Moving Charges and Magnetism
'Electricity and magnetism are two faces of the SAME coin — a moving charge creates a magnetic field, and a magnetic field affects moving charges.'
1. Chapter Overview
This chapter reveals the CONNECTION between electricity and magnetism. Topics include: the MAGNETIC FIELD due to a current-carrying wire (Biot-Savart law), AMPERE'S CIRCUITAL LAW (magnetic equivalent of Gauss's law), the FORCE on a moving charge in a magnetic field (Lorentz force), the FORCE between two parallel current-carrying wires, the TORQUE on a current loop in a magnetic field, and the MOVING COIL GALVANOMETER.
2. Biot-Savart Law
- Magnetic field due to a current element: dB⃗ = (μ₀/4π) (I dl⃗ × r̂)/r²
- μ₀ = 4π × 10⁻⁷ T·m/A (permeability of free space).
Magnetic Field Due to a Long Straight Wire
- B = μ₀I/(2πr). 'The field circles the wire — strength decreases as 1/r.'
Magnetic Field at the Centre of a Circular Loop
- B = μ₀I/(2R) (for a single turn). B = μ₀nI/(2R) for n turns.
- 'Right-hand rule: Curl your fingers along B, thumb in direction of current.'
Magnetic Field on the Axis of a Circular Loop
- B = μ₀IR²/[2(R² + x²)³/²] — at centre (x=0): B = μ₀I/(2R).
3. Ampere's Circuital Law
- ∮ B⃗ · dl⃗ = μ₀ I_enc. 'The line integral of B around a closed loop equals μ₀ times the current enclosed.'
- 'Ampere's law is to magnetism what Gauss's law is to electrostatics — a powerful symmetry tool.'
Applications
| Configuration | Magnetic Field | Amperian Loop |
|---|---|---|
| Infinite straight wire | B = μ₀I/(2πr) | Circle centred on wire |
| Solenoid (ideal) | B = μ₀nI inside, 0 outside | Rectangular loop |
| Toroid | B = μ₀NI/(2πr) | Circle inside toroid |
4. Force on a Moving Charge — Lorentz Force
- F = q(E + v × B). 'The total electromagnetic force on a charge.'
- Magnetic force only: F = q(v × B). Magnitude: F = qvB sin θ.
- Direction: PERPENDICULAR to BOTH v and B (right-hand rule).
- 'The magnetic force NEVER does work — it always acts perpendicular to velocity, changing only DIRECTION, not speed.'
Motion in a Uniform Magnetic Field
- v ⟂ B: CIRCULAR motion. Radius r = mv/(qB). Period T = 2πm/(qB).
- v ∥ B: STRAIGHT LINE (no force).
- v at angle θ: HELICAL path.
5. Force on a Current-Carrying Conductor
- F = I(L × B). F = ILB sin θ.
- Force between two parallel wires: F/L = μ₀I₁I₂/(2πd).
- SAME direction → ATTRACT. OPPOSITE direction → REPEL.
- 'This force DEFINES the Ampere — the unit of current.'
6. Torque on a Current Loop
- τ = N I A B sin θ (where θ is angle between normal to loop and B).
- Magnetic dipole moment: m = N I A (direction: along the normal per right-hand rule).
- τ = m × B. 'A current loop behaves like a magnetic dipole — it ALIGNS with the external field.'
7. Moving Coil Galvanometer (MCG)
- Principle: Torque on a current-carrying coil in a magnetic field.
- Current sensitivity: S_I = θ/I = NBA/k (k = torsion constant).
- Voltage sensitivity: S_V = θ/V = NBA/(kR_g).
