Faraday's law of induction

In the field of electromagnetism, Faraday's law of induction elucidates the mechanism by which a fluctuating magnetic field generates an electric current within an electrical circuit. This phenomenon, termed electromagnetic induction, constitutes the foundational operational principle for devices such as transformers, inductors, and various categories of electric motors, generators, and solenoids.

In electromagnetism, Faraday's law of induction describes how a changing magnetic field can induce an electric current in a circuit. This phenomenon, known as electromagnetic induction, is the fundamental operating principle of transformers, inductors, and many types of electric motors, generators and solenoids.

Within scientific literature, the designation Faraday's law commonly refers to two distinct yet intimately related physical principles. The first is the Maxwell–Faraday equation, a constituent of Maxwell's equations, which posits that a magnetic field undergoing temporal variation invariably produces an associated circulating electric field. This particular law pertains directly to the fields themselves, obviating the necessity for a tangible electrical circuit.

Conversely, the second principle is Faraday's flux rule, also known as the Faraday–Lenz law, which establishes a correlation between the electromotive force (emf) developed within a closed conducting loop and the temporal rate of alteration of magnetic flux traversing that loop. This flux rule delineates two primary mechanisms for emf generation. In the case of transformer emf, a time-dependent magnetic field induces an electric field, consistent with the Maxwell–Faraday equation, which subsequently propels a current through the loop. For motional emf, the circuit's movement through a magnetic field results in an emf originating from the magnetic component of the Lorentz force exerted upon the charge carriers within the conductor.

Historically, the divergent explanations for motional electromotive force (emf) and transformer emf presented a significant conceptual challenge, given that the observed current is solely contingent upon relative motion, yet the underlying physical interpretations varied between the two scenarios. Within the framework of special relativity, this differentiation is comprehended as being frame-dependent: a phenomenon perceived as a magnetic force in one inertial reference frame may manifest as an induced electric field in another.

Historical Overview

In 1820, Hans Christian Ørsted experimentally demonstrated that an electric current generates a magnetic field, illustrating this by the deflection of a compass needle positioned near a conductor carrying current. This seminal discovery subsequently stimulated scientific inquiry into the inverse relationship: specifically, whether a magnetic field could, in turn, induce an electric current.

Early experimental investigations indicated that a stationary magnetic field exerted no influence on an adjacent circuit; merely positioning a magnet near a wire loop failed to generate any current. The pivotal advancement occurred in 1831, when Michael Faraday conclusively demonstrated that a dynamic, or changing, magnetic field was capable of inducing an electric current within a circuit. Joseph Henry independently made analogous observations in 1832; however, Faraday holds precedence for the publication of these discoveries.

Faraday's laboratory notebook entry from August 29, 1831, details an experimental demonstration of electromagnetic induction. He constructed a rudimentary toroidal transformer by winding two coils of wire around opposing sections of an iron ring. Upon connecting one coil to a battery, he noted a momentary deflection in a galvanometer linked to the second coil. From this observation, he deduced that the fluctuating current in the primary coil established a varying magnetic field within the iron ring, which subsequently induced a current in the secondary coil. He characterized this phenomenon as a "wave of electricity" propagating through the iron.

During the subsequent months, Faraday uncovered additional manifestations of electromagnetic induction. He documented transient currents when a bar magnet was swiftly inserted into or withdrawn from a wire coil. Furthermore, he engineered a device, presently recognized as Faraday's disk or a homopolar generator, which generated a continuous (DC) current through the rotation of a copper disk within a static magnetic field, utilizing a sliding electrical contact.

Faraday elucidated these phenomena through the conceptual framework of lines of force. Nevertheless, his theoretical propositions encountered skepticism due to their lack of mathematical formalization. James Clerk Maxwell subsequently provided mathematical articulation for Faraday's insights, integrating them into his comprehensive electromagnetic theory during the early 1860s.

Within Maxwell's published works, the temporal variation aspect of electromagnetic induction is articulated as a differential equation. Oliver Heaviside subsequently designated this equation as Faraday's law, despite its divergence from Faraday's original formulation and its inability to account for motional electromotive force (emf). Heaviside's rendition is the form currently acknowledged within the set of equations collectively known as Maxwell's equations.

