10.03.02 · em-sr / electrodynamics

Faraday's law in integral and differential form: mutual and self-inductance, energy in inductors

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Anchor (Master): Jackson, Classical Electrodynamics, 3rd ed. (1999), Ch. 6; Smythe, Static and Dynamic Electricity, 3rd ed. (1968), Ch. 8

Intuition Beginner

Michael Faraday discovered in 1831 that a changing magnetic field creates an electric field. If you move a magnet through a coil of wire, a voltage appears across the wire — even though there is no battery. This is electromagnetic induction, and Faraday's law quantifies it.

The electromotive force (EMF) around a closed loop equals the negative rate of change of magnetic flux through the loop:

E = - \frac{d Φ _{B}}{d t}

The minus sign is Lenz's law: the induced EMF always opposes the change that created it. If you push a magnet toward a coil, the induced current creates its own magnetic field that pushes back. This is nature's way of resisting change — the electromagnetic equivalent of inertia.

Mutual inductance ( $M$ ) measures how much flux from circuit 1 links circuit 2. A changing current in circuit 1 induces an EMF in circuit 2: $E_{2} = - M d I_{1} / d t$ . Self-inductance ( $L$ ) is the special case where circuit 1 and circuit 2 are the same: a changing current in a coil induces an EMF in itself: $E = - L d I / d t$ .

Visual Beginner

Quantity	Definition	Units
Magnetic flux $Φ_{B}$	B-field through a surface	Weber (Wb)
EMF $E$	$\oint E \cdot d l$ (voltage around a loop)	Volts (V)
Mutual inductance $M$	$Φ_{2} = M I_{1}$	Henry (H)
Self-inductance $L$	$Φ = L I$	Henry (H)
Energy in inductor	$W = \frac{1}{2} L I^{2}$	Joules (J)

Worked example Beginner

A circular loop of radius $a$ sits in a uniform magnetic field $B (t) = B_{0} t \hat{z}$ that increases linearly with time. Find the induced EMF.

The flux through the loop is $Φ_{B} = B (t) \cdot π a^{2} = B_{0} π a^{2} t$ . Faraday's law gives $E = - d Φ_{B} / d t = - B_{0} π a^{2}$ . The induced EMF is constant (since $B$ changes at a constant rate) and drives a current that creates a field opposing the increase (Lenz's law). The induced current flows clockwise (when viewed from above) to create a downward field that opposes the increasing upward field.

Check your understanding Beginner

Formal definition Intermediate+

Faraday's law in integral form.

E = \oint_{C} E \cdot d l = - \frac{d}{d t} \int_{S} B \cdot d a

where $C$ is a closed curve bounding the surface $S$ , and the total time derivative acts on the flux (including changes in $B$ and in the geometry of $S$ if the loop moves or deforms).

Faraday's law in differential form. For a stationary loop:

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

This follows from the integral form by Stokes' theorem: $\oint E \cdot d l = \int_{S} (\nabla \times E) \cdot d a$ and comparing with $- \int_{S} (\partial B / \partial t) \cdot d a$ for arbitrary $S$ .

The flux rule (motional EMF). For a moving loop, the total time derivative of the flux includes both the time variation of $B$ and the motion of the boundary $C$ :

\frac{d}{d t} \int_{S} B \cdot d a = \int_{S} \frac{\partial B}{\partial t} \cdot d a + \oint_{C} (v \times B) \cdot d l

where $v$ is the velocity of the boundary element $d l$ . The total EMF is:

E = \oint_{C} (E + v \times B) \cdot d l

Mutual inductance (Neumann formula). The mutual inductance between two circuits $C_{1}$ and $C_{2}$ is:

M_{12} = \frac{μ _{0}}{4 π} \oint_{C_{1}} \oint_{C_{2}} \frac{d l _{1} \cdot d l _{2}}{∣ r _{1} - r _{2} ∣}

This shows $M_{12} = M_{21}$ (reciprocity): the mutual inductance is a geometric property of the two circuits, independent of which carries the current.

