Faraday's law and electromagnetic induction

Intuition [Beginner]

In electrostatics, the electric field is created by charges. Faraday discovered something deeper: a changing magnetic field creates an electric field too — even when no charges are present at all. This is electromagnetic induction.

The key quantity is magnetic flux: how much magnetic field passes through a loop. Think of it as the number of magnetic field lines threading the loop. If that number changes — because the field gets stronger, or because the loop moves, or because the loop rotates — an induced EMF (voltage) appears around the loop.

Faraday's law: the induced EMF equals the rate at which the magnetic flux through the loop changes. Lenz's law adds a minus sign: the induced current flows in the direction that opposes the change in flux. If flux is increasing, the induced current creates a field that tries to decrease it. If flux is decreasing, the induced current tries to increase it. Nature resists change — and this resistance is what conserves energy.

Examples: push a bar magnet through a coil and a current flows. Change the current in one wire and a nearby wire feels a voltage. Spin a coil in a magnetic field and you get an alternating voltage — this is how generators work. Run AC through a primary coil wrapped around an iron core and a secondary coil picks up voltage — this is how transformers work.

Visual [Beginner]

The right-hand rule for induced current: curl your fingers around the loop in the direction of the induced current, and your thumb points in the direction of the magnetic field the induced current creates. Lenz's law says this field opposes whatever change is happening to the original flux.

Worked example [Beginner]

A circular loop of wire with radius $r=0.1$ m sits in a uniform magnetic field $\mathbf{B}$ perpendicular to the loop. The field strength increases at a steady rate $dB/dt=0.5$ T/s. Compute the induced EMF.

Step 1. Magnetic flux through the loop. The field is uniform and perpendicular, so the flux equals the field strength times the area of the loop. The area of a circle of radius $r$ is $\pi r^2$:

$${\Phi }_B=B\cdot \pi r^2.$$

Step 2. Rate of change of flux. The radius is fixed (the loop does not move or change size), so only $B$ is changing:

$$\frac{d{\Phi }_B}{dt}=\pi r^2\frac{dB}{dt}.$$

Step 3. Induced EMF. By Faraday's law (with Lenz's minus sign), the induced EMF has magnitude:

$$\mathcal{E}=\pi r^2\frac{dB}{dt}=\pi (0.1{)}^2(0.5)=\pi \cdot 0.01\cdot 0.5=0.005\pi \approx 0.016\text{V}.$$

The induced EMF is about 16 millivolts. By Lenz's law, the induced current flows in the direction that creates a magnetic field opposing the increase — so the induced field points opposite to $\mathbf{B}$.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $S$ be an oriented surface with boundary curve $\mathcal{C}=\partial S$, and let $\mathbf{B}(\mathbf{r},t)$ be a magnetic field. The magnetic flux through $S$ is

$${\Phi }_B:={\int }_S\mathbf{B}\cdot d\mathbf{a}.$$

Faraday's law (integral form) states that the EMF around the boundary curve equals the negative rate of change of flux:

$$\boxed{\mathcal{E}:={\oint }_{\mathcal{C}}\mathbf{E}\cdot d\mathbf{l}=-\frac{d{\Phi }_B}{dt}=-\frac{d}{dt}{\int }_S\mathbf{B}\cdot d\mathbf{a}.}$$

The minus sign is Lenz's law: the induced EMF drives current in the direction that opposes the change in flux.

When the curve $\mathcal{C}$ is stationary (not moving or deforming), the time derivative passes through the integral, and applying Stokes' theorem to the left-hand side yields the differential form:

$$\boxed{\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}.}$$

This is the third Maxwell equation. It says: a time-varying magnetic field is a source of curl for the electric field. In electrostatics, $\partial \mathbf{B}/\partial t=0$ and $\nabla \times \mathbf{E}=0$, recovering the conservative electric field. In electrodynamics, the electric field is no longer conservative — its curl is proportional to how fast $\mathbf{B}$ is changing.

