
| T = 2πm/qB | mass m, charge q, field B | velocity v radius r |
| f = qB/2πm | charge q, field B, mass m | velocity v ← KEY to cyclotron! |
| ω = qB/m | charge q, field B, mass m | velocity v |
| Angle θ | Path | v⊥ | v∥ | Example |
|---|---|---|---|---|
| 0° | Straight line | 0 | v | Charge along B field lines |
| 90° | Perfect circle | v | 0 | Cyclotron, CRT |
| Between | Helix | v sinθ | v cosθ | Aurora Borealis, Van Allen belts |
| 180° | Straight line | 0 | v (opposite) | Antiparallel motion |
Magnetic field dB due to infinitesimal current element Idl at distance r:
| Symbol | Meaning | Unit |
|---|---|---|
| dB | Magnetic field due to element dl | Tesla (T) |
| μ₀ | Permeability of free space = 4π × 10⁻⁷ | T·m/A (H/m) |
| I | Current in conductor | Ampere (A) |
| dl | Length of current element | metre (m) |
| θ | Angle between dl⃗ and r̂ | degrees/radians |
| r | Distance from element to field point | metre (m) |
| Feature | Biot–Savart Law | Coulomb's Law |
|---|---|---|
| Source | Current element Idl | Point charge dq |
| Field produced | Magnetic field B | Electric field E |
| Distance dependence | 1/r² | 1/r² |
| Direction | ⊥ to plane of dl and r | Along r (radial) |
| Constant | μ₀/4π = 10⁻⁷ T·m/A | 1/4πε₀ = 9×10⁹ N·m²/C² |
| Current Configuration | Magnetic Field Formula | Direction Rule |
|---|---|---|
| Infinite straight wire | B = μ₀I / 2πr | Right-hand thumb rule; concentric circles |
| Semi-infinite wire (from end) | B = μ₀I / 4πr | Same as infinite wire, half value |
| Finite wire (general) | B = μ₀I(sinφ₁+sinφ₂)/4πd | d = perpendicular distance from wire |
| Circular loop (centre, 1 turn) | B = μ₀I / 2R | Right-hand rule for loop |
| Circular loop (centre, N turns) | B = Nμ₀I / 2R | Amplified by N turns |
| Circular loop (on axis) | B = μ₀IR²/2(R²+x²)^(3/2) | Along axis; max at centre |
| Semicircular arc (at centre) | B = μ₀I / 4R | Half of full loop |
| Position x | Field B |
|---|---|
| x = 0 (centre) | μ₀I/2R (maximum) |
| x >> R (far) | μ₀IR²/2x³ ∝ 1/x³ |
If current flows anticlockwise when viewed from a face → B points towards you (out of page). Use right-hand rule: curl fingers along current direction → thumb points in B direction.
The line integral of B around any closed Amperian loop equals μ₀ × total current enclosed by that loop.
Choose circular Amperian loop of radius r. By symmetry, B is constant and tangential:
n = N/L = turns per unit length
B = 0 outside toroid AND in the central hole. N = total turns, r = mean radius of ring.
| Feature | Biot–Savart Law | Ampere's Circuital Law |
|---|---|---|
| Best for | Any current distribution | Highly symmetric configurations |
| Method | Integration over elements | Line integral around closed loop |
| Complexity | High (integration needed) | Low (algebraic for symmetric cases) |
| Applicable to | Finite wires, arcs, loops | Infinite wire, solenoid, toroid |
| Analogous to | Coulomb's Law (for E) | Gauss's Law (for E) |
| Angle θ | Force F | Situation |
|---|---|---|
| 0° or 180° | F = 0 | Current ∥ B — no force |
| 90° | F = BIL (max) | Current ⊥ B — max force |
| Any θ | BIL sinθ | General case |
LEFT hand: Forefinger → B (field), Middle finger → I (current), Thumb → F (force/motion). "FBI always uses the LEFT hand for motors!"
| Current Direction | Force |
|---|---|
| Same direction (parallel) | ATTRACTIVE |
| Opposite direction (anti-parallel) | REPULSIVE |
SALA — Same Attracts, Like Appels! Opposite of electric charges — same direction currents attract, opposite direction currents repel.
