1.3 Faraday's law

Faraday's law of induction explains how changing magnetic fields create electric fields (and, in conductors, electric currents). This principle underlies generators, transformers, and inductors, and it provides one of the four Maxwell's equations. For an Electromagnetism II course, you need to be comfortable not just with the integral statement of the law but also with its differential form, its connection to Lenz's law, and how it fits into the full Maxwell framework.

Faraday's Law of Induction

Faraday's law connects two seemingly different phenomena: time-varying magnetic fields and induced electric fields. Unlike the electrostatic case where $\oint \vec{E} \cdot d\vec{l} = 0$, a changing magnetic flux makes the electric field non-conservative. The circulation of $\vec{E}$ around a closed loop is no longer zero; instead, it equals the negative time rate of change of the magnetic flux through any surface bounded by that loop.

This is what distinguishes electrostatics from electrodynamics. Once fields vary in time, electric and magnetic fields become coupled, and Faraday's law is the first place you see that coupling explicitly.

Magnetic Flux and Flux Linkage

Magnetic Flux Through a Surface

Magnetic flux quantifies how much magnetic field threads through a given surface. It depends on three things: the field strength, the surface area, and the angle between the field and the surface normal.

$\Phi_B = \int_S \vec{B} \cdot d\vec{A}$

• $d\vec{A}$ is the vector area element, directed along the outward normal to the surface
• The dot product means only the component of $\vec{B}$ perpendicular to the surface contributes
• Units: weber (Wb), equivalent to T·m²

For a uniform field through a flat surface of area $A$, this simplifies to $\Phi_B = BA\cos\theta$, where $\theta$ is the angle between $\vec{B}$ and the surface normal. Keep in mind that the choice of surface matters: Faraday's law holds for any surface bounded by the loop, but you'll typically pick the simplest one.

Flux Linkage in a Coil

For a coil with $N$ turns, the total flux linkage is:

$\lambda = N\Phi_B$

This assumes the same flux threads every turn. In practice, not all turns may link the same flux (especially in loosely wound coils), but for tightly wound coils this is an excellent approximation. Flux linkage has units of Wb·turns, and it's the quantity that directly determines the induced emf in multi-turn coils.

Induced Electromotive Force (emf)

Faraday's Experiments

Faraday's key experimental observations (1831) can be summarized in three points:

1. Moving a magnet toward or away from a coil produces a current in the coil, but only while the magnet is moving.
2. The faster the relative motion (i.e., the faster the flux changes), the larger the induced current.
3. More turns in the coil produce a proportionally larger effect.

The critical insight is that it's the change in flux that matters, not the flux itself. A steady, unchanging magnetic field through a loop induces nothing.

Mathematical Formulation of Faraday's Law

Integral form:

$\oint_{\partial S} \vec{E} \cdot d\vec{l} = -\frac{d}{dt}\int_S \vec{B} \cdot d\vec{A}$

The left side is the emf around the closed loop $\partial S$. For a coil with $N$ identical turns:

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

Differential form (using Stokes' theorem):

$\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$

This is the form you'll use most often in Electromagnetism II. It tells you that a time-varying $\vec{B}$ at a point in space produces a curling $\vec{E}$ field at that same point. No conductor is needed; the induced electric field exists in free space.

The negative sign encodes Lenz's law and is required for consistency with energy conservation.

Lenz's Law and Conservation of Energy

Direction of Induced Current

Lenz's law states: the induced current flows in the direction that creates a magnetic field opposing the change in flux that produced it.

• If the flux through a loop is increasing, the induced current flows in the direction that creates a magnetic field opposing the increase (i.e., opposing the external field).
• If the flux is decreasing, the induced current creates a field that tries to maintain the flux.

To apply Lenz's law in practice:

1. Determine whether the magnetic flux through the loop is increasing or decreasing.
2. The induced current must produce a magnetic field that opposes that change.
3. Use the right-hand rule to find the current direction that produces the opposing field.