Lenz's law, formally articulated by Emil Lenz in 1834, characterizes the "flux through the circuit" and specifies the direction of the induced electromotive force (emf) and current that arise from electromagnetic induction.

In 1845, Franz Ernst Neumann formalized the laws governing the induction of electric currents into a mathematical framework.

Albert Einstein posited that Faraday's 1834 law of induction laid significant foundational principles for his own theory of special relativity.

The Flux Rule

Faraday's law of induction, alternatively referred to as the flux rule, flux law, or Faraday–Lenz law, postulates that the electromotive force (emf) generated within a closed circuit is directly proportional to the negative rate at which the magnetic flux traversing the circuit changes. This principle applies universally to any circuit composed of thin wire, encompassing flux alterations resulting from magnetic field variations, circuit motion, or shape deformation. Lenz's law dictates the orientation of the induced emf, specifying that the induced current will establish a magnetic field that counteracts the initial change in magnetic flux.

In the International System of Units (SI), the mathematical representation of this law is given by: $${\displaystyle {\mathcal {E}}=-{\frac {\mathrm {d} \Phi _{B}}{\mathrm {d} t}},}$$ Here, ${\displaystyle {\mathcal {E}}}$ denotes the electromotive force (emf), and Φ_B signifies the magnetic flux penetrating the circuit. The magnetic flux itself is formally defined as the surface integral of the magnetic field B across a time-variant surface Σ(t), whose boundary is delineated by the wire loop: $${\displaystyle \Phi _{B}=\iint _{\Sigma (t)}\mathbf {B} (t)\cdot \mathrm {d} \mathbf {A} \,,}$$ In this context, dA represents an infinitesimal area vector oriented perpendicularly to the surface. The scalar product B · dA quantifies the magnetic flux passing through this differential area element. Conceptually, magnetic flux can be understood as being directly proportional to the density of magnetic field lines intersecting the loop.

A change in magnetic flux induces an electromotive force (emf) within the circuit loop. This induced emf represents the energy expended per unit charge to complete one full traversal of the loop. For a basic circuit possessing a resistance ${\displaystyle R}$, an emf ${\displaystyle {\mathcal {E}}}$ generates a current ${\displaystyle I}$, as defined by Ohm's Law: ${\displaystyle {\mathcal {E}}=IR}$. Alternatively, if the circuit loop is interrupted to form an open circuit and a voltmeter is connected across the resulting terminals, the emf is equivalent to the voltage reading obtained across these open ends.

For a tightly wound coil consisting of N identical turns, the magnetic field lines traverse the surface N times. Consequently, Faraday's law of induction is expressed as: $${\displaystyle {\mathcal {E}}=-N{\frac {\mathrm {d} \Phi _{B}}{\mathrm {d} t}}}$$, where N denotes the number of wire turns and Φ_B represents the magnetic flux passing through a single loop. The term NΦ_B is referred to as the linked flux.

Magnetic flux can vary due to either the temporal movement or deformation of the loop, or the inherent temporal variation of the magnetic field itself. These two distinct scenarios align with the mechanisms delineated by the flux rule:

• Motional emf: This phenomenon occurs when the electrical circuit traverses a magnetic field that is static yet spatially non-uniform.
• Transformer emf: This arises when the circuit maintains a stationary position, but the surrounding magnetic field undergoes temporal variation.

Motional Electromotive Force

The fundamental principle of motional electromotive force is exemplified by a conductive rod translating through a magnetic field oriented perpendicularly to both the rod and its trajectory. As the conductor moves within the magnetic field, its mobile electrons are subjected to the magnetic component (qv × B) of the Lorentz force, which propels them along the rod's axis. This action results in a charge separation across the rod's extremities. Under steady-state conditions, the electric field generated by the accumulated charge counteracts the magnetic force.

