7.4 Induced electric fields

A changing magnetic field doesn't just affect nearby circuits; it actually creates an electric field in the surrounding space. This induced electric field is fundamentally different from the electrostatic fields you've seen before, and understanding it is essential for grasping how generators, transformers, and electromagnetic waves work.

Faraday's law of induction

Faraday's law is the central idea of this unit: a changing magnetic field produces an electric field, and that electric field can drive current through a circuit. Everything else in this section builds on it.

Magnetic flux

Magnetic flux measures how much magnetic field passes through a given area. Think of it as counting the number of field lines threading through a 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 is measured in webers (Wb), where 1 Wb = 1 T·m².

Three things can change the flux through a loop: the field strength $B$, the loop area $A$, or the angle $\theta$ between them. Any of these changes will induce an EMF.

Rate of change of flux

The induced EMF depends on how quickly the flux changes:

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

A faster change in flux produces a larger EMF. For example, yanking a loop out of a magnetic field quickly generates a bigger voltage spike than pulling it out slowly. You can change the flux by:

• Increasing or decreasing the magnetic field strength
• Changing the area of the loop (stretching it, or moving a sliding rail)
• Rotating the loop so $\theta$ changes over time

Lenz's law

The negative sign in Faraday's law isn't just math. Lenz's law says the induced current always flows in a 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 flux is decreasing, the induced current tries to maintain it.

This is really conservation of energy at work. If the induced current helped the change instead of opposing it, you'd get runaway energy creation from nothing.

Practical examples include electromagnetic braking (where eddy currents oppose the motion of a conductor) and induction cooktops (where induced currents in a metal pan generate heat).

Induced electric fields

Here's the deeper point: Faraday's law isn't just about circuits and wires. A changing magnetic field creates an electric field in space itself, whether or not a conductor is present. If you place a wire there, charges will move. But the field exists regardless.

Non-conservative nature

Electrostatic fields (from point charges) are conservative: the work done moving a charge around any closed loop is zero, and you can define a potential $V$ at every point. Induced electric fields are non-conservative. The work done around a closed loop is not zero; that's exactly what drives a current around a circuit.

This means you can't describe an induced electric field with a simple scalar potential. The line integral around a closed path gives you the EMF:

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

This is what makes continuous current generation in a closed loop possible.

Curl of induced E-field

The differential form of Faraday's law is the Maxwell-Faraday equation:

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

For electrostatic fields, $\nabla \times \vec{E} = 0$. For induced fields, the curl is non-zero wherever $\vec{B}$ is changing in time. This non-zero curl is the mathematical signature that distinguishes induced electric fields from electrostatic ones, and it's one of the four Maxwell's equations that predict electromagnetic waves.

Relationship to changing B-field

The induced electric field lines form closed loops around the region where $\vec{B}$ is changing. This is very different from electrostatic field lines, which start on positive charges and end on negative charges.

The field strength is proportional to the rate of change $\frac{\partial \vec{B}}{\partial t}$. For a cylindrical region of changing magnetic field, you can find the induced $E$ using symmetry and the integral form of Faraday's law, much like using Ampère's law to find $B$ around a current-carrying wire.

Motional EMF

When a conductor physically moves through a magnetic field, an EMF appears across it. This is called motional EMF, and you can understand it from two equivalent perspectives.

Moving conductor in B-field

When a straight conductor of length $l$ moves with velocity $v$ perpendicular to a uniform field $B$, the free charges inside experience a magnetic force ($\vec{F} = q\vec{v} \times \vec{B}$). This pushes positive and negative charges to opposite ends, creating a potential difference:

$\varepsilon = Blv$

For example, a 0.5 m rod moving at 3 m/s through a 0.2 T field produces $\varepsilon = (0.2)(0.5)(3) = 0.3$ V.

This principle is used in electric generators and magnetic flow meters (which measure fluid velocity by detecting the EMF produced by a conducting fluid moving through a magnetic field).

Flux rule vs. Lorentz force

You can calculate motional EMF two ways:

• Flux rule approach: Track the area swept by the moving conductor, calculate the change in flux $\Phi_B$ through the circuit, and apply $\varepsilon = -\frac{d\Phi_B}{dt}$. This gives you the big-picture view of what happens to the circuit.
• Lorentz force approach: Focus on the force $q\vec{v} \times \vec{B}$ on individual charge carriers inside the conductor. The EMF equals the work done per unit charge as carriers move along the conductor's length.

Both approaches always give the same answer. The flux rule is usually easier for circuit problems, while the Lorentz force approach gives more physical insight into why the charges move.