Self-inductance. For a single circuit carrying current $I$ :

L = \frac{Φ _{B}}{I} = \frac{1}{I} \int_{S} B \cdot d a

where $Φ_{B}$ is the total flux linking the circuit. For a solenoid of length $ℓ$ , cross-section $A$ , and $n$ turns per unit length: $L = μ_{0} n^{2} A ℓ$ .

Magnetic energy. The energy stored in a system of inductors is:

W = \frac{1}{2} i \sum L_{i} I_{i}^{2} + i < j \sum M_{ij} I_{i} I_{j}

In terms of the magnetic field:

W = \frac{1}{2 μ _{0}} \int ∣ B ∣^{2} d^{3} r = \frac{1}{2} \int H \cdot B d^{3} r

Key derivation Intermediate+

Derivation (Neumann formula and reciprocity).

Theorem. The mutual inductance $M_{12}$ of circuit $C_{2}$ with respect to circuit $C_{1}$ equals $M_{21}$ , and is given by the Neumann formula.

Proof. When current $I_{1}$ flows in $C_{1}$ , it creates a vector potential $A_{1} (r) = \frac{μ _{0} I _{1}}{4 π} \oint_{C_{1}} \frac{d l _{1}}{∣ r - r _{1} ∣}$ . The flux through $C_{2}$ is:

Φ_{12} = \int_{S_{2}} B_{1} \cdot d a_{2} = \oint_{C_{2}} A_{1} \cdot d l_{2} = \frac{μ _{0} I _{1}}{4 π} \oint_{C_{1}} \oint_{C_{2}} \frac{d l _{1} \cdot d l _{2}}{∣ r _{1} - r _{2} ∣}

So $M_{12} = Φ_{12} / I_{1} = \frac{μ _{0}}{4 π} \oint\oint \frac{d l _{1} \cdot d l _{2}}{∣ r _{1} - r _{2} ∣}$ . The double integral is symmetric under $1 \leftrightarrow 2$ , so $M_{12} = M_{21}$ . $□$

Bridge. Faraday's law is the third of Maxwell's equations (after Gauss and Ampere) and provides the crucial link between electricity and magnetism: changing magnetic fields create electric fields. The foundational insight is that induction is a local phenomenon: the curl of $E$ at each point equals minus the time derivative of $B$ at that point. The central message is that inductance is a purely geometric quantity: $M$ and $L$ depend only on the shapes and positions of the circuits, not on the currents. This geometric character makes inductance a powerful tool in circuit analysis and motivates the vector potential formulation 10.02.02. Putting these together, Faraday's law provides the time-dependent coupling between electric and magnetic fields that, together with the displacement current 10.03.04, completes the set of equations that describe all classical electromagnetic phenomena.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Compute the mutual inductance of two concentric coplanar circular loops with radii $a$ (small) and $b$ (large, $b ≫ a$ ).

Hint

Use $M = Φ_{12} / I_{1}$ . For a small loop inside a large one, the field from the large loop is approximately uniform over the small loop. $Φ_{12} = B_{1} \cdot π a^{2}$ where $B_{1}$ is the on-axis field of the large loop at its center.

Answer

For the large loop (radius $b$ , current $I_{b}$ ): the field at its center is $B = μ_{0} I_{b} / (2 b)$ . This is approximately uniform over the small loop (radius $a ≪ b$ ). The flux through the small loop is $Φ_{12} = B π a^{2} = μ_{0} I_{b} π a^{2} / (2 b)$ . So $M = Φ_{12} / I_{b} = μ_{0} π a^{2} / (2 b)$ .

By reciprocity, $M_{21} = M_{12} = μ_{0} π a^{2} / (2 b)$ . For the reverse calculation (current in the small loop, flux through the large loop), we would use $Φ_{21} = M I_{a}$ , which gives $Φ_{21} / I_{a} = μ_{0} π a^{2} / (2 b)$ . The Neumann formula would give the same result (but requires evaluating an elliptic integral).

Exercise 5 (hard, symbolic).

Derive the motional EMF formula from the Lorentz force law: a conductor element $d l$ moving with velocity $v$ in a field $B$ experiences a force $d F = q v \times B$ on each charge carrier. Show that the EMF around a moving loop equals $- \frac{d Φ _{B}}{d t}$ .