- Conversion: Galvanometer → Ammeter: Add a LOW RESISTANCE SHUNT in PARALLEL. Galvanometer → Voltmeter: Add a HIGH RESISTANCE in SERIES.
| Instrument | How to Convert | Key Formula |
|---|---|---|
| Galvanometer to Ammeter | Shunt (low R) in parallel | S = I_g·G/(I − I_g) |
| Galvanometer to Voltmeter | High R in series | R = V/I_g − G |
8. Comparison Table: Electric Field vs Magnetic Field
| Property | Electric Field (E) | Magnetic Field (B) |
|---|---|---|
| Source | Stationary charges | Moving charges / currents |
| Force on charge q | F = qE (parallel to E) | F = q(v×B) (perp to v and B) |
| Lines | Start/end at charges | CLOSED LOOPS (no monopoles) |
| Work done | Can do work | ZERO work (perp to velocity) |
| Flux law | Gauss: Φ_E = q/ε₀ | Gauss: Φ_B = 0 (no monopoles) |
| Circulation law | Conservative: ∮E·dl = 0 | Ampere: ∮B·dl = μ₀I |
9. Common Mistakes
- Direction of magnetic force: Use the RIGHT-HAND RULE. Thumb = velocity (or current), fingers = B, palm = force. For negative charges, REVERSE the direction.
- Magnetic field direction around a wire: NOT radial — it is CIRCULAR (circles around the wire). Unlike E which is radial.
- Solenoid formula: B = μ₀nI, where n = N/L (turns per unit length). Do NOT use N alone.
- Ammeter and voltmeter connections: Ammeter in SERIES (low resistance). Voltmeter in PARALLEL (high resistance). Shunt for ammeter, multiplier for voltmeter.
10. CBSE Exam Focus
- Biot-Savart law — field due to straight wire, circular loop
- Ampere's circuital law — field due to solenoid, toroid
- Lorentz force — force on a moving charge, cyclotron radius
- Force between two parallel current-carrying wires
- Torque on a current loop — magnetic dipole moment
- Galvanometer — conversion to ammeter and voltmeter (SHUNT and MULTIPLIER)
11. Self-Test
Q1: A long straight wire carries 5 A. Find B at a distance of 10 cm from the wire. A1: B = μ₀I/(2πr) = (4π×10⁻⁷)(5)/(2π×0.1) = (2×10⁻⁷×5)/0.1 = 10⁻⁵ T.
Q2: A proton (q=1.6×10⁻¹⁹ C, m=1.67×10⁻²⁷ kg) moves perpendicular to a 0.5 T field at 2×10⁶ m/s. Find the radius of its path. A2: r = mv/(qB) = (1.67×10⁻²⁷×2×10⁶)/(1.6×10⁻¹⁹×0.5) = (3.34×10⁻²¹)/(8×10⁻²⁰) = 0.04175 m = 4.18 cm.
Q3: A solenoid has 1000 turns per metre and carries 2 A. Find B inside. A3: B = μ₀nI = (4π×10⁻⁷)(1000)(2) = 8π×10⁻⁴ T ≈ 2.51×10⁻³ T.
Q4: A galvanometer has G = 50 Ω and gives full-scale deflection for I_g = 5 mA. Convert it to an ammeter of range 5 A. A4: S = I_g·G/(I − I_g) = (5×10⁻³×50)/(5 − 0.005) = 0.25/4.995 = 0.05 Ω ≈ 0.05 Ω (shunt).
Q5: Two long parallel wires carry 3 A and 4 A in the SAME direction, 5 cm apart. Find the force per unit length between them. A5: F/L = μ₀I₁I₂/(2πd) = (4π×10⁻⁷×3×4)/(2π×0.05) = (4π×10⁻⁷×12)/(0.1π) = 48×10⁻⁷/0.1 = 4.8×10⁻⁵ N/m (ATTRACTIVE).
12. Conclusion
Moving charges and magnetism UNIFY electricity and magnetism:
- BIOT-SAVART: 'The building block — current elements produce magnetic fields.'
- AMPERE'S LAW: 'Sum up the field around a closed path — equals μ₀ × current enclosed.'
- LORENTZ FORCE: 'The force on a moving charge — the basis of electric motors and generators.'
- GALVANOMETER: 'Current measured through torque on a coil — the heart of analog meters.'
'Where charges move, magnetic fields appear. Where magnetic fields change, currents flow. This INSEPARABLE pair is electromagnetism.'
