7.1 Faraday's law

Faraday's Law Fundamentals

Faraday's law describes how a changing magnetic field creates an electric field, and more specifically, how it induces a voltage in a conductor. This single idea connects electricity and magnetism and is the operating principle behind generators, transformers, and wireless chargers.

Electromagnetic Induction Basics

Electromagnetic induction is the process where a changing magnetic field produces an electromotive force (emf) in a conductor. That induced emf can then drive a current through a closed circuit.

Induction happens in two main scenarios:

• A stationary conductor sits in a magnetic field that's changing over time
• A conductor moves through a magnetic field that may itself be constant

In both cases, what matters is that the magnetic flux through the conductor is changing.

Magnetic Flux

Before you can use Faraday's law, you need to understand magnetic flux. Flux measures how much magnetic field passes through a given surface. Think of it as "how many field lines thread through your loop."

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

Here $\vec{B}$ is the magnetic field, $\vec{A}$ is the area vector (perpendicular to the surface), and $\theta$ is the angle between them. Flux can change if any of these three things change:

• The magnetic field strength $B$ increases or decreases
• The area $A$ of the loop changes (e.g., a loop being stretched or compressed)
• The orientation $\theta$ between the field and the surface changes (e.g., a coil rotating)

Lenz's Law

Lenz's law tells you the direction of the induced current: it always flows in the direction that opposes the change in flux that caused it. If the flux through a loop is increasing, the induced current creates its own magnetic field to fight that increase. If the flux is decreasing, the induced current tries to maintain it.

This is where the negative sign in Faraday's law comes from. It also reflects conservation of energy: if the induced current helped the change instead of opposing it, you'd get energy from nothing.

Induced EMF Calculation

Faraday's Law Equation

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

• $\varepsilon$ = induced emf (in volts)
• $N$ = number of turns in the coil
• $\frac{d\Phi_B}{dt}$ = rate of change of magnetic flux through one turn
• The negative sign encodes Lenz's law (the opposition to change)

For a coil with multiple turns, each turn contributes the same flux change, so the total emf scales linearly with $N$.

Factors Affecting Induced EMF

Several variables control how large the induced emf will be:

• Rate of flux change: Faster changes produce larger emf. Yanking a magnet quickly through a coil induces more voltage than moving it slowly.
• Number of turns: A coil with 100 turns produces 100 times the emf of a single loop, all else being equal.
• Field strength: A stronger $B$ field means more flux to begin with, so changes are larger.
• Loop area: A bigger loop intercepts more field lines.
• Orientation: Rotating a coil so that $\theta$ changes alters the flux even if $B$ and $A$ stay constant.

Units and Dimensions

• Induced emf: volts (V)
• Magnetic flux: webers (Wb), where $1 \text{ Wb} = 1 \text{ T} \cdot \text{m}^2$
• Time: seconds (s)

A quick dimensional check confirms the equation works out:

$[\varepsilon] = \frac{[\text{Wb}]}{[\text{s}]} = \frac{[\text{T}][\text{m}^2]}{[\text{s}]} = \text{V}$

Applications of Faraday's Law

Electric Generators

Generators convert mechanical energy into electrical energy by rotating a coil inside a magnetic field. As the coil spins, the angle $\theta$ between $\vec{B}$ and $\vec{A}$ continuously changes, producing a sinusoidally varying emf. This is how most AC electricity is produced, whether the mechanical energy comes from falling water, wind, or steam turbines.

Transformers

Transformers change voltage levels in AC circuits. They consist of two coils (primary and secondary) wrapped around a shared iron core.

1. AC current in the primary coil creates a changing magnetic field in the core.
2. That changing flux passes through the secondary coil.
3. Faraday's law induces an emf in the secondary coil.

The voltage ratio depends on the turns ratio: $\frac{V_s}{V_p} = \frac{N_s}{N_p}$. Step-up transformers (more secondary turns) increase voltage for efficient long-distance transmission. Step-down transformers reduce it for household use.

Induction Cooktops

An induction cooktop generates a high-frequency alternating magnetic field just below its surface. When you place ferromagnetic cookware on it, the changing field induces eddy currents in the pan's bottom. Those currents encounter resistance in the metal and generate heat directly in the cookware. The cooktop surface itself stays relatively cool, and only compatible (ferromagnetic) pots and pans will work.

Motional EMF

Motional emf is a specific case of Faraday's law where the conductor itself moves through a magnetic field, even if that field is constant and uniform.

Moving Conductors in Magnetic Fields

When a conducting rod moves through a magnetic field, the magnetic force pushes charge carriers inside the rod to one end. Positive charges accumulate at one end, negative at the other, creating a potential difference across the rod. No changing external field is required; the motion alone does the job.

Flux Change vs. Conductor Motion

These two situations look different but are unified by Faraday's law:

• Stationary conductor, changing field: The flux changes because $B$ is changing over time.
• Moving conductor, constant field: The flux changes because the area of the circuit intercepting the field is changing.

Both produce an induced emf, and both are described by $\varepsilon = -N\frac{d\Phi_B}{dt}$.

Calculating Motional EMF

For a straight conductor of length $l$ moving with velocity $v$ perpendicular to a uniform field $B$:

$\varepsilon = Blv$

This formula applies when the motion, the conductor, and the field are all mutually perpendicular. For example, a 0.5 m rod moving at 2 m/s through a 0.3 T field produces $\varepsilon = (0.3)(0.5)(2) = 0.3 \text{ V}$. More complex geometries (curved conductors, non-perpendicular motion) require integrating the force along the conductor's length.

