Electromagnetic induction: An introduction

Introduction

Electromagnetic induction is the production of voltage across a conductor situated in a changing magnetic field or a conductor moving through a stationary magnetic field.

Michael Faraday is generally credited with the discovery of the induction phenomenon in 1831 though it may have been anticipated by the work of Francesco Zantedeschi

in 1829. Around 1830 to 1832 Joseph Henry made a similar discovery, but did not publish his findings until later

History

Faraday’s law was originally an experimental law based upon observations. Later it was formalized, and along with the other laws of electromagnetism a partial time derivative restricted version of it was incorporated into the modern Heaviside versions of Maxwell’s equations.

Faraday’s law of induction is based on Michael Faraday’s experiments in 1831. The effect was also discovered by Joseph Henry at about the same time, but Faraday published first.

Lenz’s law, formulated by Baltic German physicist Heinrich Lenz in 1834, gives the direction of the induced electromotive force and current resulting from electromagnetic induction[2]

Technical details

Faraday found that the electromotive force (EMF) produced around a closed path is proportional to the rate of change of the magnetic flux through any surface bounded by that path.

In practice, this means that an electrical current will be induced in any closed circuit when the magnetic flux through a surface bounded by the conductor changes. This applies whether the field itself changes in strength or the conductor is moved through it.

Electromagnetic induction underlies the operation of generators, all electric motors, transformers, induction motors, synchronous motors, solenoids, and most other electrical machines.

Faraday’s law of electromagnetic induction states that:

Thus:

is the electromotive force (emf) in volts

ΦB is the magnetic flux in webers

For the common but special case of a coil of wire, composed of N loops with the same area, Faraday’s law of electromagnetic induction states that

where

is the electromotive force (emf) in volts

N is the number of turns of wire

ΦB is the magnetic flux in webers through a single loop.

A corollary of Faraday’s Law, together with Ampere’s and Ohm’s laws is Lenz’s law:

The emf induced in an electric circuit always acts in such a direction that the current it drives around.

Consider the case in Figure 3 of a closed rectangular loop of wire in the xy-plane translated in the x-direction at velocity v. Thus, the center of the loop at xC satisfies v = dxC / dt. The loop has length ℓ in the y-direction and width w in the x-direction. A time-independent but spatially varying magnetic field B(x) points in the z-direction. The magnetic field on the left side is B( xC − w / 2), and on the right side is B( xC + w / 2). The electromotive force is to be found directly and by using Faraday’s law above.

Lorentz force law method

A charge q in the wire on the left side of the loop experiences a Lorentz force q v × B k = −q v B(xC − w / 2) j ( j, k unit vectors in the y- and z-directions; see vector cross product), leading to an EMF (work per unit charge) of v ℓ B(xC − w / 2) along the length of the left side of the loop. On the right side of the loop the same argument shows the EMF to be v ℓ B(xC + w / 2). The two EMF’s oppose each other, both pushing positive charge toward the bottom of the loop. In the case where the B-field increases with increase in x, the force on the right side is largest, and the current will be clockwise: using the right-hand rule, the B-field generated by the current opposes the impressed fieldThe EMF driving the current must increase as we move counterclockwise (opposite to the current). Adding the EMF’s in a counterclockwise tour of the loop we find

Faraday’s law method

At any position of the loop the magnetic flux through the loop is

The sign choice is decided by whether the normal to the surface points in the same direction as B, or in the opposite direction. If we take the normal to the surface as pointing in the same direction as the B-field of the induced current, this sign is negative. The time derivative of the flux is then (using the chain rule of differentiation or the general form of Leibniz rule for differentiation of an integral):

(where v = dxC / dt is the rate of motion of the loop in the x-direction ) leading to:

as before.

The equivalence of these two approaches is general and, depending on the example, one or the other method may prove more practical.

Moving loop in uniform B-field

Figure 4: Rectangular wire loop rotating at angular velocity ω in radially outward pointing magnetic field B of fixed magnitude. Current is collected by brushes attached to top and bottom discs, which have conducting rims.

Lorentz force law method

In this case the Lorentz force drives the current in the two vertical arms of the moving loop downward, so current flows from the top disc to the bottom disc. In the conducting rims of the discs, the Lorentz force is perpendicular to the rim, so no EMF is generated in the rims, nor in the horizontal portions of the moving loop. Current is transmitted from the bottom rim to the top rim through the external return loop, which is oriented so the B-field is in its plane. Thus, the Lorentz force in the return loop is perpendicular to the loop, and no EMF is generated in this return loop. Traversing the current path in the direction opposite to the current flow, work is done against the Lorentz force only in the vertical arms of the moving loop, where

Consequently the EMF is

where ℓ is the vertical length of the loop, and the velocity is related to the angular rate of rotation by v = r ω, with r = radius of cylinder. Notice that the same work is done on any path that rotates with the loop and connects the upper and lower rim.