Motional EMF. When the loop moves through a non-uniform field, or a conductor moves with velocity $\mathbf{v}$ through $\mathbf{B}$, the free charges inside the conductor experience a magnetic force $q\mathbf{v}\times \mathbf{B}$ that acts as an effective electric field ${\mathbf{E}}_m=\mathbf{v}\times \mathbf{B}$. The motional EMF around a moving loop $\mathcal{C}(t)$ is

$${\mathcal{E}}_{\text{mot}}={\oint }_{\mathcal{C}(t)}(\mathbf{v}\times \mathbf{B})\cdot d\mathbf{l}.$$

The flux rule (Faraday's law with the total time derivative of ${\Phi }_B$) accounts for both the changing-field and moving-loop contributions. The combined result is

$$\mathcal{E}=-\frac{d{\Phi }_B}{dt}=-{\int }_S\frac{\partial \mathbf{B}}{\partial t}\cdot d\mathbf{a}+{\oint }_{\mathcal{C}}(\mathbf{v}\times \mathbf{B})\cdot d\mathbf{l},$$

where the first term is the transformer EMF (changing field, stationary loop) and the second is the motional EMF (moving loop in static field). The universal flux rule $\mathcal{E}=-d{\Phi }_B/dt$ subsumes both mechanisms into a single expression.

Displacement current (Maxwell's correction to Ampere's law). Ampere's law in magnetostatics reads $\nabla \times \mathbf{B}={\mu }_0\mathbf{J}$. Taking the divergence of both sides gives $\nabla \cdot (\nabla \times \mathbf{B})=0={\mu }_0\nabla \cdot \mathbf{J}$, which contradicts charge conservation $\nabla \cdot \mathbf{J}=-\partial \rho /\partial t$ when charges are moving. Maxwell's correction adds the displacement current ${ϵ}_0\partial \mathbf{E}/\partial t$:

$$\boxed{\nabla \times \mathbf{B}={\mu }_0\mathbf{J}+{\mu }_0{ϵ}_0\frac{\partial \mathbf{E}}{\partial t}.}$$

The divergence of both sides now gives $0={\mu }_0(\nabla \cdot \mathbf{J}+\partial \rho /\partial t)$, using Gauss's law ${ϵ}_0\nabla \cdot \mathbf{E}=\rho$. This is the continuity equation — charge conservation is built in.

The complete Maxwell equations (vector form). At this point all four equations are in hand:

$$\nabla \cdot \mathbf{E}=\frac{\rho }{{ϵ}_0},\nabla \cdot \mathbf{B}=0,$$ $$\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t},\nabla \times \mathbf{B}={\mu }_0\mathbf{J}+{\mu }_0{ϵ}_0\frac{\partial \mathbf{E}}{\partial t}.$$

These four equations, plus the Lorentz force law $\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times \mathbf{B})$, constitute classical electrodynamics.

Inductance. For a coil of $N$ turns carrying current $I$, the total flux linkage is $\Lambda =N{\Phi }_B$. Self-inductance $L$ is defined by $\Lambda =LI$, and the induced EMF is $\mathcal{E}=-LdI/dt$. Mutual inductance $M$ between two circuits satisfies ${\Lambda }_2=M{I}_1$ and ${\mathcal{E}}_2=-Md{I}_1/dt$. The energy stored in the magnetic field of an inductor is $U=\frac{1}{2}L{I}^2$; more generally, the total magnetic field energy is

$$U_B=\frac{1}{2{\mu }_0}{\int }_V|\mathbf{B}{|}^{2}d\tau .$$

Counterexamples to common slips

• The flux rule $\mathcal{E}=-d{\Phi }_B/dt$ applies when the loop is a single connected curve. For a conductor that breaks (e.g., a sliding bar that leaves the rail), the EMF still exists but you must be careful about what "the loop" is — Griffiths Example 5.11 (the Faraday disc / homopolar generator) is the standard cautionary example.
• Faraday's law $\nabla \times \mathbf{E}=-\partial \mathbf{B}/\partial t$ is always true. The flux rule $\mathcal{E}=-d{\Phi }_B/dt$ has subtleties when the loop deforms or when the "circuit" is not well-defined. The differential form is the reliable statement; the flux rule is a computational shortcut that works in most practical situations.
• The displacement current ${ϵ}_0\partial \mathbf{E}/\partial t$ is not a flow of charge. It is the rate of change of the electric field, and it has the dimensions of current density. It produces magnetic fields in the same way as a real current — this is what makes electromagnetic waves possible.

