Magnetics & Maxwell | EE-I Exam Prep

Magnetostatics Overview

What Produces a Magnetic Field?

A constant current (charges moving at constant velocity) produces a magnetostatic field — analogous to how static charges produce an electrostatic field.

⚡ Exam Key

Static charges → Electrostatic field (E)
Steady current → Magnetostatic field (H, B)
Time-varying currents → Electromagnetic waves
Electric charges can be isolated; magnetic poles always come in pairs (N-S)

Electrostatics vs Magnetostatics

Electrostatics	Magnetostatics
Coulomb's Law	Biot-Savart Law
Gauss's Law	Ampere's Law
Polarization & D	Magnetization & B
Boundary conditions	Boundary conditions
Capacitance C	Inductance L

🎯 Tricky MCQ — H vs μ

For an infinite wire: H = I/(2πρ) — H is independent of μ (material doesn't affect H from a wire).
But B = μH — so B is proportional to μ. This distinction appears in MCQs!

Magnetic Flux Density & Permeability

Key Formulas

Flux Density

B = μH [Tesla = Wb/m²]

μ = μᵣ · μ₀, μ₀ = 4π×10⁻⁷ H/m

Magnetic Flux

Φ = B · A [Webers, Wb]

For uniform B perpendicular to area A

General (surface integral)

Φ = ∮ B · dS

Flux lines are always closed — no monopoles!

Classification of Magnetic Materials

Type	μᵣ	Example
Diamagnetic	μᵣ < 1 (≈1)	Gold (0.99996), Water (0.99999)
Paramagnetic	μᵣ > 1 (≈1)	Air (1.000004), Al (1.00002)
Ferromagnetic	μᵣ >> 1	Iron (4000–5000)

💡 Memory Trick

Non-magnetic materials have μᵣ = 1. Ferromagnetics have μᵣ thousands of times larger — great for transformer cores to concentrate flux!

📌 Key Properties of Magnetic Flux Lines

Always closed loops — no beginning or end (no magnetic monopoles)
Never cross each other regardless of current distribution
Direction: inside magnet from S→N, outside from N→S
Denser lines = stronger field

Maxwell's Equations

Four equations that describe all classical electromagnetism — both static and time-varying fields.

Static Fields (Electrostatics + Magnetostatics)

1. Gauss's Law (E)

∇ · D = ρᵥ

Divergence of D equals volume charge density. D = εE. Charges are sources of E field.

2. No Magnetic Monopoles

∇ · B = 0

Divergence of B is always zero — magnetic field lines have no source or sink.

3. Conservativeness of E

∇ × E = 0

Curl of electrostatic field is zero — it's conservative (path-independent work).

4. Ampere's Law (Static)

∇ × H = J

Curl of H equals current density J. Current is the source of magnetic field.

Time-Varying Fields (Electromagnetic Waves)

1. Gauss's Law (unchanged)

∇ · D = ρᵥ

Same as static — charges still source E field.

2. No Magnetic Monopoles (unchanged)

∇ · B = 0

Same — still no magnetic monopoles.

3. Faraday's Law ✨ NEW

∇ × E = −∂B/∂t

Changing B creates E field — replaces ∇×E = 0

4. Ampere's Law ✨ NEW

∇ × H = J + ∂D/∂t

Added displacement current ∂D/∂t — enables EM waves!

🎯 What Changes Between Static and Time-Varying?

Equations 1 & 2 are identical (∇·D = ρᵥ and ∇·B = 0)
∇×E = 0 (static) becomes ∇×E = −∂B/∂t (dynamic) — Faraday coupling
∇×H = J (static) becomes ∇×H = J + ∂D/∂t (dynamic) — Maxwell added the ∂D/∂t displacement current term
Static fields are independent; time-varying E and B are interdependent (waves!)

Faraday's Law & Lenz's Law

Faraday's Law — Induced EMF

V_emf = −N · dΦ/dt

N = number of turns · Φ = flux through each turn (Wb) · V_emf in Volts

Faraday's law: changing B through a coil induces an opposing EMF (Lenz's Law)

Lenz's Law — Direction of Induced Current

The negative sign in Faraday's Law is Lenz's Law: the induced current creates a magnetic field that opposes the change in flux.

Switch Closed (t=0)

Flux increases →

Induced current opposes increase → field opposing B

After Steady State

Flux constant →

dΦ/dt = 0 → No induced current!

Switch Opened

Flux decreases →

Induced current maintains flux → field in same direction as B

✏️ Tutorial 9 — Problem 1

Given: Coil with N = 200 turns, square side = 18 cm = 0.18 m. B changes linearly from 0 → 0.50 T in 0.80 s. Total R = 2 Ω.

