faraday's experiment on electromagnetic induction

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• Coulomb’s law
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• Faraday’s law of induction
• Self-inductance and mutual inductance
• Effects of varying electric fields
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• Formulation of the quantitative laws of electrostatics and magnetostatics
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Faraday’s discovery of electric induction

Maxwell’s unified theory of electromagnetism, discovery of the electron and its ramifications.

• Special theory of relativity
• Development of electromagnetic technology

• What was Carl Friedrich Gauss’s childhood like?
• What awards did Carl Friedrich Gauss win?
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• How did James Clerk Maxwell die?
• What was Michael Faraday’s childhood like?

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• Table Of Contents

Faraday, the greatest experimentalist in electricity and magnetism of the 19th century and one of the greatest experimental physicists of all time, worked on and off for 10 years trying to prove that a magnet could induce electricity. In 1831 he finally succeeded by using two coils of wire wound around opposite sides of a ring of soft iron ( Figure 7 ). The first coil was attached to a battery; when a current passed through the coil, the iron ring became magnetized. A wire from the second coil was extended to a compass needle a metre away, far enough so that it was not affected directly by any current in the first circuit . When the first circuit was turned on, Faraday observed a momentary deflection of the compass needle and its immediate return to its original position. When the primary current was switched off, a similar deflection of the compass needle occurred but in the opposite direction. Building on this observation in other experiments, Faraday showed that changes in the magnetic field around the first coil are responsible for inducing the current in the second coil. He also demonstrated that an electric current can be induced by moving a magnet, by turning an electromagnet on and off, and even by moving an electric wire in Earth’s magnetic field . Within a few months, Faraday built the first, albeit primitive, electric generator .

Henry had discovered electric induction quite independently in 1830, but his results were not published until after he had received news of Faraday’s 1831 work , nor did he develop the discovery as fully as Faraday. In his paper of July 1832, Henry reported and correctly interpreted self-induction . He had produced large electric arcs from a long helical conductor when it was disconnected from a battery. When he had opened the circuit, the rapid decrease in the current had caused a large voltage between the battery terminal and the wire. As the wire lead was pulled away from the battery, the current continued to flow for a short time in the form of a bright arc between the battery terminal and the wire.

Faraday’s thinking was permeated by the concept of electric and magnetic lines of force . He visualized that magnets, electric charges, and electric currents produce lines of force. When he placed a thin card covered with iron filings on a magnet, he could see the filings form chains from one end of the magnet to the other. He believed that these lines showed the directions of the forces and that electric current would have the same lines of force. The tension they build explains the attraction and repulsion of magnets and electric charges. Faraday had visualized magnetic curves as early as 1831 while working on his induction experiments; he wrote in his notes, “By magnetic curves I mean lines of magnetic forces which would be depicted by iron filings.” Faraday opposed the prevailing idea that induction occurred “at a distance”; instead, he held that induction occurs along curved lines of force because of the action of contiguous particles. Later he explained that electricity and magnetism are transmitted through a medium that is the site of electric or magnetic “fields,” which make all substances magnetic to some extent.

Faraday was not the only researcher laying the groundwork for a synthesis between electricity, magnetism, and other areas of physics . On the continent of Europe , primarily in Germany , scientists were making mathematical connections between electricity, magnetism, and optics . The work of the physicists Franz Ernst Neumann , Wilhelm Eduard Weber , and H.F.E. Lenz belongs to this period. At the same time, Helmholtz and the English physicists William Thomson (later Lord Kelvin) and James Prescott Joule were clarifying the relationship between electricity and other forms of energy . Joule investigated the quantitative relationship between electric currents and heat during the 1840s and formulated the theory of the heating effects that accompany the flow of electricity in conductors. Helmholtz, Thomson, Henry, Gustav Kirchhoff , and Sir George Gabriel Stokes also extended the theory of the conduction and propagation of electric effects in conductors. In 1856 Weber and his German colleague, Rudolf Kohlrausch , determined the ratio of electric and magnetic units and found that it has the same dimensions as light and that it is almost exactly equal to its velocity . In 1857 Kirchhoff used this finding to demonstrate that electric disturbances propagate on a highly conductive wire with the speed of light .

The final steps in synthesizing electricity and magnetism into one coherent theory were made by Maxwell. He was deeply influenced by Faraday’s work, having begun his study of the phenomena by translating Faraday’s experimental findings into mathematics. (Faraday was self-taught and had never mastered mathematics .) In 1856 Maxwell developed the theory that the energy of the electromagnetic field is in the space around the conductors as well as in the conductors themselves. By 1864 he had formulated his own electromagnetic theory of light, predicting that both light and radio waves are electric and magnetic phenomena. While Faraday had discovered that changes in magnetic fields produce electric fields , Maxwell added the converse: changes in electric fields produce magnetic fields even in the absence of electric currents. Maxwell predicted that electromagnetic disturbances traveling through empty space have electric and magnetic fields at right angles to each other and that both fields are perpendicular to the direction of the wave . He concluded that the waves move at a uniform speed equal to the speed of light and that light is one form of electromagnetic wave . Their elegance notwithstanding, Maxwell’s radical ideas were accepted by few outside England until 1886, when the German physicist Heinrich Hertz verified the existence of electromagnetic waves traveling at the speed of light; the waves he discovered are known now as radio waves .

Maxwell’s four field equations represent the pinnacle of classical electromagnetic theory. Subsequent developments in the theory have been concerned either with the relationship between electromagnetism and the atomic structure of matter or with the practical and theoretical consequences of Maxwell’s equations . His formulation has withstood the revolutions of relativity and quantum mechanics . His equations are appropriate for distances as small as 10 −10 centimetres —100 times smaller than the size of an atom . The fusion of electromagnetic theory and quantum theory, known as quantum electrodynamics , is required only for smaller distances.