Should the rod be integrated into a closed conducting loop traversing a non-uniform magnetic field, an analogous effect can induce a current within the circuit. Consider, for example, a scenario where the magnetic field is spatially restricted, and the loop initially resides outside this defined region. Upon entry into the field, the portion of the loop encompassing magnetic flux expands, thereby inducing an electromotive force. From the standpoint of the Lorentz force, this occurs because the field exerts a magnetic force on the charge carriers within the segments of the loop entering the region. However, once the entire loop is situated within a uniform magnetic field and maintains a constant velocity, the total magnetic flux enclosed remains invariant, causing the induced electromotive force to dissipate. In such circumstances, the magnetic forces acting on opposing sides of the loop effectively neutralize each other.

Transformer Electromotive Force

A distinct yet complementary phenomenon is transformer electromotive force, which manifests when a conductive loop remains static while the magnetic flux passing through it varies as a result of a time-dependent magnetic field. This variation can arise through two primary mechanisms: either the source generating the magnetic field undergoes movement, consequently modifying the field distribution across the stationary loop, or the intensity of the magnetic field itself fluctuates temporally at a fixed point, exemplified by the operation of an energized electromagnet.

Under either of these conditions, no magnetic force is exerted upon the charges; instead, the induced electromotive force originates exclusively from the electric component (qE) of the Lorentz force. As stipulated by the Maxwell–Faraday equation, a magnetic field that changes over time generates a circulating electric field, which subsequently propels current within the loop. This principle is fundamental to the functioning of various electrical machinery, including synchronous generators. The electric field induced through this mechanism is inherently non-conservative, implying that its line integral over any closed loop will not equate to zero.

Determining the Direction of Induced Current

The direction of the electromotive force (emf) can be ascertained directly from Faraday's law, obviating the need to invoke Lenz's law. A left-hand rule provides a practical method for this determination, as outlined below:

• Position the curved fingers of the left hand to align with the loop (indicated by the yellow line).
• Extend the thumb. The extended thumb then signifies the direction of n (brown), which represents the normal vector to the area encompassed by the loop.
• The determination of the sign for ΔΦ_B, which signifies the change in magnetic flux, involves calculating the difference between the initial and final flux values. These flux values are defined relative to the normal vector n, conventionally oriented according to the right-hand rule, where the stretched thumb indicates its direction.
• A positive change in magnetic flux, denoted as ΔΦ_B, corresponds to an electromotive force (EMF) whose direction is represented by the curved fingers, aligning with the yellow arrowheads.
• Conversely, a negative value for ΔΦ_B implies that the electromotive force's direction is contrary to that indicated by the curved fingers and the yellow arrowheads.

Maxwell–Faraday Equation

As one of the four fundamental Maxwell's equations, the Maxwell–Faraday equation holds a pivotal position in the theoretical framework of classical electromagnetism. It postulates that a time-dependent magnetic field invariably induces a spatially varying, non-conservative electric field. Expressed in its differential form and utilizing SI units, the equation is presented as follows:

In this formulation, ∇ × denotes the curl operator, E(r, t) represents the electric field, and B(r, t) signifies the magnetic field. Both fields are typically dependent on spatial position r and temporal variable t.

Alternatively, the equation can be expressed in an integral form, derived through the application of the Kelvin–Stokes theorem:

Within this integral representation, as illustrated in the accompanying figure, Σ designates a surface delimited by the closed loop ∂Σ. The term dl represents an infinitesimal vector element along this loop. The vector area element dA is defined as perpendicular to the surface and its orientation adheres to the right-hand rule: the thumb's direction aligns with dA, while the curled fingers indicate the corresponding direction of dl along the boundary.

The left-hand side of this equation quantifies the circulation of the electric field along the closed loop ∂Σ. In the case of static electric fields, this circulation is zero because such fields can be derived from the gradient of a scalar potential. Conversely, a time-varying magnetic field generates a non-conservative electric field, resulting in a non-zero circulation. When this induced electric field interacts with a conducting loop, it drives an electric current within that loop.