Step 1 — Area: A = (0.18)² = 0.0324 m²

Step 2 — ΔΦ: ΔΦ = A · ΔB = 0.0324 × (0.5 − 0) = 0.0162 Wb

Step 3 — EMF: V_emf = N · |ΔΦ/Δt| = 200 × (0.0162/0.80)

V_emf = 4.05 V | I = V/R = 4.05/2 = 2.025 A

⚠️ If coil ends are open (no circuit), current = 0 regardless of EMF!

✏️ Tutorial 9 — Problem 3

Given: Air-core solenoid, N = 300 turns, l = 25 cm = 0.25 m, A = 4.00 cm² = 4×10⁻⁴ m². Current decreasing at 50.0 A/s.

Part A — Inductance:
L = μ₀N²A/l = (4π×10⁻⁷ × 300² × 4×10⁻⁴) / 0.25

Part B — Self-induced EMF:
di/dt = −50 A/s (negative = decreasing)
V_L = −L · (di/dt) = −0.181×10⁻³ × (−50)

L = 0.181 mH | V_L = 9.05 mV

Inductance of a Solenoid

Air-core solenoid — inductance proportional to N², μ, A and inversely to length l

Inductance Formula Derivation

Magnetic field inside solenoid

B = μNI/l

Flux through each turn

Φ = BA = μNIA/l

Total flux linkage → Inductance

L = NΦ/I = μN²A/l

Valid when l >> r (long solenoid assumption)

How to Increase Inductance

🔧 Design Tips

More turns N — L ∝ N² (doubling N quadruples L!)
Use ferromagnetic core — iron has μᵣ = 4000-5000, multiplying L by thousands
Larger cross-section A — L ∝ A
Shorter length l — L ∝ 1/l

Self-induced EMF

V_L = −L · di/dt

Negative sign = opposes change in current (Lenz!)

Transformers

Ideal transformer — voltage ratio equals turns ratio

Turns Ratio

Transformer Voltage Ratio

V₁/V₂ = N₁/N₂

V in volts · N = number of turns

Type	Condition	Use
Step-Up	V₂ > V₁	Power transmission (11kV → 220kV)
Step-Down	V₂ < V₁	Substations (220kV → 220V home)

How Transformers Work

Primary coil connected to AC source (V₁) carries alternating current
Changing current creates a time-varying magnetic flux Φ in the core
Faraday's Law: changing flux induces EMF in secondary coil (V₂)
Turns ratio determines voltage step-up or step-down

⚠️ Important

Transformers only work with AC — DC produces constant flux so dΦ/dt = 0 and no EMF is induced in secondary!

Real-World Applications

🔋

Wireless Charger

Uses Faraday's law: alternating current in transmitter coil creates changing B, inducing EMF in phone receiver coil.

⚡

Power Transformer

Step-up at generation (11kV → 440kV) for efficient long-distance transmission; step-down at substations for safety.

🎙️

Microphone

Sound moves coil in magnetic field → changing flux → induced EMF → electrical signal (moving-coil mic).

📻

RF Choke (Decoupling Inductor)

Inductor in series with load: blocks AC at high frequencies (Z_L = jωL → ∞), passes DC to load.

📡 Tutorial 9 — Problem 4: AC Frequency & Inductor

Lightbulb in series with inductor — brightest at low frequencies.
At low ω: Z_L = jωL → 0 (short circuit) — full voltage across bulb.
At high ω: Z_L = jωL → ∞ (open circuit) — no current flows, bulb off.

Magnetics Quick Reference

Quantity	Symbol	Formula	Units
Magnetic field intensity	H	I / (2πρ) — infinite wire	A/m
Permeability (free space)	μ₀	4π × 10⁻⁷	H/m
Flux density	B	μH = μᵣμ₀H	T (Tesla)
Magnetic flux	Φ	B · A	Wb (Weber)
Faraday EMF	V_emf	−N · dΦ/dt	V
Solenoid inductance	L	μN²A / l	H (Henry)
Self-induced voltage	V_L	−L · di/dt	V
Transformer ratio	N₁/N₂	V₁/V₂	dimensionless

🎯 Magnetics Exam Tips

H vs B for infinite wire: H = I/2πρ (independent of μ!) — B = μH (depends on μ). MCQ trap!
Lenz's Law sign: The −N in V_emf = −N·dΦ/dt means induced current OPPOSES the change — always state this when asked for direction.
Open circuit EMF: EMF can be induced, but current = 0 if no closed path! V_emf exists, I = 0.
Solenoid L formula: L = μN²A/l — double N → 4× L. Double l → half L. Use μ = μ₀ for air-core.
Transformer = AC only: DC steady state → constant flux → dΦ/dt = 0 → no secondary voltage.
Maxwell's changed equations: Static: ∇×E=0 and ∇×H=J. Dynamic: add −∂B/∂t and +∂D/∂t respectively.