Hint

The EMF is $\oint (E + v \times B) \cdot d l$ where $E$ is the electric field in the lab frame. Use the convective derivative of the flux.

Answer

For charges in a conductor moving with velocity $v$ , the force per unit charge is $f = E + v \times B$ (Lorentz force). The EMF around a closed loop of the conductor is:

E = \oint_{C} (E + v \times B) \cdot d l

Using Faraday's law for the electric field part: $\oint_{C} E \cdot d l = - \int_{S} \frac{\partial B}{\partial t} \cdot d a$ . For the motional part, use the identity: $\oint_{C} (v \times B) \cdot d l = - \int_{S} \frac{\partial}{\partial t} (B \cdot d a)_{convective}$ . Together:

E = - \int_{S} \frac{\partial B}{\partial t} \cdot d a - \int_{S} v \cdot (curl E \times B) \cdot d a = - \frac{d Φ _{B}}{d t}

where the total derivative includes both the changing field and the moving boundary. This is the flux rule: the EMF equals the negative total rate of change of flux, regardless of whether the change comes from a time-varying $B$ , a moving loop, or a deforming loop.

Exercise 6 (hard, symbolic).

Two coupled circuits have self-inductances $L_{1}$ , $L_{2}$ and mutual inductance $M$ . Derive the coefficient of coupling $k = M / L_{1} L_{2}$ and show that $0 \leq k \leq 1$ .

Hint

The energy $W = \frac{1}{2} L_{1} I_{1}^{2} + M I_{1} I_{2} + \frac{1}{2} L_{2} I_{2}^{2} \geq 0$ for all currents. This is a positive-definite quadratic form.

Answer

The magnetic energy of two coupled circuits is $W = \frac{1}{2} L_{1} I_{1}^{2} + M I_{1} I_{2} + \frac{1}{2} L_{2} I_{2}^{2}$ . Since energy is always non-negative, this quadratic form must be positive semi-definite: $W \geq 0$ for all $I_{1}, I_{2}$ . The matrix $(L_{1} M M L_{2})$ must be positive semi-definite, which requires $L_{1} \geq 0$ , $L_{2} \geq 0$ , and $L_{1} L_{2} - M^{2} \geq 0$ . The last condition gives $M^{2} \leq L_{1} L_{2}$ , i.e., $∣ M ∣ \leq L_{1} L_{2}$ .

Defining $k = M / L_{1} L_{2}$ (the coupling coefficient): $- 1 \leq k \leq 1$ , and since $M$ can be taken positive by appropriate choice of current direction: $0 \leq k \leq 1$ . $k = 0$ : no coupling (distant circuits). $k = 1$ : perfect coupling (all flux from one circuit links the other, e.g., a tightly wound transformer).

Exercise 7 (medium, symbolic).

A conducting bar of length $ℓ$ slides along two parallel rails separated by distance $ℓ$ , in a uniform field $B = B_{0} \hat{z}$ perpendicular to the rail plane. The bar moves with velocity $v$ to the right. Find the induced EMF and the current if the circuit has resistance $R$ .

Hint

The area of the circuit changes as the bar moves: $A = ℓ \cdot x (t)$ where $x (t) = v t$ . The flux is $Φ = B_{0} ℓ v t$ .

Answer

The area enclosed by the circuit increases as the bar moves: $A (t) = ℓ \cdot x = ℓ v t$ . The flux is $Φ = B_{0} A = B_{0} ℓ v t$ . By Faraday's law: $E = - d Φ/ d t = - B_{0} ℓ v$ . The induced EMF is constant (since the bar moves at constant velocity). The current is $I = E / R = B_{0} ℓ v / R$ .

The force on the current-carrying bar in the magnetic field is $F = I ℓ B_{0} = B_{0}^{2} ℓ^{2} v / R$ (opposing the motion, by Lenz's law). The power dissipated is $P = I^{2} R = B_{0}^{2} ℓ^{2} v^{2} / R$ . The mechanical power input is $F v = B_{0}^{2} ℓ^{2} v^{2} / R$ . Energy is conserved: the mechanical work done pushing the bar equals the electrical energy dissipated in the resistance.

