Unit 4: Electromagnetism

Magnetic flux measures the total B-field piercing a surface:

For uniform field and flat surface: \(\Phi_B = BA\cos\theta\), where \(\theta\) is the angle between \(\vec{B}\) and the surface normal \(\hat{n}\). Units: webers (Wb = T·m²).

• \(\theta = 0°\) (B ∥ normal): max flux \(\Phi_B = BA\)
• \(\theta = 90°\) (B ∥ surface): zero flux
• \(\theta = 180°\) (B anti-parallel): \(\Phi_B = -BA\)

Changing magnetic flux through a loop induces an EMF:

For a coil with N turns: \(\mathcal{E} = -N\frac{d\Phi_B}{dt}\)

Flux can change through three mechanisms:

1. Changing B: moving a magnet, turning an electromagnet on/off
2. Changing area: deforming loop, sliding rod on rails
3. Changing orientation: rotating the loop (electric generators!)

Lenz's Law: The induced current's magnetic flux opposes the change in the original flux. This is the minus sign in Faraday's law — nature resists changes in magnetic flux.

3-step strategy:

1. Identify direction of original \(\Phi_B\) and whether it's increasing or decreasing
2. The induced B-field must oppose the change:
   • \(\Phi_B\) increasing → induced B opposite to original B
   • \(\Phi_B\) decreasing → induced B same direction as original B
3. Use RHR to find current direction that creates the induced B

alen: N pole approaching loop → increasing flux into loop → induced B points out → current = CCW as viewed from magnet.

When a conductor moves through B, charges experience magnetic force → separation → induced EMF:

Sliding rod on rails: The rod's motion changes loop area → changing flux:

Magnetic force on induced current opposes motion: \(F_B = I\ell B = B^2\ell^2 v/R\). To maintain constant v, apply external force \(F_{\text{ext}} = F_B\).

Electric Generators: A coil rotating in a uniform B-field produces sinusoidal EMF:

where N = turns, A = coil area, ω = angular frequency. Peak EMF: \(\mathcal{E}_{\text{max}} = NBA\omega\). This is the principle behind AC generators (alternators).

Self-inductance L describes opposition to current change:

Units: henries (H = V·s/A). For a solenoid: \(L = \mu_0 n^2 A\ell = \mu_0 N^2 A/\ell\).

Inductor behavior at extremes:

• t = 0 (switch closed): inductor = open circuit (I cannot jump)
• t → ∞ (steady DC): inductor = short circuit (V_L = 0)

Energy stored: \(U = \frac{1}{2}LI^2\) — dual to capacitor \(U = \frac{1}{2}CV^2\).

Energy density of the magnetic field:

This is the magnetic analog of \(u_E = \frac{1}{2}\varepsilon_0 E^2\). For a solenoid: \(U = u_B \cdot A\ell = \frac{B^2}{2\mu_0}A\ell = \frac{1}{2}LI^2\).

Mutual Inductance (M): When changing current in coil 1 induces EMF in coil 2:

M depends on geometry of both coils and their relative position. Units: henries (H). The basis for transformers.

RL Circuits — time constant \(\tau_L = L/R\):

Current growth (connected to battery):

Current decay (battery removed):

RL and RC are mathematically identical — I in RL ↔ V in RC. Both first-order exponentials.

LC Circuits — no resistance, perpetual energy oscillation:

\(q(t) = Q_0\cos(\omega_0 t)\), \(I(t) = -\omega_0 Q_0\sin(\omega_0 t)\). Energy: \(U_{\text{total}} = \frac{Q_0^2}{2C} = \frac{1}{2}LI_{\text{max}}^2\) (constant).

Mechanical analogy: L ↔ mass (inertia), C ↔ 1/k (spring), \(\omega_0 = \sqrt{k/m} \leftrightarrow \omega_0 = 1/\sqrt{LC}\)

Transformers: Two coils wound on a shared iron core. AC in the primary creates changing flux, inducing EMF in the secondary:

For an ideal transformer: \(P_p = P_s \Rightarrow V_p I_p = V_s I_s\). Step-up: \(N_s > N_p\) (higher voltage, lower current). Step-down: \(N_s < N_p\). Used in power distribution.

Eddy Currents: Changing flux through bulk conductors induces circulating currents that dissipate energy as heat (\(I^2R\)). Applications: induction braking (trains), metal detectors, induction cooktops. Lamination of iron cores reduces eddy current losses.

Adding resistance to an LC circuit produces damped oscillation. Current oscillates at:

(derived from LRC differential equation; AP expects qualitative understanding only).

• Underdamped (small R): oscillations decay exponentially
• Critically damped (R² = 4L/C): fastest return to equilibrium without oscillation
• Overdamped (large R): slow, non-oscillatory decay

Resonance: At the natural frequency \(\omega_0 = 1/\sqrt{LC}\), the impedance is minimized and current amplitude is maximized — the basis for radio tuning.