Faraday’s law method

An intuitively appealing but mistaken approach to using the flux rule would say the flux through the circuit was just ΦB = B w ℓ, where w = width of the moving loop. This number is time-independent, so the approach predicts incorrectly that no EMF is generated. The flaw in this argument is that it fails to consider the entire current path, which is a closed loop.

To use the flux rule, we have to look at the entire current path, which includes the path through the rims in the top and bottom discs. We can choose an arbitrary closed path through the rims and the rotating loop, and the flux law will find the EMF around the chosen path. Any path that has a segment attached to the rotating loop captures the relative motion of the parts of the circuit.

As an example path, let’s traverse the circuit in the direction of rotation in the top disc, and in the direction opposite to the direction of rotation in the bottom disc (shown by arrows in Figure 4). In this case, for the moving loop at an angle θ from the collecting loop, a portion of the cylinder of area A = r ℓ θ is part of the circuit. This area is perpendicular to the B-field, and so contributes to the flux an amount:

where the sign is negative because the right-hand rule suggests the B-field generated by the current loop is opposite in direction to the applied B field. As this is the only time-dependent portion of the flux, the flux law predicts an EMF of

in agreement with the Lorentz force law calculation.

Now let’s try a different path. Follow a path traversing the rims via the opposite choice of segments. Then the coupled flux would decrease as θ increased, but the right-hand rule would suggest the current loop added to the applied B-field, so the EMF around this path is the same as for the first path. Any mixture of return paths leads to the same result for EMF, so it is actually immaterial which path is followed.

The use of a closed path to find EMF as done above appears to depend upon details of the path geometry. In contrast, the Lorentz-law approach is independent of such restrictions. A discussion follows intended to understand better the equivalence of paths and escape the particulars of path selection when using the flux law.

Figure 5 is an idealization of with the cylinder unwrapped onto a plane. The same path-related analysis works, but a simplification is suggested. The time-independent aspects of the circuit cannot affect the time-rate-of-change of flux. For example, at a constant velocity of sliding the loop, the details of current flow through the loop are not time dependent. Instead of concern over details of the closed loop selected to find the EMF, one can focus on the area of B-field swept out by the moving loop. This suggestion amounts to finding the rate at which flux is cut by the circuit. That notion provides direct evaluation of the rate of change of flux, without concern over the time-independent details of various path choices around the circuit. Just as with the Lorentz law approach, it is clear that any two paths attached to the sliding loop, but differing in how they cross the loop, produce the same rate-of-change of flux.

In Figure 5 the area swept out in unit time is simply dA / dt = v ℓ, regardless of the details of the selected closed path, so Faraday’s law of induction provides the EMF as:

This path independence of EMF shows that if the sliding loop is replaced by a solid conducting plate, or even some complex warped surface, the analysis is the same: find the flux in the area swept out by the moving portion of the circuit. In a similar way, if the sliding loop in the drum generator of Figure 4 is replaced by a 360° solid conducting cylinder, the swept area calculation is exactly the same as for the case with only a loop. That is, the EMF predicted by Faraday’s law is exactly the same for the case with a cylinder with solid conducting walls or, for that matter, a cylinder with a cheese grater for walls. Notice, though, that the current that flows as a result of this EMF will not be the same because the resistance of the circuit determines the current[3]

Applications

The principles of electromagnetic induction are applied in many devices and systems, including:

Induction Sealing
Induction motors
Electrical generators
Transformers
Contactless charging of rechargeable batteries
The 6.6kW Magne Charge system for Battery electric vehicles
Induction cookers
Induction welding
Inductors
Electromagnetic forming
Magnetic flow meters
Transcranial magnetic stimulation
Faraday Flashlight
Graphics tablet
Wireless energy transfer
Electric Guitar Pickups
Hall effect meters
Current transformer meters
Clamp meter
Audio and video tapes

the circuit opposes the change in magnetic flux which produces the emf.

The direction mentioned in Lenz’s law can be thought of as the result of the minus sign in the above equation

Eddy currents

An eddy current is a swirling current set up in a conductor in response to a changing magnetic field. By Lenz¹s law, the current swirls in such a way as to create a magnetic field opposing the change; to do this in a conductor, electrons swirl in a plane perpendicular to the magnetic field.

Because of the tendency of eddy currents to oppose, eddy currents cause energy to be lost. More accurately, eddy currents transform more useful forms of energy, such as kinetic energy, into heat, which is generally much less useful. In many applications the loss of useful energy is not particularly desirable, but there are some practical applications. One is in the brakes of some trains. During braking, the metal wheels are exposed to a magnetic field from an electromagnet, generating eddy currents in the wheels. The magnetic interaction between the applied field and the eddy currents acts to slow the wheels down. The faster the wheels are spinning, the stronger the effect, meaning that as the train slows the braking force is reduced, producing a smooth stopping motion.