Energy Considerations in Induction

Lenz's law is really a statement about energy conservation. If the induced current aided the flux change instead of opposing it, you'd get a runaway process: more flux → more current → more flux → and so on, creating energy from nothing.

• The mechanical work you do pushing a magnet toward a coil gets converted into electrical energy in the circuit (and ultimately into heat via Joule dissipation, $P = I^2R$).
• In a generator, mechanical work against the back-emf is converted to electrical energy.
• In a motor, electrical energy does work against mechanical loads.

Energy is always conserved. The induced emf acts as the intermediary that converts between mechanical and electrical energy.

Applications of Faraday's Law

Generators and Alternators

A generator works by rotating a coil of area $A$ with $N$ turns in a uniform magnetic field $B$. The flux through the coil varies as:

$\Phi_B = BA\cos(\omega t)$

Applying Faraday's law gives the induced emf:

$\mathcal{E} = NBA\omega\sin(\omega t)$

This is a sinusoidal AC voltage with peak value $\mathcal{E}_0 = NBA\omega$. Alternators use slip rings to maintain continuous contact with the rotating coil, producing AC output. DC generators use a split-ring commutator to rectify the output.

Transformers and Power Transmission

A transformer consists of two coils (primary and secondary) wound on a shared magnetic core. An AC voltage in the primary coil creates a time-varying flux in the core, which induces an emf in the secondary coil.

For an ideal transformer (no flux leakage, no resistive losses):

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

Power is conserved ($V_p I_p = V_s I_s$), so stepping up the voltage steps down the current by the same ratio. This is why power lines operate at high voltage: for a given power $P = IV$, higher voltage means lower current, which means lower $I^2R$ losses in the transmission lines.

Eddy Currents and Magnetic Braking

When a bulk conductor (not just a wire loop) is exposed to a changing magnetic field, currents circulate within the conductor itself. These are eddy currents.

• They obey Lenz's law, so they oppose the change that created them. A conducting plate moving through a magnetic field experiences a braking force.
• Eddy currents are useful in magnetic braking (trains, roller coasters) and induction heating.
• They're a source of energy loss in transformer cores, which is why cores are made of laminated sheets or ferrite materials to limit the circulation paths.

Maxwell's Correction to Ampère's Law

Displacement Current

Ampère's original law, $\nabla \times \vec{B} = \mu_0 \vec{J}$, works fine for magnetostatics but fails for time-varying fields. Maxwell noticed that it's inconsistent with the continuity equation $\nabla \cdot \vec{J} = -\frac{\partial \rho}{\partial t}$ (taking the divergence of both sides of Ampère's law gives zero on the left but not necessarily zero on the right).

Maxwell's fix was to add a displacement current density:

$\vec{J}_D = \varepsilon_0 \frac{\partial \vec{E}}{\partial t}$

(In a linear dielectric medium, this generalizes to $\vec{J}_D = \frac{\partial \vec{D}}{\partial t}$.)

The classic example is a charging capacitor: no conduction current flows between the plates, yet the magnetic field circulates as if a current were there. The changing electric field between the plates acts as the source. The displacement current "completes the circuit" in a way that makes Ampère's law consistent.

Maxwell's Equations in Integral Form

The four Maxwell's equations in integral form (in vacuum, with sources):

• Gauss's law (electric): $\oint_S \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}$
• Gauss's law (magnetic): $\oint_S \vec{B} \cdot d\vec{A} = 0$
• Faraday's law: $\oint_{\partial S} \vec{E} \cdot d\vec{l} = -\frac{d}{dt}\int_S \vec{B} \cdot d\vec{A}$
• Ampère-Maxwell law: $\oint_{\partial S} \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}} + \mu_0\varepsilon_0\frac{d}{dt}\int_S \vec{E} \cdot d\vec{A}$

Notice the symmetry: a changing $\vec{B}$ produces a curling $\vec{E}$ (Faraday), and a changing $\vec{E}$ produces a curling $\vec{B}$ (Ampère-Maxwell). This mutual coupling is what allows electromagnetic waves to propagate.

