Changing magnetic flux induces electromotive force (emf).

Motional emf (conductor moving in static field): $\mathcal{E} = \oint (\vec{v} \times \vec{B}) \cdot d\vec{l}$

For rod of length L moving with velocity v perpendicular to B: $\mathcal{E} = BLv$

Induced emf (changing B, stationary conductor): $\mathcal{E} = -\frac{d\Phi_B}{dt}$

General case (both motion and changing B): $\mathcal{E} = \oint (\vec{v} \times \vec{B}) \cdot d\vec{l} - \frac{d\Phi_B}{dt}$

Self-inductance L:

• Changing current in coil induces emf in same coil
• $\mathcal{E} = -L\frac{dI}{dt}$

Mutual inductance M:

• Changing current in coil 1 induces emf in coil 2
• $\mathcal{E}_2 = -M\frac{dI_1}{dt}$

Reciprocity: $M_{12} = M_{21}$

$\Phi_B = \int \vec{B} \cdot d\vec{A} = BA\cos\theta$

Where:

• $\vec{B}$ = magnetic field
• $d\vec{A}$ = area element
• $\theta$ = angle between B and area normal

Units: Weber (Wb) = Tesla - meter2 (T - m2)

For N-turn coil: $\Phi_B = N \int \vec{B} \cdot d\vec{A}$

The induced emf in a closed loop equals negative rate of change of magnetic flux.

$\mathcal{E} = -\frac{d\Phi_B}{dt}$

For N-turn coil: \mathcal{E} = -N\frac{d\Phi_B{dt}}

Physical meaning:

• Changing flux -> induced emf
• Negative sign: Lenz's law (opposition to change)

Ways to change flux:

1. Change B (vary magnetic field strength)
2. Change A (move coil, change area)
3. Change $\theta$ (rotate coil)

The direction of induced current creates magnetic field opposing the change in flux that produced it.

Physical basis:

• Consistency with conservation of energy
• Prevents runaway induced current

Procedure:

1. Determine direction of flux change (increasing/decreasing)
2. Induced B must oppose this change
3. Use right-hand rule to find current direction

Applications:

• Braking in eddy current devices
• Direction determination in circuits

EMF induced by conductor moving through magnetic field.

$\mathcal{E} = \oint (\vec{v} \times \vec{B}) \cdot d\vec{l}$

For straight rod of length L: $\mathcal{E} = BLv\sin\theta$

Where:

• B = magnetic field
• L = length in field
• v = velocity
• $\theta$ = angle between v and B

Direction: same as force on positive charges in moving conductor

Applications:

• Generators
• Railguns
• Velocity sensors

Changing magnetic field produces non-conservative electric field.

$\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}$

Key difference from electrostatics:

• E-field is NOT conservative (curl nonzero)
• Cannot define scalar potential globally
• E-field forms closed loops around changing B

Applications:

• Transformers
• Induction heating
• Betatron

Property of coil opposing change in its own current.

$\mathcal{E} = -L\frac{dI}{dt}$

Where L = self-inductance (Henry, H)

Physical meaning:

• Inductor has "electrical inertia"
• Larger L = harder to change current

Unit of inductance.

$1\ \text{H} = 1\ \text{V - s/A} = 1\ \text{Wb/A} = 1\ \text{Ω - s}$

Physical meaning:

• 1 H: changing current at 1 A/s produces 1 V of opposing emf

$L = \frac{\mu_0 N^2 A}{l} = \mu_0 n^2 Al = \mu_0 n^2 V$

Where:

• N = number of turns
• A = cross-sectional area
• l = length
• n = N/l (turns per unit length)
• V = Al (volume)

Energy stored in magnetic field of inductor:

$U_L = \frac{1}{2}LI^2$

Derivation (integrating power): $U = \int_0^I P\,dt = \int_0^I \mathcal{E}I\,dt = \int_0^I L I \frac{dI}{dt}\,dt = \frac{LI^2}{2}$

Magnetic energy density: $u_B = \frac{U}{Volume} = \frac{B^2}{2\mu_0}$

For solenoid: $U = \frac{B^2}{2\mu_0} \cdot Al = \frac{1}{2}LI^2$

Maxwell's addition to Ampere's law.

Problem with original Ampere's law: fails for capacitor charging (current flows but no enclosed current between plates).