One Ampere is that constant current which, when maintained in two straight parallel conductors of infinite length placed 1 metre apart in vacuum, produces a force of 2 × 10⁻⁷ N/m between them.
| θ | τ | U | Equilibrium |
|---|---|---|---|
| 0° | 0 | −MB (min) | STABLE |
| 90° | MB (max) | 0 | Not equilibrium |
| 180° | 0 | +MB (max) | UNSTABLE |
Magnetic dipole in B field behaves EXACTLY like electric dipole in E field: τ = p×E (electric) ↔ τ = M×B (magnetic). U = −p·E ↔ U = −M·B.
| # | Formula | Expression | Unit | Condition / Notes |
|---|---|---|---|---|
| 1 | Lorentz Force | F = q(E + v×B) | Newton | Total electromagnetic force |
| 2 | Magnetic Force (scalar) | F = qvB sinθ | Newton | θ = angle between v and B |
| 3 | Radius of circular orbit | r = mv/qB | metre | v ⊥ B; mass m, charge q |
| 4 | Time Period (circular) | T = 2πm/qB | second | ⭐ Independent of v and r! |
| 5 | Cyclotron frequency | f = qB/2πm | Hz | ⭐ Independent of v — basis of cyclotron |
| 6 | Angular velocity | ω = qB/m | rad/s | Independent of v |
| 7 | Pitch of helix | p = v cosθ × T | metre | θ = angle between v and B |
| 8 | Biot–Savart Law | dB = μ₀Idl sinθ/4πr² | Tesla | For current element Idl |
| 9 | Infinite straight wire | B = μ₀I/2πr | Tesla | r = perpendicular distance from wire |
| 10 | Finite wire (general) | B = μ₀I(sinφ₁+sinφ₂)/4πd | Tesla | d = perpendicular distance |
| 11 | Circular loop (centre) | B = μ₀I/2R | Tesla | R = radius; single turn |
| 12 | N-turn loop (centre) | B = Nμ₀I/2R | Tesla | N = number of turns |
| 13 | Loop on axis | B = μ₀IR²/2(R²+x²)^(3/2) | Tesla | x = axial distance from centre |
| 14 | Ampere's Circuital Law | ∮B·dl = μ₀I_enc | T·m | Closed Amperian loop |
| 15 | Solenoid (inside) | B = μ₀nI | Tesla | n = N/L = turns per unit length |
| 16 | Toroid (inside) | B = μ₀NI/2πr | Tesla | N = total turns, r = mean radius |
| 17 | Force on conductor | F = BIL sinθ | Newton | θ = angle between I and B |
| 18 | Parallel wire force/length | F/L = μ₀I₁I₂/2πd | N/m | d = separation between wires |
| 19 | Magnetic dipole moment | M = NIA | A·m² | N turns, current I, area A |
| 20 | Torque on current loop | τ = MB sinθ = NIAB sinθ | N·m | θ = angle between M and B |
| 21 | Potential energy of dipole | U = −MB cosθ = −M·B | Joule | Min at θ=0° (stable), max at θ=180° |
| 22 | Velocity selector | v = E/B | m/s | Balanced electric and magnetic forces |
| 23 | Cyclotron max energy | KE_max = q²B²R²/2m | Joule | R = radius of dee |
| 24 | Radius (same KE) | r = √(2mK)/qB | metre | K = kinetic energy |
| 25 | Permeability of free space | μ₀ = 4π × 10⁻⁷ T·m/A | T·m/A | Fundamental constant |
| Graph | Equation | Shape | Key Feature | Exam Significance |
|---|---|---|---|---|
| r vs v | r = mv/qB | Straight line through origin | Slope = m/qB | r increases with v; cyclotron uses fixed r |
| r vs B | r = mv/qB = const/B | Rectangular hyperbola | rB = constant | Higher B = tighter circular orbit |
| T vs v | T = 2πm/qB = constant | Horizontal line | T independent of v! | KEY: cyclotron principle |
| f vs B | f = qB/2πm | Straight line through origin | Slope = q/2πm | Cyclotron frequency ∝ B |
| F vs θ | F = qvB sinθ | Sine curve | Max at 90°, zero at 0°,180° | Direction sensitivity of force |
| τ vs θ | τ = MB sinθ | Sine curve | Max at θ=90° | Galvanometer design |
| U vs θ | U = −MB cosθ | Cosine (inverted) | Min at θ=0°, max at θ=180° | Stability analysis |
| B vs r (wire) | B = μ₀I/2πr | Rectangular hyperbola | B→0 as r→∞ | Field weakens with distance |
| B vs x (loop axis) | B = μ₀IR²/2(R²+x²)^(3/2) | Bell curve | Max at x=0 (centre) | MRI coil design |
Right-Hand Thumb Rule: Thumb along current → fingers curl in direction of B (concentric circles)
Cross-product: Fingers along v, curl to B → palm direction = v×B force
Fleming's LHR: Left hand FBI: F(thumb) B(forefinger) I(middle finger)
Loop right-hand rule: Curl fingers along current direction → thumb points in M direction
Step 3: sinθ=1 because v⊥B. Step 4: Standard circular motion. Step 5: Only force acting is F_B, so it must equal F_c. Step 6: Cancel one factor of v from both sides.