Key theorem with proof [Intermediate+]

Theorem (Equivalence of the flux rule and the differential Faraday law for a stationary loop). Let $S$ be a fixed (time-independent) oriented surface with boundary $\mathcal{C}=\partial S$, and let $\mathbf{B}(\mathbf{r},t)$ be a smooth magnetic field. Then

$${\oint }_{\mathcal{C}}\mathbf{E}\cdot d\mathbf{l}=-\frac{d}{dt}{\int }_S\mathbf{B}\cdot d\mathbf{a}$$

if and only if $\nabla \times \mathbf{E}=-\partial \mathbf{B}/\partial t$.

Proof. Since $S$ is fixed, the time derivative passes through the spatial integral:

$$\frac{d}{dt}{\int }_S\mathbf{B}\cdot d\mathbf{a}={\int }_S\frac{\partial \mathbf{B}}{\partial t}\cdot d\mathbf{a}.$$

By Stokes' theorem applied to the left-hand side:

$${\oint }_{\mathcal{C}}\mathbf{E}\cdot d\mathbf{l}={\int }_S(\nabla \times \mathbf{E})\cdot d\mathbf{a}.$$

Substituting both into the flux rule gives

$${\int }_S(\nabla \times \mathbf{E})\cdot d\mathbf{a}=-{\int }_S\frac{\partial \mathbf{B}}{\partial t}\cdot d\mathbf{a},$$

or equivalently

$${\int }_S\left(\nabla \times \mathbf{E}+\frac{\partial \mathbf{B}}{\partial t}\right)\cdot d\mathbf{a}=0.$$

Since this must hold for every fixed surface $S$, the integrand vanishes pointwise:

$$\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}.\text{qed}$$

Corollary. For a moving loop $\mathcal{C}(t)$ sweeping out surface $S(t)$, the total time derivative of the flux picks up an additional boundary term that is precisely the motional EMF:

$$\frac{d}{dt}{\int }_{S(t)}\mathbf{B}\cdot d\mathbf{a}={\int }_{S(t)}\frac{\partial \mathbf{B}}{\partial t}\cdot d\mathbf{a}+{\oint }_{\mathcal{C}(t)}(\mathbf{v}\times \mathbf{B})\cdot d\mathbf{l}.$$

This is the Leibniz rule (transport theorem) for flux integrals. The flux rule $\mathcal{E}=-d{\Phi }_B/dt$ unifies both contributions.

Worked example: mutual inductance of coaxial solenoids

A long solenoid of radius $a$, $n_1$ turns per unit length, carries current $I_1$. A second solenoid of radius $b>a$, $n_2$ turns per unit length, is wound coaxially around the first, over a length $\ell$.

Inside the inner solenoid the field is $B={\mu }_0{n}_1{I}_1$ (from Ampere's law); outside it is negligible for an ideal solenoid. The flux through one turn of the outer solenoid is ${\Phi }_{22}=B\cdot \pi a^2={\mu }_0{n}_1{I}_1\pi a^2$ (only the cross-section of the inner solenoid carries flux). The total flux linkage in the outer solenoid is ${\Lambda }_2=n_2\ell \cdot {\Phi }_{22}={\mu }_0{n}_1{n}_2\ell \pi a^2{I}_1$, so

$$M=\frac{{\Lambda }_2}{I_1}={\mu }_0{n}_1{n}_2\ell \pi a^2.$$

Note that $M$ depends on the geometry (lengths, radii, turn densities) but not on the currents. Reversing the argument (driving $I_2$ through the outer solenoid and computing flux linkage in the inner) gives the same $M$ — mutual inductance is symmetric: $M_{12}=M_{21}$.