Provided that the surface Σ remains constant over time, the right-hand side of the equation simplifies to the time-derivative of the magnetic flux Φ_B passing through that surface: $${\displaystyle \oint _{\partial \Sigma }\mathbf {E} \cdot \mathrm {d} \mathbf {l} =-{\frac {\mathrm {d} }{\mathrm {d} t}}\iint _{\Sigma }\mathbf {B} \cdot \mathrm {d} \mathbf {A} =-{\frac {\mathrm {d} \Phi _{B}}{\mathrm {d} t}}.}$$When the left-hand side is interpreted as the work per unit charge performed by the electric field on charges within a stationary conducting loop, this equation effectively reiterates the flux rule, specifically for a circuit that is not in motion.

Within the non-relativistic approximation, the solenoidal component of the induced electric field is calculable through the following volume integral: E s ( r , t ) ≈<!-- ≈ --> −<!-- − --> §3637§ §3940§ π ∭ V   ( ∂ B ( r ′ , t ) ∂ t ) × ( r − r ′ ) | r − r ′ | §156157§ d §166167§ r ′ {\displaystyle \mathbf {E} _{s}(\mathbf {r} ,t)\approx -{\frac {1}{4\pi }}\iiint _{V}\ {\frac {\left({\frac {\partial \mathbf {B} (\mathbf {r} ',t)}{\partial t}}\right)\times \left(\mathbf {r} -\mathbf {r} '\right)}{|\mathbf {r} -\mathbf {r} '|^{3}}}d^{3}\mathbf {r'} } This formulation demonstrates the spatial dependence of the induced electric field, illustrating how variations in the magnetic field contribute to its magnitude at a specific location, with each contribution inversely proportional to the cube of the distance.

Derivation of the Flux Rule from Microscopic Equations

Classical electromagnetism is fundamentally established by the four Maxwell's equations, complemented by the Lorentz force law. From this foundational set of principles, Faraday's law can be directly deduced.

The derivation commences by examining the temporal derivative of the magnetic flux across a surface Σ(t), which is permitted to change over time.

However, the term within the integral does not represent the complete Lorentz force per unit charge, as the velocity ${\displaystyle \mathbf {v} _{c}}$ pertains to the motion of the loop boundary rather than the actual velocity of the charge carriers. To accurately determine the physical electromotive force, it is essential to differentiate between these distinct velocities. By selecting an integration path that corresponds to the physical circuit, the velocity of a charge carrier within the conductor can be expressed as $${\displaystyle \mathbf {v} (\mathbf {r} ,t)=\mathbf {v} _{c}(\mathbf {r} ,t)+\mathbf {v} _{d}(\mathbf {r} ,t),}$$ where ${\displaystyle \mathbf {v} _{c}}$ denotes the velocity of the conductor (specifically, the ions within the material), and ${\displaystyle \mathbf {v} _{d}}$ represents the drift velocity of electrons relative to the material. This decomposition relies on the assumption of nonrelativistic (Galilean) velocity addition.

The electromotive force (EMF) ${\displaystyle {\mathcal {E}}}$, which is linked to the Lorentz force, is formally defined by the following integral expression: $${\displaystyle {\mathcal {E}}=\oint _{\partial \Sigma (t)}\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)\cdot \mathrm {d} \mathbf {l} .}$$

By substituting the derived expression for the carrier velocity and incorporating the preceding result, the following equation is obtained:

This phenomenon can be equivalently represented by the following mathematical expression:

For circuits composed of slender, one-dimensional conductors, the drift velocity vector aligns with the wire's axis, and consequently, with the infinitesimal line element ⁠${\displaystyle \mathrm {d} \mathbf {l} }$⁠. Under these conditions, the cross product ${\textstyle \mathbf {v} _{d}\times \mathbf {B} }$ becomes orthogonal to ⁠${\displaystyle \mathrm {d} \mathbf {l} }$⁠, causing the component proportional to the drift velocity to become zero. This simplification yields the conventional formulation of Faraday's law.

Limitations of the Flux Rule

A common but often erroneous generalization of Faraday's law posits that: If ∂Σ is any arbitrary closed loop in space whatsoever, then the total time derivative of magnetic flux through Σ equals the emf around ∂Σ. However, this assertion does not universally hold. As previously established, the applicability of Faraday's law is contingent upon the abstract curve's velocity, ∂Σ, aligning with the actual velocity of the electrically conductive material. Furthermore, if the conductor is not an infinitesimally thin wire, the relative velocity of charges within the material must also be considered. Subsequent examples demonstrate that an overly broad application of Faraday's law frequently yields inaccurate outcomes.