The r in numerator and denominator cancel perfectly, leaving f with no dependence on speed. This is WHY cyclotrons work — the alternating voltage can stay synchronized with the particle regardless of how fast it gets!
An electron (m = 9.1×10⁻³¹ kg, q = 1.6×10⁻¹⁹ C) moves at 3×10⁷ m/s perpendicular to B = 0.3 T. Find radius, time period, and frequency.
r = mv/qB = (9.1×10⁻³¹ × 3×10⁷)/(1.6×10⁻¹⁹ × 0.3) = 5.7×10⁻⁴ m ≈ 0.57 mm
T = 2πm/qB = 2π×9.1×10⁻³¹/(4.8×10⁻²⁰) = 1.19×10⁻¹⁰ s
f = 1/T = 8.4×10⁹ Hz
Using T = 2πr/v without substituting v — leads to unnecessary complexity. Use T = 2πm/qB directly!
A 0.5 m wire carrying 2 A is placed at 30° to B = 0.4 T. Find the force.
F = BIL sinθ = 0.4 × 2 × 0.5 × sin30° = 0.4 × 2 × 0.5 × 0.5 = 0.2 N
Two parallel wires 0.1 m apart carry 5 A and 10 A in the same direction. Find F/L and state attraction/repulsion.
F/L = μ₀I₁I₂/2πd = (4π×10⁻⁷×5×10)/(2π×0.1) = 10⁻⁴ N/m
ATTRACTIVE (same direction currents)
E = 1.5×10⁵ V/m, B = 0.015 T. Find speed of ions passing straight through.
qE = qvB → v = E/B = 1.5×10⁵/0.015 = 10⁷ m/s
Only particles with v = E/B pass straight. Faster particles curve toward B side, slower toward E side.
Circular coil: 30 turns, radius 8 cm, current 6 A, B = 1 T. Find maximum torque.
A = π(0.08)² = 0.02011 m²
M = NIA = 30 × 6 × 0.02011 = 3.62 A·m²
τ_max = MB sin90° = 3.62 × 1 = 3.62 N·m
Solenoid: length 0.5 m, 1000 turns, current 5 A. Find B inside.
n = N/L = 1000/0.5 = 2000 turns/m
B = μ₀nI = 4π×10⁻⁷×2000×5 = 4π×10⁻³ ≈ 12.57 mT
(a) State and derive Biot–Savart Law. (b) Using it, derive B at centre of circular current loop of radius R carrying current I.
(a) State dB = (μ₀/4π)(Idl sinθ/r²). Explain each symbol. (b) At centre, every element dl is perpendicular to r (θ=90°, sinθ=1) and all dB point in same direction. Integrate: B = ∮dB = ∮(μ₀I/4πR²)dl = (μ₀I/4πR²)(2πR) = μ₀I/2R ✓. Always draw labeled diagram for full marks!
Proton (m=1.67×10⁻²⁷kg, q=1.6×10⁻¹⁹C) enters B=0.5T field at 30° to field with v=10⁶ m/s. Find pitch and radius of helix.
v_perp = v sin30° = 5×10⁵ m/s
v_para = v cos30° = 8.66×10⁵ m/s
r = mv_perp/qB = (1.67×10⁻²⁷×5×10⁵)/(1.6×10⁻¹⁹×0.5) ≈ 1.04 cm
T = 2πm/qB = 1.31×10⁻⁷ s
Pitch = v_para×T = 8.66×10⁵×1.31×10⁻⁷ ≈ 11.3 cm
A proton and alpha particle enter perpendicular B field with same KE. Ratio of radii r_p : r_α = ?
(A) 1:1 (B) 1:√2 (C) 1:2 (D) √2:1
r = √(2mK)/qB. For proton: r_p = √(2m_p K)/eB. For alpha (mass=4m_p, charge=2e): r_α = √(2×4m_p K)/2eB = √(8m_p K)/2eB = r_p. They are EQUAL!