Motional emf and Lorentz Force

Moving Conductors in Magnetic Fields

When a conductor moves through a magnetic field, the free charges inside it experience a Lorentz force $\vec{F} = q\vec{v} \times \vec{B}$. This force separates charges, creating a potential difference across the conductor.

For a straight conductor of length $l$ moving with velocity $\vec{v}$ through a uniform field $\vec{B}$:

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

For the simple case where $\vec{v}$, $\vec{B}$, and $\vec{l}$ are mutually perpendicular, this reduces to $\mathcal{E} = Blv$.

An important subtlety: motional emf and Faraday's law always give the same answer, but they offer different physical pictures. Motional emf comes from the Lorentz force on charges in a moving conductor. Faraday's law comes from the changing flux through a circuit. For moving conductors, both approaches work. For a stationary loop in a time-varying field, only Faraday's law (with the induced electric field) applies directly.

Hall Effect and Its Applications

When current flows through a conductor in the presence of a perpendicular magnetic field, the Lorentz force deflects charge carriers to one side, building up a transverse voltage called the Hall voltage:

$V_H = \frac{IB}{nqt}$

where $n$ is the carrier density, $q$ is the carrier charge, and $t$ is the thickness of the conductor in the direction of the magnetic field.

• The sign of $V_H$ reveals the sign of the charge carriers (this is how we know current in p-type semiconductors is carried by positive holes).
• Hall probes are widely used as magnetic field sensors.
• The Hall effect is also used for measuring carrier density and mobility in semiconductor characterization.

Inductance and Mutual Inductance

Self-Inductance of a Coil

Self-inductance $L$ measures how much flux linkage a coil produces per unit current through itself:

$L = \frac{N\Phi_B}{I}$

Equivalently, the emf induced in the coil by a changing current is:

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

The inductance depends on geometry (number of turns, cross-sectional area, length) and the core material. For a solenoid with $N$ turns, length $\ell$, cross-sectional area $A$, and core permeability $\mu$:

$L = \frac{\mu N^2 A}{\ell}$

Units: henry (H). One henry means that a current changing at 1 A/s induces 1 V.

Mutual Inductance Between Coils

Mutual inductance $M$ quantifies the flux linkage in one coil due to current in another:

$M = \frac{N_2 \Phi_{12}}{I_1}$

where $\Phi_{12}$ is the flux through a single turn of coil 2 due to current $I_1$ in coil 1. The Neumann formula guarantees that $M_{12} = M_{21} = M$, so mutual inductance is symmetric regardless of coil geometry.

The emf induced in coil 2 by a changing current in coil 1 is:

$\mathcal{E}_2 = -M\frac{dI_1}{dt}$

Mutual inductance is the operating principle behind transformers and coupled inductors. The coupling coefficient $k = M/\sqrt{L_1 L_2}$ ranges from 0 (no coupling) to 1 (perfect coupling).

Energy Stored in Magnetic Fields

The energy stored in an inductor carrying current $I$ is:

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

This energy resides in the magnetic field. You can express it in terms of the field itself by considering a solenoid:

$u_B = \frac{B^2}{2\mu_0}$

where $u_B$ is the energy density (energy per unit volume). This expression is general and applies to any magnetic field configuration, not just solenoids.

When the current through an inductor changes, the stored energy changes too. This is why inductors resist sudden changes in current: the energy has to come from or go somewhere.

AC Circuits and Resonance

RLC Circuits and Impedance

In an AC circuit driven at angular frequency $\omega$, each element has a characteristic impedance:

• Resistor: $Z_R = R$
• Inductor: $Z_L = j\omega L$
• Capacitor: $Z_C = \frac{1}{j\omega C} = -\frac{j}{\omega C}$

For a series RLC circuit, the total impedance is:

$Z = R + j\left(\omega L - \frac{1}{\omega C}\right)$

The magnitude is $|Z| = \sqrt{R^2 + \left(\omega L - \frac{1}{\omega C}\right)^2}$, and the phase angle between voltage and current is $\theta = \arctan\left(\frac{\omega L - 1/\omega C}{R}\right)$.

