Faraday's law of induction


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.
Faraday's law is used in the literature to refer to two closely related but physically distinct statements. One is the Maxwell–Faraday equation, one of Maxwell's equations, which states that a time-varying magnetic field is always accompanied by a circulating electric field. This law applies to the fields themselves and does not require the presence of a physical circuit.
The other is Faraday's flux rule, or the Faraday–Lenz law, which relates the electromotive force around a closed conducting loop to the time rate of change of magnetic flux through the loop. The flux rule accounts for two mechanisms by which an emf can be generated. In transformer emf, a time-varying magnetic field induces an electric field as described by the Maxwell–Faraday equation, and the electric field drives a current around the loop. In motional emf, the circuit moves through a magnetic field, and the emf arises from the magnetic component of the Lorentz force acting on the charges in the conductor.
Historically, the differing explanations for motional emf and transformer emf posed a conceptual problem, since the observed current depends only on relative motion, but the physical explanations were different in the two cases. In special relativity, this distinction is understood as frame-dependent: what appears as a magnetic force in one frame may appear as an induced electric field in another.

History

In 1820, Hans Christian Ørsted demonstrated that an electric current produces a magnetic field, showing that a compass needle could be deflected by a nearby current-carrying wire. This discovery prompted scientists to ask whether the reverse was also true—whether a magnetic field could generate an electric current.
Initial experiments revealed that a static magnetic field had no effect on a nearby circuit: simply placing a magnet near a wire loop produced no current. The breakthrough came in 1831, when Michael Faraday demonstrated that a changing magnetic field could indeed induce an electric current in a circuit. Independently, Joseph Henry made similar observations in 1832, though Faraday was the first to publish his findings.
Faraday's notebook on August 29, 1831 describes an experimental demonstration of induction. He wrapped two coils of wire around opposite sides of an iron ring, forming a primitive toroidal transformer. When he connected one coil to a battery, he observed a brief deflection in a galvanometer attached to the second coil. He concluded that a changing current in the first coil created a changing magnetic field in the ring, which in turn induced a current in the second coil. He described this as a "wave of electricity" propagated through the iron.
Over the following months, Faraday discovered other manifestations of electromagnetic induction. He observed transient currents when a bar magnet was rapidly moved into or out of a coil of wire. He also built a device, now known as Faraday's disk or homopolar generator, that produced a steady current by rotating a copper disk in the presence of a stationary magnet, using a sliding electrical contact.
File:Faraday disk generator.jpg|thumb|Faraday's disk, the first electric generator, a type of homopolar generator|left
Faraday explained these phenomena using the concept of lines of force. However, his theoretical ideas were met with skepticism, as they were not formulated mathematically. James Clerk Maxwell later gave Faraday's insights mathematical expression, incorporating them into his broader electromagnetic theory in the early 1860s.
In Maxwell's papers, the time-varying aspect of electromagnetic induction is expressed as a differential equation which Oliver Heaviside referred to as Faraday's law even though it is different from the original version of Faraday's law, and does not describe motional emf. Heaviside's version is the form recognized today in the group of equations known as Maxwell's equations.
Lenz's law, formulated by Emil Lenz in 1834, describes "flux through the circuit", and gives the direction of the induced emf and current resulting from electromagnetic induction.
The laws of induction of electric currents in mathematical form were established by Franz Ernst Neumann in 1845.
According to Albert Einstein, much of the groundwork and discovery of his special relativity theory was presented by this law of induction by Faraday in 1834.

Flux rule

Faraday's law of induction, also known as the flux rule, flux law, and FaradayLenz law, states that the electromotive force around a closed circuit is equal to the negative rate of change of the magnetic flux through the circuit. This rule holds for any circuit made of thin wire and accounts for changes in flux due to variations in the magnetic field, movement of the circuit, or deformation of its shape. The direction of the induced emf is given by Lenz's law, which states that the induced current will flow in such a way that its magnetic field opposes the change in the original magnetic flux.
Mathematically, in SI units, the law is expressed as
where is the electromotive force and is the magnetic flux through the circuit. The magnetic flux is defined as the surface integral of the magnetic field over a time-dependent surface, whose boundary is the wire loop:
where is an infinitesimal area vector normal to the surface. The dot product represents the flux through the differential area element.
In more visual terms, the magnetic flux is proportional to the number of magnetic field lines passing through the loop.
When the flux changes, an emf is induced around the loop. This emf corresponds to the energy per unit charge required to move it once around the loop. In a simple circuit with resistance, an emf gives rise to a current according to the Ohm's law. Equivalently, if the loop is broken to form an open circuit and a voltmeter is connected across the terminals, the emf is equal to the voltage measured across the open ends.
For a tightly wound coil of wire, composed of identical turns, the same magnetic field lines cross the surface times. In this case, Faraday's law of induction states that
where is the number of turns of wire and is the magnetic flux through a single loop. The product is known as linked flux.
The flux can change either because the loop moves or deforms over time, or because the field itself varies in time. These two possibilities correspond to the two mechanisms described by the flux rule:
  • Motional emf: The circuit moves through a static but non-uniform magnetic field.
  • Transformer emf: The circuit remains stationary while the magnetic field changes over time.

    Motional emf

The basic mechanism behind motional emf is illustrated by a conducting rod moving through a magnetic field that is perpendicular to both the rod and its direction of motion. Due to movement in magnetic field, the mobile electrons of the conductor experience the magnetic component of the Lorentz force that drives them along the length of the rod. This leads to a separation of charge between the two ends of the rod. In the steady state, the electric field from the accumulated charge balances the magnetic force.
If the rod is part of a closed conducting loop moving through a nonuniform magnetic field, the same effect can drive a current around the circuit. For instance, suppose the magnetic field is confined to a limited region of space, and the loop initially lies outside this region. As it moves into the field, the area of the loop that encloses magnetic flux increases, and an emf is induced. From the Lorentz force perspective, this is because the field exerts a magnetic force on charge carriers in the parts of the loop entering the region. Once the entire loop lies in a uniform magnetic field and continues at constant speed, the total enclosed flux remains constant, and the emf vanishes. In this situation, magnetic forces on opposite sides of the loop cancel out.

Transformer emf

A complementary case is transformer emf, which occurs when the conducting loop remains stationary but the magnetic flux through it changes due to a time-varying magnetic field. This can happen in two ways: either the source of the magnetic field moves, altering the field distribution through the fixed loop, or the strength of the magnetic field changes over time at a fixed location, as in the case of a powered electromagnet.
In either situation, no magnetic force acts on the charges, and the emf is entirely due to the electric component of the Lorentz force. According to the Maxwell–Faraday equation, a time-varying magnetic field produces a circulating electric field, which drives current in the loop. This phenomenon underlies the operation of electrical machines such as synchronous generators. The electric field induced in this way is non-conservative, meaning its line integral around a closed loop is not zero.

Direction of the induced current

It is possible to find out the direction of the electromotive force directly from Faraday's law, without invoking Lenz's law. A left hand rule helps doing that, as follows:
  • Align the curved fingers of the left hand with the loop.
  • Stretch your thumb. The stretched thumb indicates the direction of , the normal to the area enclosed by the loop.
  • Find the sign of, the change in flux. Determine the initial and final fluxes with respect to the normal, as indicated by the stretched thumb.
  • If the change in flux,, is positive, the curved fingers show the direction of the electromotive force.
  • If is negative, the direction of the electromotive force is opposite to the direction of the curved fingers.