Eddy Currents

When a bulk piece of conducting material (not just a wire loop) sits in a changing magnetic field, Faraday's law induces circulating loops of current throughout the material. These are eddy currents.

Formation and Characteristics

• Eddy currents flow in closed loops within the conductor, oriented perpendicular to the changing field.
• Their strength depends on the material's electrical conductivity and how fast the field is changing.
• By Lenz's law, they produce their own magnetic fields that oppose the flux change.
• They dissipate energy as heat due to the material's resistance. This is why transformers use laminated cores: thin insulated sheets reduce eddy current paths and cut energy losses.

Eddy Current Braking

Eddy current brakes slow a moving conductor without any physical contact. A magnetic field (from a permanent magnet or electromagnet) is positioned near a moving conductive surface, such as a metal rail or disk. The relative motion induces eddy currents, which create an opposing force that decelerates the conductor. These brakes are used in trains, roller coasters, and some exercise equipment. They provide smooth, wear-free braking.

Electromagnetic Damping

The same principle damps unwanted oscillations. In a galvanometer, for instance, the coil swings through a magnetic field, and the induced eddy currents resist the motion, bringing the needle to rest quickly. This technique also appears in vibration damping for vehicles and sensitive instruments.

Faraday's Law in Everyday Life

Magnetic Card Readers

When you swipe a credit card, the magnetic strip moves past a small coil in the reader. The varying magnetization along the strip creates a changing flux through the coil, inducing a voltage pattern that encodes the card's data. The card must move relative to the reader for induction to occur.

Metal Detectors

A metal detector has a transmitter coil that generates an alternating magnetic field. When a metallic object enters this field, eddy currents are induced in the object, which in turn create their own magnetic field. A separate receiver coil picks up the disturbance as a change in flux, and signal processing identifies the presence and rough location of the metal.

Wireless Charging

Wireless chargers use a transmitter coil in the charging pad and a receiver coil in the device. AC current in the transmitter coil produces a changing magnetic field, which induces an emf in the receiver coil. That emf charges the battery. Efficiency drops significantly if the coils are misaligned or too far apart, which is why most wireless chargers require you to place the device in a specific spot.

Limitations and Considerations

Non-Uniform Magnetic Fields

The simple form $\Phi_B = BA\cos\theta$ assumes a uniform field over the entire area. Real magnetic fields often vary across space, so you need to integrate $\vec{B} \cdot d\vec{A}$ over the surface. This can make flux calculations significantly more complex, and numerical methods are sometimes necessary.

Time-Varying Fields at High Frequencies

Faraday's law as presented here uses a quasi-static approximation: it assumes the fields change slowly enough that you can ignore the time it takes for changes to propagate. At very high frequencies, this breaks down. Rapidly changing electric fields produce displacement currents (Maxwell's correction to Ampère's law), and you need the full set of Maxwell's equations. This regime is where electromagnetic wave propagation becomes important.

Relativistic Effects

The standard formulation assumes all speeds involved are much less than the speed of light. For conductors or charges moving at relativistic speeds, electric and magnetic fields mix and transform between reference frames. These corrections matter in particle accelerators and astrophysical contexts but are well beyond the scope of this course.

Historical Context

Faraday's Experiments

Michael Faraday demonstrated electromagnetic induction in 1831 through a series of experiments. His most famous involved an iron ring with two separate wire coils wound around it. When he connected and disconnected a battery to one coil, a brief current appeared in the other. He also showed that moving a magnet into and out of a coil induced current. These experiments led him to build the first electromagnetic generator, known as the Faraday disk.

Connection to Maxwell's Equations

Faraday's law became one of the four Maxwell's equations, which James Clerk Maxwell assembled into a unified theory of electromagnetism in the 1860s. In differential form, it states that the curl of the electric field equals the negative time derivative of the magnetic field. Maxwell's framework predicted electromagnetic waves, unifying electricity, magnetism, and optics into a single theory.

Advanced Concepts

Differential Form of Faraday's Law

The integral form $\varepsilon = -N\frac{d\Phi_B}{dt}$ generalizes to a differential (point-by-point) form:

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

This says that a time-changing magnetic field at any point in space creates a curling electric field at that point. This form is essential for understanding electromagnetic wave propagation and phenomena like the skin effect, where high-frequency currents concentrate near the surface of a conductor.

Superconductors and Faraday's Law

Superconductors have zero electrical resistance, which creates some striking consequences for Faraday's law. Induced currents in a superconducting loop persist indefinitely, and the magnetic flux through a superconducting ring becomes "trapped" at a fixed value. In fact, flux in superconducting loops is quantized in discrete units. The Meissner effect (expulsion of magnetic fields from a superconductor's interior) and applications like magnetic levitation and SQUIDs (superconducting quantum interference devices) all connect back to how Faraday's law operates in zero-resistance materials.

Quantum Mechanical Connections

At the quantum level, Faraday's law connects to phenomena like the Aharonov-Bohm effect, where a charged particle is influenced by a magnetic vector potential even in regions where the magnetic field itself is zero. Magnetic flux quantization in superconductors is a macroscopic quantum effect. These ideas feed into active research areas including topological insulators and quantum computing.