Exercises [Intermediate+]

Retarded potentials and gauge freedom [Master]

The Maxwell equations in the potential formulation use the scalar potential $\varphi$ and vector potential $\mathbf{A}$ with $\mathbf{B}=\nabla \times \mathbf{A}$ and $\mathbf{E}=-\nabla \varphi -\partial \mathbf{A}/\partial t$. Substituting into the two inhomogeneous Maxwell equations gives

$${\nabla }^2\varphi +\frac{\partial }{\partial t}(\nabla \cdot \mathbf{A})=-\frac{\rho }{{ϵ}_0},$$ $${\nabla }^2\mathbf{A}-{\mu }_0{ϵ}_0\frac{{\partial }^2\mathbf{A}}{\partial t^2}-\nabla \left(\nabla \cdot \mathbf{A}+{\mu }_0{ϵ}_0\frac{\partial \varphi }{\partial t}\right)=-{\mu }_0\mathbf{J}.$$

These are coupled and messy. Gauge freedom simplifies them. The potentials are not unique: the transformation

$$\varphi \to {\varphi }^{\prime }=\varphi -\frac{\partial \Lambda }{\partial t},\mathbf{A}\to {\mathbf{A}}^{\prime }=\mathbf{A}+\nabla \Lambda ,$$

for any smooth function $\Lambda (\mathbf{r},t)$, leaves $\mathbf{E}$ and $\mathbf{B}$ unchanged. This is because $\mathbf{B}=\nabla \times \mathbf{A}$ is invariant under $\mathbf{A}\to \mathbf{A}+\nabla \Lambda$ (curl of gradient is zero), and the change in $\mathbf{E}$ from the $\partial \mathbf{A}/\partial t$ term cancels the change from the $\nabla \varphi$ term.

Lorenz gauge. Choose $\Lambda$ so that

$$\nabla \cdot \mathbf{A}+{\mu }_0{ϵ}_0\frac{\partial \varphi }{\partial t}=0.$$

The two potential equations decouple into wave equations with sources:

$$\boxed{{\nabla }^2\varphi -{\mu }_0{ϵ}_0\frac{{\partial }^2\varphi }{\partial t^2}=-\frac{\rho }{{ϵ}_0},{\nabla }^2\mathbf{A}-{\mu }_0{ϵ}_0\frac{{\partial }^2\mathbf{A}}{\partial t^2}=-{\mu }_0\mathbf{J}.}$$

These are solved by the retarded potentials:

$$\varphi (\mathbf{r},t)=\frac{1}{4\pi {ϵ}_0}\int \frac{\rho ({\mathbf{r}}^{\prime },t_r)}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}{d}^3{r}^{\prime },\mathbf{A}(\mathbf{r},t)=\frac{{\mu }_0}{4\pi }\int \frac{\mathbf{J}({\mathbf{r}}^{\prime },t_r)}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}{d}^3{r}^{\prime },$$

where $t_r=t-|\mathbf{r}-{\mathbf{r}}^{\prime }|/c$ is the retarded time. The fields at point $\mathbf{r}$ and time $t$ are determined by the sources at the retarded time — the information travels at speed $c$. This is the precise sense in which electromagnetic influences propagate at the speed of light: the potentials are retarded, and the fields inherit this causality.

Coulomb gauge. The alternative choice $\nabla \cdot \mathbf{A}=0$ gives ${\nabla }^2\varphi =-\rho /{ϵ}_0$ (Poisson's equation — the scalar potential is instantaneous), but the vector potential satisfies a more complicated equation with an additional non-local term. The Coulomb gauge is useful in quantum mechanics (radiation problems); the Lorenz gauge is natural in relativistic formulations.