Exercise 8 (medium, symbolic).

An RL circuit has resistance $R$ and inductance $L$ . A battery of voltage $V_{0}$ is connected at $t = 0$ . Find the current as a function of time.

Hint

The circuit equation is $V_{0} = I R + L d I / d t$ . This is a first-order linear ODE with initial condition $I (0) = 0$ .

Answer

$V_{0} = I R + L d I / d t$ . Rearranging: $d I / d t + (R / L) I = V_{0} / L$ . The integrating factor is $e^{R t / L}$ . Multiplying: $d / d t (I e^{R t / L}) = (V_{0} / L) e^{R t / L}$ . Integrating from 0 to $t$ : $I (t) e^{R t / L} - I (0) = (V_{0} / R) (e^{R t / L} - 1)$ . With $I (0) = 0$ : $I (t) = (V_{0} / R) (1 - e^{- R t / L})$ .

The time constant is $τ = L / R$ . After one time constant: $I = 0.632 \times V_{0} / R$ . The current approaches the steady-state value $V_{0} / R$ exponentially. The inductor initially opposes the change in current (back-EMF $L d I / d t$ ) and eventually acts like a wire (no voltage drop at steady state).

Lean formalization Intermediate+

Mathlib has line integrals and Stokes' theorem but not Faraday's law, the concept of EMF, inductance, or magnetic energy. The time-dependent coupling between electric and magnetic fields is a physical law, not a mathematical theorem, and would require physical axioms beyond pure mathematics. lean_status: none.

Advanced results Master

The flux rule controversy. The "flux rule" $E = - d Φ_{B} / d t$ is not universally valid. Feynman (Lectures on Physics, Vol. II, Ch. 17) identified cases where the flux rule fails: it gives wrong answers for circuits that change topology, or for situations involving both moving and stationary parts. The correct approach is to use the Lorentz force law $f = E + v \times B$ directly, integrating around the circuit. The flux rule is a convenient heuristic that works for most practical situations but is not a fundamental law.

Self-inductance and the Neumann formula. The Neumann formula for self-inductance involves a divergent integral (the denominator $∣ r_{1} - r_{2} ∣$ vanishes when the two line elements coincide). This divergence is resolved by giving the wire a finite radius $a$ and computing the internal inductance separately. For a round wire of radius $a$ : the internal contribution is $μ_{0} ℓ / (8 π)$ (independent of $a$ ) and the external contribution depends logarithmically on $a$ .

Magnetic energy and force. The force between two circuits can be computed from the energy: $F = - \frac{\partial W}{\partial x}$ where $x$ is the separation. For two coupled circuits: $F = - I_{1} I_{2} \frac{\partial M}{\partial x}$ . This provides a convenient way to compute forces without integrating the Lorentz force density.

Synthesis. Faraday's law is the dynamical heart of electromagnetism: it introduces time-dependence and creates the coupling between electric and magnetic fields that makes wave propagation possible. The foundational insight is that a changing magnetic field creates a curling electric field — not a conservative field that can be written as the gradient of a potential, but a circulating field with non-zero line integral around any loop enclosing the changing flux. The central message is that inductance — both mutual and self — is a purely geometric property of circuits, determined by the Neumann formula and characterised by the coupling coefficient $k$ . Putting these together, Faraday's law provides the third Maxwell equation ( $curl E = - \partial B / \partial t$ ), which together with the Ampere-Maxwell law 10.03.04 forms the dynamical pair that supports electromagnetic waves 10.04.02.

Full proof set Master

Proposition (Reciprocity of mutual inductance). $M_{12} = M_{21}$ .

Proof. From the Neumann formula: $M_{12} = \frac{μ _{0}}{4 π} \oint_{C_{1}} \oint_{C_{2}} \frac{d l _{1} \cdot d l _{2}}{∣ r _{1} - r _{2} ∣}$ . The dot product and the distance are symmetric under interchange of indices 1 and 2. Therefore $M_{12} = M_{21}$ . $□$

Proposition (Energy stored in an inductor). The work done to establish a current $I$ in an inductor $L$ is $W = \frac{1}{2} L I^{2}$ .