Most students use r = mv/qB directly. Alpha has mass 4m_p and charge 2e — these exactly cancel for same KE! Always use r = √(2mK)/qB for same-KE comparisons.
Two parallel wires carry currents in ratio 1:2. Ratio of force/length on wire 1 to wire 2 = ?
(A) 1:2 (B) 2:1 (C) 1:1 (D) 1:4
By Newton's 3rd Law, force on wire 1 due to wire 2 = force on wire 2 due to wire 1 in magnitude. F/L = μ₀I₁I₂/2πd — same formula for both. Equal and opposite!
An electron moves in circular orbit of radius R in uniform B. If KE is doubled, find new radius and ratio of new to old time period.
r = √(2mKE)/qB ∝ √KE → r_new = R√2 (radius increases by factor √2)
T = 2πm/qB — completely independent of KE! → T_new/T_old = 1:1 (unchanged)
A cyclotron has dee radius R = 0.5 m and magnetic field B = 1.5 T. Find maximum kinetic energy of protons (m = 1.67×10⁻²⁷ kg, q = 1.6×10⁻¹⁹ C).
KE_max = q²B²R²/2m = (1.6×10⁻¹⁹)²×(1.5)²×(0.5)²/(2×1.67×10⁻²⁷)
= (2.56×10⁻³⁸×2.25×0.25)/(3.34×10⁻²⁷)
= 1.44×10⁻³⁸/3.34×10⁻²⁷ = 4.31×10⁻¹² J ≈ 26.9 MeV
| # | Assertion (A) | Reason (R) | Answer | Explanation |
|---|---|---|---|---|
| 1 | A magnetic field can accelerate a charged particle | Magnetic force is always perpendicular to velocity | D A false, R true | Magnetic force changes direction, not speed. So it cannot accelerate (change KE). Reason correctly explains why Assertion is false. |
| 2 | Cyclotron is not suitable for accelerating electrons | Cyclotron frequency is independent of speed | B Both true, R doesn't explain A | Electrons quickly become relativistic → mass increases → f = qB/2πm fails. The stated Reason is a property, not the explanation. |
| 3 | Two parallel wires attract when currents flow in same direction | Magnetic field created by one wire acts on the other | A Both true, R correctly explains A | Wire 1 creates B at wire 2's location; force on wire 2 in this field pulls it toward wire 1 → attraction. |
| 4 | Net force on a current loop in uniform B is zero | Forces on opposite sides of loop are equal and opposite | A Both true, R correctly explains A | In uniform field: opposite side forces cancel → zero net force. But they create a couple (torque ≠ 0). |
"Magnetic force does work on charges"
Cause: Force exists, so it must do work.
Correction: F always ⊥ v → W = F·v·dt = 0. Magnetic force only changes direction, never speed or KE.
"Magnetic force is the lazy force — it works but does no work!"
"Solenoid B depends on total length L"
Cause: More turns = more field, longer solenoid has more turns.
Correction: B = μ₀nI = μ₀(N/L)I — only turns per length matter. Doubling both N and L keeps B the same!
Always write n = N/L separately before substituting.
"Parallel currents should repel (like charges)"
Cause: Analogizing with Coulomb's law.
Correction: Magnetic interaction is fundamentally different. Same direction currents ATTRACT. Proven by F/L formula and confirmed experimentally.
Draw field lines of wire 1 at wire 2's location to determine force direction directly.
"Cyclotron can accelerate electrons to any energy"
Cause: Cyclotron principle seems universal.
Correction: At relativistic speeds, mass increases → f = qB/2πm fails. Synchrotron solves this by varying B with energy.
Note: T = 2πm/qB assumes classical (non-relativistic) mechanics only.
"Right hand for current force direction"
Cause: Confusion between two different right-hand rules.
Correction: RIGHT hand → direction of B due to current. LEFT hand (Fleming's) → force on current in B field.
FBI mnemonic with LEFT hand: F(thumb) B(forefinger) I(middle finger).
"Torque is maximum when M is parallel to B"
Cause: Confusion between equilibrium (τ=0) and maximum torque.
Correction: τ = MB sinθ: maximum at θ=90°; zero at 0° (stable) and 180° (unstable).
Energy and torque are different: minimum energy → stable equilibrium → zero torque.
"Wrong direction for cross product v × B"
Cause: Difficulty with 3D spatial visualization.
Correction: Right-hand rule: fingers from v curling to B, thumb = v×B. For negative charge, reverse the result.
Set up coordinate axes and verify with determinant method for a few examples until it becomes intuitive.