Solution: displacement current: $I_d = \varepsilon_0\frac{d\Phi_E}{dt}$

Where $\Phi_E$ = electric flux

Physical meaning:

• Changing electric field acts like current
• Produces magnetic field
• Completes symmetry with Faraday's law

Electromagnetic waves propagating through space.

Wave equation solutions:

• Propagation speed: $c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} = 3.00 \times 10^8$ m/s
• E and B perpendicular to each other and to propagation direction
• Transverse waves
• $E/B = c$ (ratio of field magnitudes)

Properties:

• Self-sustaining: changing E produces B, changing B produces E
• Carry energy: $S = \frac{EB}{\mu_0}$ (Poynting vector)
• Carry momentum: $p = E/c^2$
• Can be polarized

Spectrum: radio, microwave, infrared, visible, ultraviolet, X-ray, gamma

Modified Ampere's law including displacement current.

$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc} + \mu_0\varepsilon_0\frac{d\Phi_E}{dt}$

Differential form: $\nabla \times \vec{B} = \mu_0\vec{J} + \mu_0\varepsilon_0\frac{\partial\vec{E}}{\partial t}$

Key addition: $\mu_0\varepsilon_0\frac{d\Phi_E}{dt}$ (displacement current term)

$I_d = \varepsilon_0\frac{d\Phi_E}{dt} = \varepsilon_0 A \frac{dE}{dt}$

Units: Amperes (A)

Physical significance:

• Allows electromagnetic wave propagation
• Symmetric with Faraday's law
• Essential for consistency with charge conservation

Integral form:

1. Gauss's Law for E: $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\varepsilon_0}$
2. Gauss's Law for B: $\oint \vec{B} \cdot d\vec{A} = 0$ (no magnetic monopoles)
3. Faraday's Law: $\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}$
4. Ampere-Maxwell Law: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc} + \mu_0\varepsilon_0\frac{d\Phi_E}{dt}$

Differential form:

1. $\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}$
2. $\nabla \cdot \vec{B} = 0$
3. $\nabla \times \vec{E} = -\frac{\partial\vec{B}}{\partial t}$
4. $\nabla \times \vec{B} = \mu_0\vec{J} + \mu_0\varepsilon_0\frac{\partial\vec{E}}{\partial t}$

These equations completely describe classical electromagnetism.

From Kirchhoff's loop rule:

$\mathcal{E} - IR - L\frac{dI}{dt} = 0$

$L\frac{dI}{dt} + RI = \mathcal{E}$

First-order linear differential equation.

Charging (switch closed, I(0) = 0): $I(t) = \frac{\mathcal{E}}{R}\left(1 - e^{-Rt/L}\right)$

Discharging (source removed, I(0) = I0): $I(t) = I_0 e^{-Rt/L}$

$\tau_L = \frac{L}{R}$

Units: seconds

Physical meaning: characteristic time for current changes.

At t = τ: reaches 63.2% of final value At t = 5τ: essentially complete (99.3%)

Undamped LC circuit (no resistance):

Differential equation: $L\frac{d^2Q}{dt^2} + \frac{Q}{C} = 0$

This is harmonic oscillator equation!

Angular frequency: $\omega = \frac{1}{\sqrt{LC}}$

Frequency: $f = \frac{\omega}{2\pi} = \frac{1}{2\pi\sqrt{LC}}$

Charge (analogous to displacement): $Q(t) = Q_{max}\cos(\omega t + \phi)$

Current (analogous to velocity): $I(t) = -Q_{max}\omega\sin(\omega t + \phi)$

Energy oscillates between electric (capacitor) and magnetic (inductor).

$U_{total} = U_C + U_L = \frac{Q^2}{2C} + \frac{1}{2}LI^2 = \text{constant}$

Energy exchange:

• Q = max, I = 0: all energy in capacitor
• Q = 0, I = max: all energy in inductor
• Continuous exchange at frequency ω

RLC circuit (damped): $L\frac{d^2Q}{dt^2} + R\frac{dQ}{dt} + \frac{Q}{C} = 0$

Damping regimes:

• Underdamped ($R < 2\sqrt{L/C}$): oscillations decay
• Critically damped ($R = 2\sqrt{L/C}$): fastest approach to equilibrium
• Overdamped ($R > 2\sqrt{L/C}$): no oscillations