Electrical generator

Faraday’s disc electric generator. The disc rotates with angular rate ω, sweeping the conducting radius circularly in the static magnetic field B. The magnetic Lorentz force v × B drives the current along the conducting radius to the conducting rim, and from there the circuit completes through the lower brush and the axle supporting the disc. Thus, current is generated from mechanical motion.

The EMF generated by Faraday’s law of induction due to relative movement of a circuit and a magnetic field is the phenomenon underlying electrical generators. When a permanent magnet is moved relative to a conductor, or vice versa, an electromotive force is created. If the wire is connected through an electrical load, current will flow, and thus electrical energy is generated, converting the mechanical energy of motion to electrical energy. For example, the drum generator is based upon . A different implementation of this idea is the Faraday’s disc, shown in simplified form in Figure 8. Note that either the analysis of Figure 5, or direct application of the Lorentz force law, shows that a solid conducting disc works the same way.

In the Faraday’s disc example, the disc is rotated in a uniform magnetic field perpendicular to the disc, causing a current to flow in the radial arm due to the Lorentz force. It is interesting to understand how it arises that mechanical work is necessary to drive this current. When the generated current flows through the conducting rim, a magnetic field is generated by this current through Ampere’s circuital law (labeled “induced B” in Figure 8). The rim thus becomes an electromagnet that resists rotation of the disc (an example of Lenz’s law). On the far side of the figure, the return current flows from the rotating arm through the far side of the rim to the bottom brush. The B-field induced by this return current opposes the applied B-field, tending to decrease the flux through that side of the circuit, opposing the increase in flux due to rotation. On the near side of the figure, the return current flows from the rotating arm through the near side of the rim to the bottom brush. The induced B-field increases the flux on this side of the circuit, opposing the decrease in flux due to rotation. Thus, both sides of the circuit generate an emf opposing the rotation. The energy required to keep the disc moving, despite this reactive force, is exactly equal to the electrical energy generated (plus energy wasted due to friction, Joule heating, and other inefficiencies). This behavior is common to all generators converting mechanical energy to electrical energy.

Although Faraday’s law always describes the working of electrical generators, the detailed mechanism can differ in different cases. When the magnet is rotated around a stationary conductor, the changing magnetic field creates an electric field, as described by the Maxwell-Faraday equation, and that electric field pushes the charges through the wire. This case is called an induced EMF. On the other hand, when the magnet is stationary and the conductor is rotated, the moving charges experience a magnetic force (as described by the Lorentz force law), and this magnetic force pushes the charges through the wire. This case is called motional EMF. (For more information on motional EMF, induced EMF, Faraday’s law, and the Lorentz force.

Electrical motor

An electrical generator can be run “backwards” to become a motor. For example, with the Faraday disc, suppose a DC current is driven through the conducting radial arm by a voltage. Then by the Lorentz force law, this traveling charge experiences a force in the magnetic field B that will turn the disc in a direction given by Fleming’s left hand rule. In the absence of irreversible effects, like friction or Joule heating, the disc turns at the rate necessary to make d ΦB / dt equal to the voltage driving the current.

Electrical transformer

The EMF predicted by Faraday’s law is also responsible for electrical transformers. When the electric current in a loop of wire changes, the changing current creates a changing magnetic field. A second wire in reach of this magnetic field will experience this change in magnetic field as a change in its coupled magnetic flux, a d ΦB / d t. Therefore, an electromotive force is set up in the second loop called the induced EMF or transformer EMF. If the two ends of this loop are connected through an electrical load, current will flow.

Magnetic flow meter

Faraday’s law is used for measuring the flow of electrically conductive liquids and slurries. Such instruments are called magnetic flow meters. The induced voltage ε generated in the magnetic field B due to a conductive liquid moving at velocity v is thus given by:

where ℓ is the distance between electrodes in the magnetic flow meter.

Parasitic induction and waste heating

All metal objects moving in relation to a static magnetic field will experience inductive power flow, as do all stationary metal objects in relation to a moving magnetic field. These power flows are occasionally undesirable, resulting in flowing electric current at very low voltage and heating of the metal.

There are a number of methods employed to control these undesirable inductive effects.

Electromagnets in electric motors, generators, and transformers do not use solid metal, but instead use thin sheets of metal plate, called laminations. These thin plates reduce the parasitic eddy currents, as described below.

Inductive coils in electronics typically use magnetic cores to minimize parasitic current flow. They are a mixture of metal powder plus a resin binder that can hold any shape. The binder prevents parasitic current flow through the powdered metal.

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