"Forgetting sinθ or using wrong angle in F = BIL sinθ"
Cause: θ is between conductor and B, but sometimes the complement angle is given.
Correction: θ = angle between current direction (L vector) and B. If given angle with normal to B, use (90° − given angle).
Always draw a diagram. Default check: if θ=90°, F = BIL (max). If θ=0°, F = 0.
| Topic | Marks | Question Type | Priority |
|---|---|---|---|
| Biot–Savart Law + B at centre of loop | 5 | Long answer (derivation) | ⭐⭐⭐⭐⭐ Must prepare |
| Ampere's Law + Solenoid derivation | 5 | Long answer (derivation) | ⭐⭐⭐⭐⭐ Must prepare |
| Force on conductor + Fleming's rule | 3 | Short answer + diagram | ⭐⭐⭐⭐ |
| Circular motion formulas (r, T, f) | 3 | Numericals | ⭐⭐⭐⭐⭐ Easy 3 marks |
| Parallel wire force + definition of Ampere | 3 | Short answer/numerical | ⭐⭐⭐⭐ |
| Torque on current loop + galvanometer | 5 | Long answer | ⭐⭐⭐⭐⭐ Very frequently asked |
| Cyclotron principle and working | 3–5 | Short/long answer | ⭐⭐⭐⭐ |
| Case-based questions (MRI, motors) | 4 | Case study MCQ format | ⭐⭐⭐⭐ New 2026–27 pattern |
| Question Type | Frequency | Key Formulas | Difficulty |
|---|---|---|---|
| Circular motion: r, T, f for given particle | Very High | r=mv/qB, T=2πm/qB, f=qB/2πm | Easy–Medium |
| Same KE or same momentum comparison | High | r=√(2mK)/qB; r=p/qB | Medium (needs insight) |
| Force on current conductor direction | High | F=BIL sinθ, Fleming's LHR | Easy |
| Magnetic moment and torque | Medium | M=NIA, τ=MB sinθ | Medium |
| Parallel wire force direction | Medium | F/L=μ₀I₁I₂/2πd | Easy |
| B at centre of loop variants | Medium | B=μ₀I/2R, superposition | Medium |
| Trap | Wrong Approach | Correct Approach |
|---|---|---|
| Same KE comparison | Use r=mv/qB directly — unsure of v | r=√(2mK)/qB |
| Proton vs alpha | Treat alpha charge as e | Alpha: mass=4m_p, charge=2e → both cancel! |
| Force between wires | "Higher current wire gets more force" | BOTH wires get EQUAL force (Newton's 3rd Law) |
| Magnetic force and work | "Force exists → work done" | Work by magnetic force = 0 always (F⊥v) |
| Solenoid B with changed length | B unchanged if N and L both change | If L doubles with same N: n halves → B halves! |
Particle entering B field at boundary: exit angle = entry angle (reflection-like). Chord = 2r sinα where α = half the arc angle.
KE = q²B²r²/2m. Cyclotron energy limited by dee radius R. KE_max = q²B²R²/2m.
For dipole in non-uniform B: F = ∇(M·B). Dipole attracted toward stronger field region (when M aligned with B).
| Time Slot | Activity | Focus |
|---|---|---|
| 6:00–7:00 AM | Formula revision | All 25 formulas — write from memory without hints |
| 7:00–8:30 AM | Derivations | Biot–Savart for circular loop; Solenoid via Ampere's Law; Torque on loop |
| 8:30–9:30 AM | Numericals | 5 each: circular motion, force on conductor, parallel wires |
| 9:30–10:30 AM | Diagrams practice | Solenoid field, Galvanometer, Cyclotron — labeled fully |
| 10:30–11:30 AM | MCQs & assertion-reason | All 20 MCQs from this guide |
| 11:30–12:00 PM | Previous year questions | Last 5 years board questions on this chapter |
| 12:00–12:30 PM | Quick revision | Mind map, last-minute checklist |
Force (thumb) · B-field (forefinger) · Induced/current (middle finger)
Always the LEFT hand for motors!
"FBI agents always use their LEFT hand."
SALA: Same Attracts, Love Always!
Opposite of charges — same direction currents attract each other.
Opposite direction currents repel.
"T = 2πm/qB has no v — no velocity!"
Think: "The cyclone's spin rate depends on how BIG and CHARGED the storm is, not how fast it's moving."
Torque maxes at 90° → that's not equilibrium.
Equilibrium (τ=0) at 0° (stable, min energy) and 180° (unstable, max energy).