Applications of induction

Generators and alternators

Generators convert mechanical energy into electrical energy. A coil rotates in a magnetic field (or a magnet rotates inside a coil), causing the flux through the coil to change sinusoidally. This produces an alternating EMF and, therefore, alternating current (AC).

The output depends on:

• Rotation speed (faster = higher frequency and larger peak EMF)
• Magnetic field strength
• Number of turns in the coil

In practice, most large generators use a rotating electromagnet (the rotor) inside stationary coils (the stator), which makes it easier to extract high-current electrical power without slip rings on the output.

Transformers

Transformers change AC voltage levels using mutual induction. Two coils (primary and secondary) share a common magnetic core. An AC current in the primary creates a changing flux that induces an EMF in the secondary.

The voltage ratio follows the turns ratio:

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

• Step-up transformers have more secondary turns than primary turns, increasing voltage (used for long-distance power transmission).
• Step-down transformers have fewer secondary turns, decreasing voltage (used to bring household voltage down from transmission levels).

An ideal transformer conserves power: $V_p I_p = V_s I_s$. If voltage goes up, current goes down proportionally.

Eddy currents

When a bulk conductor (not just a wire loop) sits in a changing magnetic field, circulating currents called eddy currents are induced throughout the material. By Lenz's law, these currents oppose the flux change.

Eddy currents cause resistive heating, which is useful in induction heating and electromagnetic braking, but wasteful in transformer cores. To reduce eddy current losses, transformer cores are built from thin laminated sheets or ferrite materials, which limit the size of the current loops.

Inductance

Inductance is a circuit element's tendency to oppose changes in current by inducing a back-EMF. Any coil of wire has inductance, and components designed to maximize it are called inductors.

Self-inductance

When current through a coil changes, the changing magnetic flux through the coil itself induces an EMF that opposes the change. The self-inductance $L$ quantifies this:

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

The induced back-EMF is $\varepsilon = -L\frac{dI}{dt}$. A large $L$ means even small changes in current produce significant opposing voltages. Self-inductance is measured in henrys (H) and depends on the coil's geometry (number of turns, cross-sectional area, length) and core material.

Mutual inductance

When two coils are near each other, a changing current in one induces an EMF in the other. The mutual inductance $M$ between them is:

$M = \frac{N_2 \Phi_{21}}{I_1}$

where $\Phi_{21}$ is the flux through coil 2 due to current $I_1$ in coil 1. Mutual inductance is the operating principle behind transformers. It can be positive or negative depending on the relative orientation of the coils.

Energy stored in magnetic field

An inductor carrying current $I$ stores energy in its magnetic field:

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

This is analogous to the energy $\frac{1}{2}CV^2$ stored in a capacitor's electric field. When current through an inductor is interrupted suddenly, this stored energy must go somewhere, which is why you can get large voltage spikes ("inductive kick") across switches or contacts.

RL circuits

An RL circuit contains a resistor and an inductor in series. The inductor's opposition to current changes gives these circuits distinctive time-dependent behavior.

Time constant

The time constant of an RL circuit is:

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

This tells you how quickly the circuit responds to changes. After one time constant, the current has reached about 63% of its final value. After $5\tau$, it's effectively at steady state (over 99%).

A large inductance or small resistance means a longer time constant and slower response.

Transient behavior

When you suddenly connect or disconnect a voltage source, the current doesn't jump instantly. Instead, it follows an exponential curve:

$I(t) = I_f + (I_i - I_f)e^{-t/\tau}$

where $I_i$ is the initial current and $I_f$ is the final (steady-state) current.

For a circuit being energized from rest ($I_i = 0$) with a battery of voltage $V$:

$I(t) = \frac{V}{R}(1 - e^{-t/\tau})$

Watch out for inductive kick: if current through an inductor is suddenly interrupted, the inductor tries to maintain the current by producing a very large voltage spike. This can damage components or create sparks.

Steady-state behavior

Once transients die out:

• DC input: The inductor acts like a short circuit (just a wire with negligible resistance). All the voltage drops across the resistor.
• AC input: The inductor has an impedance $Z_L = j\omega L$ that increases with frequency. It introduces a 90° phase shift where voltage leads current. This affects the power factor and reactive power in AC systems.

Electromagnetic waves

Maxwell's equations predict that changing electric fields produce magnetic fields and changing magnetic fields produce electric fields. Together, these self-sustaining oscillations propagate through space as electromagnetic waves.

