Magnetic fields

Magnetic fields

1. What a Magnetic Field Is

Magnetic field $B$ (magnetic flux density, Tesla T) exerts force on moving charges: Lorentz force

So $B$ acts on currents, not static charges.

Field lines form closed loops (no magnetic monopole: $∇·B=0$).

2. Sources — Biot-Savart and Ampère

• Biot-Savart law: field due to current element $I dl$ at $r$:

• For infinite straight wire current $I$ at distance $r$:

(circles wire, right-hand rule).

• For circular loop radius $a$ on axis distance $z$:

• Ampère's law (magnetostatics):

with $I_{enc}$ current through surface bounded by $C$.

• For infinite wire: choose circle radius $r$ ⇒ $B·2πr=μ_0 I ⇒ B=μ_0 I/2πr$ (matches Biot-Savart).
• For solenoid $n$ turns/m:

inside (uniform, parallel to axis), $≈0$ outside (ideal infinite).

• For toroid:

inside.

3. Force and Torque

• Force on straight wire length $l$:

(if $l$ ∥ $B$, zero; $⊥$, maximal).

• Torque on loop magnetic moment $μ = I·Area \hat{n}$:

tends to align loop.

4. Worked Examples

(a) Infinite wire I=1 A at r=1 cm: B= μ0·1/(2π·0.01)=2e-6? Actually μ0=4πe-7 ⇒ 4πe-7·1/(2π·0.01)=2e-5 T=20 µT ✓

(b) Solenoid n=1000 m⁻¹, I=2 A: B=4πe-7·1000·2=2.5e-3 T=2.5 mT ✓

(c) Loop R=0.05 m, I=1 A at centre: B= μ0 I/(2R)=12.6 µT ✓

(d) Right-hand rule: thumb along current, fingers curl along B.

5. Exam Points

• Define Lorentz $F=q(v×B)$ — 3 marks, frequent.
• Biot-Savart and Ampère's law — 5 marks, very frequent.
• Derive $B=μ0I/2πr$ for wire and $B=μ0 nI$ for solenoid via Ampère — 7 marks, very frequent (common derivation).
• Direction via right-hand rule — 3 marks.