Electromagnetic Induction

In Chapter 5.1, we connected field laws to circuit rules and saw that moving a wire through a magnetic field drives a current (Motional EMF). But Faraday discovered something even more profound: a changing magnetic field itself creates an electric field, even if nothing is moving.

1. The General Law

This is the true form of electromagnetic induction. A time-varying magnetic field acts as a source for a "curling" electric field. This is the 3rd Maxwell Equation:

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

In a loop of wire, this induced field drives an EMF. This explains why standard generators (Motional) and transformers (Induced Field) obey the exact same "Flux Rule" for magnetic flux $\Phi_B$:

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

The Circuit-Level Consequence

The electrostatic KVL rule used in the previous lesson assumed that no changing magnetic flux linked the loop. Faraday's law now tells us how to extend it. Choose a direction around the loop and keep all voltage drops and EMFs signed in that direction. The induced EMF $\mathcal{E}_{\mathrm{ind}}=-d\Phi_B/dt$ must be included as an additional source term:

\sum V_{\mathrm{drops}}-\sum \mathcal{E}_{\mathrm{sources}} =\mathcal{E}_{\mathrm{ind}}=-\frac{d\Phi_B}{dt}

When the linked magnetic flux is constant, the right-hand side is zero and the ordinary quasistatic KVL equation is recovered. Thus Faraday's law does not discard the circuit rule; it states its range of validity and supplies the required correction.

Deep Dive: The Origins of Induction

2. Inductance: From Geometry to Inertia

One current-carrying loop can induce an EMF in another, even without any physical connection. This is Mutual Inductance. The key insight is that the magnetic flux $\Phi_{B,2}$ through loop 2 (from current $I_1$ in loop 1) is proportional to the current:

\Phi_{B,2} = M I_1

A changing current means a changing magnetic flux, which induces an EMF: $\mathcal{E}_2 = -M \frac{dI_1}{dt}$.

But a loop also links its own magnetic field. Changing its current induces an EMF in itself. We call this Self Inductance ($L$):

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

The minus sign is Lenz's law: the induced EMF opposes the change in current. In circuit notation we usually use the passive sign convention, taking current to enter the terminal labeled positive. The corresponding terminal voltage drop is then

V_L = L \frac{dI}{dt}

The terminal voltage and the back EMF describe the same effect with opposite reference directions. This is why current cannot change instantly—the inductor acts like electrical inertia. The greater the inductance, the harder it is to start or stop the current.

Deep Dive: Mutual & Self Inductance

3. Magnetic Energy

Because the inductor fights you, you must do work to get the current flowing. This work isn't lost to heat; it is stored in the magnetic field you created.

W = \frac{1}{2} L I^2

This confirms a fundamental truth: Magnetic fields store energy.

Deep Dive: Energy in Magnetic Fields

The Big Picture

We have moved from the fundamental physics of space to the practical rules of circuits:

The Physics: $\nabla \times \vec{E} = -\partial \vec{B}/\partial t$ (Nature's Law)
The Geometry: $L$ (How strict your loop is)
The Circuit: $\mathcal{E} = -L dI/dt$ (The resulting inertia)
The Consequence: $W = \frac{1}{2}LI^2$ (The energy cost)

Next, the charging-capacitor paradox returns to the current-continuity question and shows why a time-varying electric field must appear in the complete field equations.