"A door needs 90° push for max torque — but the door is at equilibrium at 0° (closed)!"
Solenoid: B = μ₀nI (open cylinder, field extends along axis)
Toroid: B = μ₀NI/2πr (closed ring, field confined inside ring only, zero outside)
"Toroid is a solenoid that ate its own tail."
μ₀ = 4π × 10⁻⁷ T·m/A
"mu-naught: four-pi-naught-naught-7"
Or: 4π times one-ten-millionth
Imagine a city where electricity and magnetism are neighbors. When Mr. CURRENT (I) walks down CONDUCTOR Street (L) in the MAGNETIC FIELD (B) district, Officer FLEMING stops him with his LEFT HAND — Forefinger pointing toward B Street, Middle finger toward Current Lane, and Thumb pointing in the direction Mr. Current gets pushed (FORCE).
In the CIRCULAR PARK, charged particles (running at speed v) can't escape — the park radius r = mv/qB keeps them going in circles. The remarkable thing? The park's rotation period T = 2πm/qB never changes no matter how fast they run — that's why the CYCLOTRON MACHINE works perfectly.
Mayor AMPERE declared the law: go around any closed loop in the city, add up all B·dl contributions, and you get exactly μ₀ times all the enclosed current. The SOLENOID Tower (n floors per meter, current I) has a perfectly uniform field inside (B = μ₀nI) but zero field outside. And the TOROID Doughnut? Its field is completely trapped inside the ring — zero everywhere else!
| CORE CONCEPT | → LEADS TO | → APPLICATION |
|---|---|---|
| Moving Charge / Current | Creates Magnetic Field B | Motors, Electromagnets, MRI |
| Lorentz Force F=qvB sinθ | Circular / Helical Motion | Cyclotron, Mass Spectrometer, CRT |
| Biot–Savart Law | B for wire, loop, arc | Galvanometer coil design |
| Ampere's Circuital Law | B for solenoid, toroid | Solenoid magnets, Transformers |
| F = BIL sinθ | Force on conductors | DC Motor, Rail gun |
| Parallel wire force F/L | Defines Ampere (SI unit) | Precision current measurement |
| Torque τ = NIAB sinθ | Rotation of current loop | Electric motor, Galvanometer |
| Magnetic Moment M = NIA | Analogous to dipole moment | Atomic magnetism, MRI contrast |
The chapter 'Moving Charges and Magnetism' establishes the profound relationship between electricity and magnetism first discovered by Oersted in 1820 when he noticed that a current-carrying wire deflected a nearby compass needle. This discovery revealed that moving electric charges — forming electric current — produce magnetic fields in the surrounding space.
The force on a moving charge in a magnetic field is described by the Lorentz Force Law: F = q(E + v × B). The magnetic part F_B = q(v × B) is fundamentally different from electric force — it acts perpendicular to both velocity and field, meaning it can never do work on a charged particle and cannot change its kinetic energy; it only changes the direction of motion.
When a charged particle moves perpendicular to a uniform magnetic field, the constant perpendicular force creates uniform circular motion. The radius r = mv/qB depends on the particle's mass, speed, and charge. The remarkable discovery: the time period T = 2πm/qB and frequency f = qB/2πm are completely independent of the particle's speed — the principle exploited by the cyclotron to accelerate particles to high energies using alternating voltage synchronized with the natural orbital frequency.
The Biot–Savart Law provides a method to calculate B produced by any current distribution: dB = (μ₀/4π)(Idl sinθ/r²). It yields B = μ₀I/2πr for an infinite straight wire and B = μ₀I/2R at the centre of a circular loop. For symmetric configurations, Ampere's Circuital Law (∮B·dl = μ₀I_enclosed) offers a simpler approach, yielding B = μ₀nI inside a solenoid and B = μ₀NI/2πr inside a toroid.
The force on a current-carrying conductor F = BIL sinθ, derived from the Lorentz force on charge carriers, forms the basis of the SI definition of the Ampere. Same-direction currents attract; opposite-direction currents repel — the opposite of electrical charges.
A current loop in a magnetic field is a magnetic dipole with moment M = NIA. In uniform field, it experiences torque τ = MB sinθ with stable equilibrium (minimum energy U = −MB) at M parallel to B. This principle underlies the moving coil galvanometer, where deflecting magnetic torque is balanced by restoring spring torque, giving deflection proportional to current. Applications permeate modern technology — from electric motors and cyclotrons to MRI machines and maglev trains.