Energy conservation: Poynting's theorem [Master]

Multiplying Faraday's law by $\mathbf{B}/{\mu }_0$ and Ampere's law (with displacement current) by ${ϵ}_0\mathbf{E}$, then subtracting, yields Poynting's theorem:

$$\boxed{\frac{\partial u}{\partial t}+\nabla \cdot \mathbf{S}=-\mathbf{J}\cdot \mathbf{E},}$$

where the electromagnetic energy density is $u=\frac{1}{2}{ϵ}_0|\mathbf{E}{|}^2+\frac{1}{2{\mu }_0}|\mathbf{B}{|}^2$ and the Poynting vector is

$$\mathbf{S}=\frac{1}{{\mu }_0}\mathbf{E}\times \mathbf{B}.$$

Poynting's theorem is the energy conservation law for electromagnetism. The term $\mathbf{J}\cdot \mathbf{E}$ is the rate at which the field does work on the charges (Joule heating, or mechanical power). If this is zero (vacuum), then $\partial u/\partial t+\nabla \cdot \mathbf{S}=0$ — energy is conserved locally, with $\mathbf{S}$ being the energy flux (power per unit area carried by the field). For electromagnetic waves, $\mathbf{S}$ points in the direction of propagation and has magnitude $cu$.

The electromagnetic field tensor (preview) [Master]

The four Maxwell equations are not four independent laws — they are the coordinate expression of a single geometric object. Define the electromagnetic field tensor $F^{\mu \nu }$ in terms of the 4-potential $A^{\mu }=(\varphi /c,\mathbf{A})$:

$$F^{\mu \nu }:={\partial }^{\mu }{A}^{\nu }-{\partial }^{\nu }{A}^{\mu }.$$

In matrix form (with metric signature $+---$):

$$F^{\mu \nu }=\left(\begin{array}{cccc}0& E_x/c& E_y/c& E_z/c\\ -E_x/c& 0& -B_z& B_y\\ -E_y/c& B_z& 0& -B_x\\ -E_z/c& -B_y& B_x& 0\end{array}\right).$$

The two homogeneous Maxwell equations ($\nabla \cdot \mathbf{B}=0$ and $\nabla \times \mathbf{E}=-\partial \mathbf{B}/\partial t$) are equivalent to the Bianchi identity ${\partial }_{\lambda }{F}_{\mu \nu }+{\partial }_{\mu }{F}_{\nu \lambda }+{\partial }_{\nu }{F}_{\lambda \mu }=0$. The two inhomogeneous equations ($\nabla \cdot \mathbf{E}=\rho /{ϵ}_0$ and $\nabla \times \mathbf{B}={\mu }_0\mathbf{J}+{\mu }_0{ϵ}_0\partial \mathbf{E}/\partial t$) become

$$\boxed{{\partial }_{\mu }{F}^{\mu \nu }={\mu }_0{J}^{\nu },}$$

where $J^{\nu }=(c\rho ,\mathbf{J})$ is the four-current. This single tensor equation replaces the four vector Maxwell equations. Gauge invariance is the statement that $F^{\mu \nu }$ is unchanged under $A^{\mu }\to A^{\mu }+{\partial }^{\mu }\Lambda$ — the field tensor is gauge-invariant even though the potentials are not. The full differential-forms treatment lives in unit 10.04.01 pending.

Connections [Master]

• Coulomb's law and Gauss's law 10.01.01 pending provides the first and third Maxwell equations ($\nabla \cdot \mathbf{E}=\rho /{ϵ}_0$, $\nabla \cdot \mathbf{B}=0$). Faraday's law ($\nabla \times \mathbf{E}=-\partial \mathbf{B}/\partial t$) is the time-dependent generalisation that makes the electric field non-conservative.

• Biot-Savart and Ampere's law 10.02.01 pending gives the magnetostatic Ampere's law $\nabla \times \mathbf{B}={\mu }_0\mathbf{J}$. Maxwell's displacement current ${\mu }_0{ϵ}_0\partial \mathbf{E}/\partial t$ extends it to time-varying fields — the fourth equation, completing the set.

• Maxwell's equations in differential forms 10.04.01 pending reformulates the four vector Maxwell equations as two differential-form equations ($dF=0$, $d\star F=J$). The Faraday law content is part of $dF=0$; the displacement current is part of $d\star F=J$.