Maxwell's equations

The four Maxwell's equations unify all of electricity and magnetism:

1. Gauss's law for electricity: Electric flux through a closed surface equals the enclosed charge divided by $\varepsilon_0$
2. Gauss's law for magnetism: Magnetic flux through any closed surface is zero (no magnetic monopoles)
3. Faraday's law: A changing magnetic field produces a circulating electric field
4. Ampère-Maxwell law: Magnetic fields are produced by currents and by changing electric fields (the displacement current term Maxwell added)

The displacement current in equation 4 was Maxwell's key addition. Without it, the equations wouldn't predict electromagnetic waves.

Wave equation

Combining Maxwell's equations in free space yields the wave equation:

$\nabla^2\vec{E} = \frac{1}{c^2}\frac{\partial^2\vec{E}}{\partial t^2}$

(with an identical equation for $\vec{B}$). This predicts waves traveling at speed:

$c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} \approx 3.0 \times 10^8 \text{ m/s}$

When Maxwell calculated this from the known values of $\mu_0$ and $\varepsilon_0$, the result matched the measured speed of light. This was strong evidence that light is an electromagnetic wave. The same equation describes all EM radiation: radio waves, microwaves, infrared, visible light, UV, X-rays, and gamma rays.

Energy and momentum of EM waves

EM waves carry both energy and momentum. The energy density at any point is:

$u = \frac{1}{2}\varepsilon_0 E^2 + \frac{1}{2\mu_0}B^2$

The rate of energy flow per unit area is given by the Poynting vector:

$\vec{S} = \frac{1}{\mu_0}\vec{E} \times \vec{B}$

EM waves also carry momentum with density $\vec{g} = \frac{\vec{S}}{c^2}$. This means light exerts pressure on surfaces it hits (radiation pressure), which is the principle behind solar sails for spacecraft propulsion.

Induction in conductors vs. dielectrics

Electromagnetic induction affects conductors and dielectrics differently because of how charges respond in each type of material.

Induced currents

In conductors, free electrons respond to the induced electric field by flowing, creating real currents. These currents obey Lenz's law and dissipate energy as heat (Joule heating). This is the basis for induction heating (used in industrial metalworking and induction cooktops) and electromagnetic braking.

Polarization effects

In dielectrics (insulators), there are no free charges to flow. Instead, the induced electric field causes bound charges and molecular dipoles to shift or rotate slightly, creating a polarization of the material. No net current flows, but the polarization field affects how EM waves propagate through the medium (changing the wave speed and wavelength). This is why the refractive index of a material depends on its dielectric properties.

Measurement techniques

Search coils

A search coil is a simple loop (or multi-turn coil) of wire used to detect changing magnetic fields. By Faraday's law, the voltage across the coil is:

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

The output voltage is proportional to the rate of change of flux, so search coils detect changing fields, not static ones. Increasing the number of turns $N$ or using a high-permeability core boosts sensitivity. They're used in geomagnetic measurements and non-destructive testing.

Hall effect sensors

Unlike search coils, Hall effect sensors can measure static magnetic fields. A current-carrying conductor placed in a magnetic field develops a voltage across it (the Hall voltage) perpendicular to both the current and the field. This voltage is proportional to the field strength $B$.

Hall sensors are widely used for position sensing in motors, current measurement in power electronics, and magnetic field mapping.

Electromagnetic induction limits

Superconductors

Below a critical temperature, superconductors have zero electrical resistance. Induced currents in a superconductor persist indefinitely with no energy loss. Superconductors also exhibit the Meissner effect: they expel magnetic flux from their interior completely (perfect diamagnetism).

These properties enable extremely strong, stable magnetic fields used in MRI machines and magnetic levitation systems. The practical limitations are the critical temperature, maximum current density, and maximum magnetic field the material can sustain.

Skin effect

At high frequencies, alternating current doesn't penetrate uniformly through a conductor. Instead, it concentrates near the surface. This is the skin effect, and it happens because the changing magnetic field inside the conductor induces opposing currents that cancel the current in the interior.

The skin depth $\delta$ (the depth at which current density falls to $1/e$ of its surface value) decreases with increasing frequency. At 60 Hz in copper, $\delta \approx 8.5$ mm. At 1 MHz, it's only about 0.066 mm. This effectively increases the conductor's resistance at high frequencies. Engineers use Litz wire (many thin, insulated strands woven together) to mitigate this in high-frequency applications.

Magnetic shielding

Magnetic shielding uses materials with high magnetic permeability (like mu-metal) to redirect magnetic field lines around a protected region. The shield doesn't absorb the field; it provides an easier path for the flux, diverting it away from the interior.

Effectiveness depends on:

• The shield material's permeability
• The geometry and thickness of the enclosure
• The frequency of the field being shielded

Applications include protecting sensitive electronics and creating low-field environments for precision experiments. At very strong fields, the shielding material can saturate, losing its effectiveness.