Such scenarios can be accurately analyzed by ensuring that the path ∂Σ maintains the same velocity as the material. The electromotive force (EMF) can consistently be determined through the integration of the Lorentz force law and the Maxwell–Faraday equation, expressed as: $${\displaystyle {\mathcal {E}}=\int _{\partial \Sigma }(\mathbf {E} +\mathbf {v} \times \mathbf {B} )\cdot \mathrm {d} \mathbf {l} =-\int _{\partial \Sigma (t)}{\frac {\partial \mathbf {B} }{\partial t}}\cdot {\rm {d}}\mathbf {A} +\oint _{\partial \Sigma (t)}\left(\mathbf {v} \times \mathbf {B} \right)\cdot \mathrm {d} \mathbf {l} ,}$$ Here, v represents the conductor's velocity within the reference frame where B is defined. Generally, the time derivative cannot be factored out of the integral because the loop's position or configuration may vary with time.

The Flux Rule and Relativity

Historically, the two distinct mechanisms subsumed by the flux rule—motional electromotive force (EMF) and transformer EMF—presented a significant conceptual challenge. James Clerk Maxwell had already identified that electromagnetic induction could originate from disparate physical processes, despite the induced EMF conforming to a singular mathematical formulation. In his 1861 publication, On Physical Lines of Force, Maxwell provided distinct physical elucidations for each phenomenon.

In 1905, Albert Einstein elucidated this asymmetry within classical electrodynamics in his seminal paper, On the Electrodynamics of Moving Bodies. He observed that the resultant physical phenomenon, specifically the induced current, is solely contingent upon the relative motion between the conductor and the magnet. However, classical theoretical frameworks offered divergent explanations based on which object was posited to be in motion. This inherent inconsistency implied the non-existence of a privileged reference frame and significantly contributed to the conceptualization of special relativity.

Contemporary physics conceptualizes electric and magnetic fields as integral components of a unified electromagnetic field tensor. When an inertial frame of reference undergoes transformation, these two fields interconvert.

Notes

Darrigol, Olivier (2000). Electrodynamics from Ampère to Einstein. Oxford; New York: Clarendon Press. ISBN 0-19-850594-9.

• Darrigol, Olivier (2000). Electrodynamics from Ampère to Einstein. Oxford; New York: Clarendon Press. ISBN 0-19-850594-9.Feynman, Richard Phillips; Leighton, Robert B.; Sands, Matthew L. (2006). The Feynman lectures on physics. Vol. 2. Pearson / Addison-Wesley. ISBN 0-8053-9047-2.Griffiths, David J. (2023). Introduction to Electrodynamics. Cambridge University Press. doi:10.1017/9781009397735. ISBN 978-1-009-39773-5.Purcell, Edward M.; Morin, David J. (2013). Electricity and Magnetism. Cambridge University Press. doi:10.1017/cbo9781139012973. ISBN 978-1-139-01297-3.Sadiku, Matthew N. O. (2018). Elements of electromagnetics (7th ed.). New York/Oxford: Oxford University Press. ISBN 978-0-19-069861-4. LCCN 2017046497.Zangwill, Andrew (2013). Modern Electrodynamics. Cambridge University Press. doi:10.1017/CBO9781139034777. ISBN 978-0-521-89697-9.Clerk Maxwell, James (1954) [1st pub. 1881]. A treatise on electricity and magnetism, Vol. II. Oxford: Clarendon Press. ch. III, sec. 530, p. 178. ISBN 0-486-60637-6. {{cite book}}: ISBN / Date incompatibility (help)
  • Clerk Maxwell, James (1954) [1st pub. 1881]. A treatise on electricity and magnetism, Vol. II. Oxford: Clarendon Press. ch. III, sec. 530, p. 178. ISBN 0-486-60637-6.{{cite book}}: ISBN / Date incompatibility (help)
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