11.3 Magnetism & Electromagnetic Induction

11.3 Magnetism & Electromagnetic Induction

Quick Answer: Magnetic fields exert F = qvB sinθ on moving charges and F = ILB sinθ on wires. Changing magnetic flux induces EMF by Faraday's law, with direction fixed by Lenz's law. A rotating coil yields alternating EMF — the AC generator principle.

This magnetism review is application-first: forces, flux, induced EMF, and the conceptual AC story that powers aircraft electrical systems at a high level.

Magnetic Field and Force on a Moving Charge

A charge $q$ moving with velocity $\vec{v}$ in magnetic field $\vec{B}$ experiences:

$$F = qvB\sin\theta$$

where θ is the angle between $\vec{v}$ and $\vec{B}$. Unit of B: tesla (T) = N·s/(C·m). If v is parallel to B, F = 0; maximum force when v ⊥ B.

Direction (positive charge): right-hand rule — fingers in $\vec{v}$ direction, curl toward $\vec{B}$, thumb gives $\vec{F}$. Negate for negative charge (or use left hand).

Magnetic force is always perpendicular to velocity, so it changes direction of motion, not speed — kinetic energy is unchanged by magnetic force alone. Uniform B perpendicular to v → circular path; radius $r = mv/(qB)$ for speed v.

Worked example — force on a charge. A proton (q = 1.6 × 10⁻¹⁹ C) moves at 2.0 × 10⁶ m/s perpendicular to B = 0.50 T. Find F.

$$F = (1.6 \times 10^{-19})(2.0 \times 10^6)(0.50)\sin 90^\circ = 1.6 \times 10^{-13}\ \mathrm{N}$$

Force on a Current-Carrying Conductor

For a straight wire of length L carrying current I in field B:

$$F = ILB\sin\theta$$

θ between current direction and B. Right-hand rule: fingers along I, curl to B, thumb along F.

Two parallel wires attract if currents are same direction, repel if opposite — definition route historically used for the ampere.

Worked example — wire force. A 25 cm wire carries 4.0 A perpendicular to B = 0.30 T. Find F.

$$F = (4.0)(0.25)(0.30) = 0.30\ \mathrm{N}$$

Magnetic Flux

Magnetic flux through a surface:

$$\Phi = BA\cos\theta$$

where θ is the angle between $\vec{B}$ and the normal to the area A. Unit: weber (Wb) = T·m². Flux is maximum when B is perpendicular to the face (θ = 0 for the normal). For a coil of N turns, total flux linkage is $N\Phi$.

Faraday's Law and Lenz's Law

Faraday's law: induced EMF equals the negative rate of change of flux linkage:

$$\varepsilon = -N \frac{d\Phi}{dt}$$

Flux can change by changing B, area A, or orientation θ (rotation).

Lenz's law: the induced current's magnetic effect opposes the change in flux that caused it. The minus sign encodes this. If flux into a loop is increasing into the page, induced current tries to create flux out of the page.

Precise phrasing: "oppose the change," not "oppose the field." If the external field is decreasing, induced current tries to maintain the field.

Worked example — Faraday. A 100-turn coil of area 0.010 m² has B perpendicular to the plane increasing from 0 to 0.40 T in 0.20 s. Find average induced EMF.

$$\Delta\Phi = A\Delta B = 0.010 \times 0.40 = 4.0 \times 10^{-3}\ \mathrm{Wb}$$ $$|\varepsilon| = N\frac{\Delta\Phi}{\Delta t} = 100\frac{4.0 \times 10^{-3}}{0.20} = 2.0\ \mathrm{V}$$

Motional EMF

A conducting rod of length ℓ moving at speed v perpendicular to B (and to its length) has induced EMF:

$$\varepsilon = B\ell v$$

Charges in the rod experience magnetic force until the electric field balances it. If the rod slides on rails forming a closed loop, induced current $I = B\ell v / R$ and magnetic drag opposes the motion (Lenz again).

Worked example — motional EMF. B = 0.80 T, ℓ = 0.50 m, v = 2.0 m/s, all mutually perpendicular. Find ε. If circuit resistance is 0.40 Ω, find I.

$$\varepsilon = (0.80)(0.50)(2.0) = 0.80\ \mathrm{V},\quad I = 0.80/0.40 = 2.0\ \mathrm{A}$$

AC Idea — Generators and Transformers

A coil rotating in a uniform magnetic field has flux $\Phi = BA\cos(\omega t)$ (depending on angle definition). Differentiating gives a sinusoidal EMF:

$$\varepsilon = NBA\omega\sin(\omega t)$$

Peak EMF $\varepsilon_0 = NBA\omega$. This is the principle of an AC generator (alternator). Aircraft electrical systems use AC generation extensively; at FSc depth you need the sinusoidal idea and that frequency relates to rotation rate and pole count, not a full machine-design treatment.

Transformers use mutual induction: changing current in a primary coil changes flux through a secondary. For an ideal transformer:

$$\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}$$

Step-up: $N_s > N_p$ raises voltage, lowers current for same power. Step-down: opposite. Core losses and non-ideal behaviour exist, but introductory exercises usually assume the ideal ratio.

Worked example — transformer. Primary 200 turns at 220 V; secondary needs 11 V. Find N_s (ideal).

$$\frac{N_s}{200} = \frac{11}{220} \Rightarrow N_s = 10\ \mathrm{turns}$$

Direction and Sign Checklist

Bridging to Aeronautical Practice

Eddy currents in conducting airframe parts near changing fields dissipate heat — laminated cores in transformers reduce them. Lightning and static involve rapid electromagnetic phenomena; grounding and bonding again connect back to potential and current paths from earlier sections.

Section Checklist

• Always check whether sinθ or cosθ is needed (force vs flux definition).
• Average EMF uses ΔΦ/Δt; instantaneous uses dΦ/dt.
• Lenz questions are qualitative — state the opposition clearly.
• Generator EMF is AC because angle (hence flux) varies continuously with time.