Proof. The EMF across the inductor is $E = - L d I / d t$ (opposing the increase). The external source must provide voltage $V = L d I / d t$ . The power supplied is $P = V I = L I d I / d t$ . The total energy stored is $W = \int_{0}^{t} P d t^{'} = \int_{0}^{I} L I^{'} d I^{'} = \frac{1}{2} L I^{2}$ . $□$

Connections Master

Vector potential 10.02.02 provides the Neumann formula for inductance and the link between flux and the line integral of $A$ .
Displacement current 10.03.04 is the fourth Maxwell equation; Faraday's law is the third. Together they form the dynamical pair.
EM waves 10.04.02 result from the coupling of Faraday's law and the Ampere-Maxwell law.
Poynting vector 10.03.05 describes energy flow; the inductor energy $L I^{2} /2$ is stored in the magnetic field $B^{2} / (2 μ_{0})$ .
Faraday tensor 10.06.01 encodes both $E$ and $B$ ; Faraday's law is one component of the exterior derivative $d F = 0$ .

Historical & philosophical context Master

Faraday's discovery of electromagnetic induction in 1831 was one of the most important experiments in the history of physics. He found that a changing magnetic field induces an electric current, connecting electricity and magnetism for the first time. Faraday had no mathematical training and described his results in terms of "lines of force" — what we now call field lines.

Maxwell formalised Faraday's results mathematically in 1865, expressing the law in differential form as $curl E = - \partial B / \partial t$ . The integral form (the flux rule) is often attributed to Faraday himself, though the precise mathematical statement was formulated later.

Lenz's law (1834) provided the sign convention and was recognised as a consequence of energy conservation. The Neumann formula for mutual inductance was derived by Franz Ernst Neumann in 1845, providing the first quantitative theory of induction.

The concept of self-inductance was introduced by Joseph Henry in 1832 (independently of Faraday), and the unit of inductance (the Henry) is named after him. Henry discovered self-inductance before Faraday published his results on mutual induction, though Faraday's work was completed first.

Bibliography Master

Faraday, M., "Experimental Researches in Electricity," Phil. Trans. R. Soc. 122, 125-162 (1832).
Lenz, E., "Ueber die Bestimmung der Richtung der durch elektodynamische Vertheilung erregten galvanischen Strome," Ann. Phys. 107, 483-494 (1834).
Neumann, F. E., "Allgemeine Gesetze der inducirten elektrischen Strome," Abh. Konigl. Akad. Wiss. Berlin (1845).
Jackson, J. D., Classical Electrodynamics, 3rd ed. (Wiley, 1999).
Griffiths, D. J., Introduction to Electrodynamics, 4th ed. (Cambridge, 2017).
Feynman, R. P., Leighton, R. B., and Sands, M., The Feynman Lectures on Physics, Vol. II (Addison-Wesley, 1964).

Prerequisites

10.03.01
10.02.02

Used in

10.03.04
10.03.05

Tier anchors

beginner: Griffiths, Introduction to Electrodynamics, 4th ed. (2017), Ch. 7.1-7.2
intermediate: Griffiths, Introduction to Electrodynamics, 4th ed. (2017), Ch. 7; Jackson, Classical Electrodynamics, 3rd ed. (1999), Ch. 6.1-6.3
master: Jackson, Classical Electrodynamics, 3rd ed. (1999), Ch. 6; Smythe, Static and Dynamic Electricity, 3rd ed. (1968), Ch. 8

References

Griffiths — Introduction to Electrodynamics, 4th ed. (Cambridge, 2017) · Ch. 7.1 Faraday's law; Ch. 7.2 The induced electric field; mutual inductance; self-inductance; energy in magnetic fields
Jackson — Classical Electrodynamics, 3rd ed. (Wiley, 1999) · Ch. 6.1-6.3 Faraday's law; Induction; Self and mutual inductance; Magnetic energy
Smythe — Static and Dynamic Electricity, 3rd ed. (McGraw-Hill, 1968) · Ch. 8 Electromagnetic induction; self and mutual inductance; Neumann formula

Estimated time

beginner: 15m
intermediate: 35m
master: 55m