• EM waves 10.04.02 pending (pending) derives wave solutions from the complete Maxwell equations. The wave equation for $\mathbf{E}$ (Exercise 8) requires both Faraday's law and the displacement current — without either, electromagnetic waves do not exist.

• Hamiltonian mechanics 09.04.02 pending uses gauge transformations in the context of constrained systems (Dirac-Bergmann algorithm). The gauge freedom $\mathbf{A}\to \mathbf{A}+\nabla \Lambda$ in electromagnetism is the physical origin of gauge symmetry, and the minimal-coupling Hamiltonian $H=|\mathbf{p}-e\mathbf{A}{|}^2/(2m)+e\varphi$ is the bridge between classical mechanics and electrodynamics.

• Special relativity (Lecture 5 in Susskind's Special Relativity and Classical Field Theory) shows that the scalar and vector potentials combine into the 4-potential $A^{\mu }$, and $\mathbf{E}$ and $\mathbf{B}$ are different components of the same tensor $F^{\mu \nu }$. Faraday's law and the absence of magnetic monopoles are the "space-space" components of the Bianchi identity; Gauss's law and Ampere's law with displacement current are the "time-space" components of the inhomogeneous equation.

Historical & philosophical context [Master]

Michael Faraday discovered electromagnetic induction in August 1831, demonstrating that a changing magnetic field induces an electric current. Joseph Henry independently discovered self-induction in 1832. Faraday had no mathematical training — he thought in terms of field lines and "electrotonic states" — but his experimental results were precise and reproducible. Maxwell (1865) translated Faraday's qualitative picture into the differential equations that bear his name, added the displacement current, and predicted electromagnetic waves travelling at $c=1/\sqrt{{\mu }_0{ϵ}_0}$. Hertz confirmed the wave prediction experimentally in 1887. The retarded-potential formulation is due to Lorenz (1867, not Lorentz of the Lorentz transformation — the two are frequently confused). The tensor formulation emerged from Minkowski (1908) after Einstein's special relativity (1905) showed that the Maxwell equations are already Lorentz-covariant.

Faraday's discovery resolved a deep asymmetry in the known laws. Oersted (1820) had shown that electric currents create magnetic fields. Ampere's law quantified this. But the reverse — do magnetic fields create electric fields? — was missing. Faraday's law closes the loop: changing magnetic fields create electric fields, and (via Maxwell's displacement current) changing electric fields create magnetic fields. This mutual coupling is what sustains electromagnetic waves in vacuum, requiring no charges or currents at all. The philosophical import is that fields are real dynamical entities, not mere bookkeeping devices for forces between charges — they carry energy, momentum, and information at a finite speed.

Bibliography [Master]

Primary literature (cite when used; not all currently in reference/):

• Faraday, M., "Experimental Researches in Electricity", Phil. Trans. Roy. Soc. 122 (1832), 125–162. [Need to source — originator paper.]
• Henry, J., "On the Production of Currents and Sparks of Electricity from Magnetism", American Journal of Science 22 (1832), 403–408.
• Maxwell, J. C., "A Dynamical Theory of the Electromagnetic Field", Phil. Trans. Roy. Soc. 155 (1865), 459–512.
• Lorenz, L., "On the Identity of the Vibrations of Light with Electrical Currents", Phil. Mag. 34 (1867), 287–301.
• Hertz, H., "On Electromagnetic Waves in Air and their Reflection", Annalen der Physik 34 (1888), 610–623.
• Minkowski, H., "Die Grundgleichungen fur die elektromagnetischen Vorgange in bewegten Korpern", Nachrichten der Gesellschaft der Wissenschaften zu Gottingen (1908), 53–111.
• Griffiths, D. J., Introduction to Electrodynamics, 4th ed. (Cambridge, 2017).
• Jackson, J. D., Classical Electrodynamics, 3rd ed. (Wiley, 1999).
• Zangwill, A., Modern Electrodynamics (Cambridge, 2013).
• Susskind, L. & Friedman, A., Special Relativity and Classical Field Theory (Basic Books, 2017).
• Tong, D., Electromagnetism (DAMTP Cambridge lecture notes, §3 "Electrodynamics").