At low frequencies, the capacitor dominates (large $Z_C$), and the circuit is capacitive. At high frequencies, the inductor dominates (large $Z_L$), and the circuit is inductive.

Resonance in AC Circuits

Resonance occurs when the inductive and capacitive reactances cancel:

$\omega L = \frac{1}{\omega C} \implies \omega_0 = \frac{1}{\sqrt{LC}}$

At resonance:

• The impedance is purely real: $Z = R$
• Current is maximum for a given driving voltage
• Voltage and current are in phase
• The voltages across $L$ and $C$ can individually be much larger than the driving voltage (voltage magnification), with the quality factor $Q = \omega_0 L / R$ characterizing how sharp the resonance peak is

A high-$Q$ circuit has a narrow resonance peak and stores energy efficiently, cycling it between the inductor and capacitor with little dissipation per cycle.

Power in AC Circuits

For a sinusoidal AC circuit, the instantaneous power fluctuates, so we work with time-averaged quantities.

• Real (active) power: $P = V_{\text{rms}} I_{\text{rms}} \cos\theta$ (measured in watts, W). This is the power actually dissipated.
• Reactive power: $Q = V_{\text{rms}} I_{\text{rms}} \sin\theta$ (measured in VAR). This represents energy sloshing back and forth between source and reactive components.
• Apparent power: $S = V_{\text{rms}} I_{\text{rms}}$ (measured in VA).

These are related by $S^2 = P^2 + Q^2$. The power factor $\text{PF} = \cos\theta = P/S$ tells you what fraction of the apparent power is doing useful work. A power factor of 1 means all the power is real; a power factor near 0 means the circuit is mostly reactive.

Electromagnetic Oscillations and Waves

LC Oscillations

An LC circuit with no resistance oscillates indefinitely (in the ideal case). Energy swings between the electric field of the capacitor ($U_E = \frac{1}{2}CV^2$) and the magnetic field of the inductor ($U_B = \frac{1}{2}LI^2$), analogous to a mass on a spring exchanging kinetic and potential energy.

The oscillation frequency is:

$f_0 = \frac{1}{2\pi\sqrt{LC}}$

In a real circuit, resistance causes the oscillations to decay exponentially (damped oscillations). Adding a driving source at the resonant frequency sustains the oscillations.

Electromagnetic Wave Equation

Combining Faraday's law and the Ampère-Maxwell law in free space (no charges, no currents), you can decouple the equations to get wave equations for both fields:

$\nabla^2 \vec{E} = \mu_0\varepsilon_0\frac{\partial^2 \vec{E}}{\partial t^2}$

$\nabla^2 \vec{B} = \mu_0\varepsilon_0\frac{\partial^2 \vec{B}}{\partial t^2}$

These have the standard form of a wave equation with propagation speed:

$c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} \approx 3.00 \times 10^8 \text{ m/s}$

This was Maxwell's great triumph: the speed predicted from purely electromagnetic constants matched the measured speed of light, revealing that light is an electromagnetic wave.

Properties of Electromagnetic Waves

• $\vec{E}$ and $\vec{B}$ are perpendicular to each other and to the direction of propagation (transverse wave)
• The fields are in phase and related by $E = cB$ in free space
• No medium is required for propagation
• They carry energy, described by the Poynting vector: $\vec{S} = \frac{1}{\mu_0}\vec{E} \times \vec{B}$
• They carry momentum, with radiation pressure $P_{\text{rad}} = S/c$ for complete absorption
• Wavelength and frequency are related by $\lambda f = c$
• The electromagnetic spectrum spans from radio waves (long $\lambda$) through microwaves, infrared, visible, ultraviolet, X-rays, to gamma rays (short $\lambda$), all governed by the same wave equation