While the mainstream of theoretical activity concerning electric and magnetic phenomena during the 19th century was devoted to showing how they are interrelated, some scientists made use of them to discover new properties of materials and heat. Weber developed Ampère’s suggestion that there are internal circulating currents of molecular size in metals. He explained how a substance loses its magnetic properties when the molecular magnets point in random directions. Under the action of an external force, they may turn to point in the direction of the force; when all point in this direction, the maximum possible degree of magnetization is reached, a phenomenon known as magnetic saturation . In 1895 Pierre Curie of France discovered that a ferromagnetic substance has a specific temperature above which it ceases to be magnetic. Finally, superconductivity was discovered in 1900 by the German physicist Heike Kammerlingh-Onnes. In superconductivity , electric conductors lose all resistance at very low temperatures .

Although little of major importance was added to electromagnetic theory in the 19th century after Maxwell, the discovery of the electron in 1898 opened up an entirely new area of study: the nature of electric charge and of matter itself. The discovery of the electron grew out of studies of electric currents in vacuum tubes . Heinrich Geissler , a glassblower who assisted the German physicist Julius Plücker , improved the vacuum tube in 1854. Four years later, Plücker sealed two electrodes inside the tube, evacuated the air, and forced electric currents between the electrodes; he attributed the green glow that appeared on the wall of the tube to rays emanating from the cathode . From then until the end of the century, the properties of cathode-ray discharges were studied intensively. The work of the English physicist Sir William Crookes in 1879 indicated that the luminescence was a property of the electric current itself. Crookes concluded that the rays were composed of electrified charged particles. In 1898 another English physicist, Sir J.J. Thomson , identified a cathode ray as a stream of negatively charged particles, each having a mass 1 / 1836 smaller than that of a hydrogen ion . Thomson’s discovery established the particulate nature of charge; his particles were later dubbed electrons .

Following the discovery of the electron, electromagnetic theory became an integral part of the theories of the atomic, subatomic, and subnuclear structure of matter. This shift in focus occurred as the result of an impasse between electromagnetic theory and statistical mechanics over attempts to understand radiation from hot bodies. Thermal radiation had been investigated in Germany by the physicist Wilhelm Wien between 1890 and 1900. Wien had virtually exhausted the resources of thermodynamics in dealing with this problem. Two British scientists, Lord Rayleigh (John William Strutt) and Sir James Hopwood Jeans , had by 1900 applied the newly developed science of statistical mechanics to the same problem. They obtained results that, though in agreement with Wien’s thermodynamic conclusions (as distinct from his speculative extensions of thermodynamics ), only partially agreed with experimental observations. The German physicist Max Planck attempted to combine the statistical approach with a thermodynamic approach. By concentrating on the necessity of fitting together the experimental data, he was led to the formulation of an empirical law that satisfied Wien’s thermodynamic criteria and accommodated the experimental data. When Planck interpreted this law in terms of Rayleigh’s statistical concepts, he concluded that radiation of frequency ν exists only in quanta of energy. Planck’s result, including the introduction of the new universal constant h in 1900, marked the foundation of quantum mechanics and initiated a profound change in physical theory ( see atom: Bohr’s shell model ).

By 1900 it was apparent that Thomson’s electrons were a universal constituent of matter and, thus, that matter is essentially electric in nature. As a result, in the early years of the 20th century, many physicists attempted to construct theories of the electromagnetic properties of metals , insulators , and magnetic materials in terms of electrons. In 1909 the Dutch physicist Hendrik Antoon Lorentz succeeded in doing so in The Theory of Electrons and Its Applications to the Phenomena of Light and Radiant Heat ; his work has since been modified by quantum theory.

Faraday’s Laws of Electromagnetic Induction: First & Second Law

What is Faraday’s Law

Faraday’s experiment.

In this experiment, Faraday takes a magnet and a coil and connects a galvanometer across the coil. At starting, the magnet is at rest, so there is no deflection in the galvanometer i.e the needle of the galvanometer is at the center or zero position. When the magnet is moved towards the coil, the needle of the galvanometer deflects in one direction.

Faraday’s First Law

Faraday’s second law, faraday law formula.

How To Increase EMF Induced in a Coil

Applications of faraday’s law.

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• Electromagnetism
• Faradays Law

Faraday’s Laws of Electromagnetic Induction

Faraday’s law of electromagnetic induction, also known as Faraday’s law, is the basic law of electromagnetism which helps us predict how a magnetic field would interact with an electric circuit to produce an electromotive force (EMF). This phenomenon is known as electromagnetic induction.

Michael Faraday proposed the laws of electromagnetic induction in the year 1831. Faraday’s law or the law of electromagnetic induction is the observation or results of the experiments conducted by Faraday. He performed three main experiments to discover the phenomenon of electromagnetic induction.

Faraday’s Laws of Electromagnetic Induction

Faraday’s Laws of Electromagnetic Induction consists of two laws. The first law describes the induction of emf in a conductor and the second law quantifies the emf produced in the conductor. In the next few sections, let us learn these laws in detail.

Faraday’s First Law of Electromagnetic Induction

The discovery and understanding of electromagnetic induction are based on a long series of experiments carried out by Faraday and Henry. From the experimental observations, Faraday concluded that an emf is induced when the magnetic flux across the coil changes with time. Therefore, Faraday’s first law of electromagnetic induction states the following:

Whenever a conductor is placed in a varying magnetic field, an electromotive force is induced. If the conductor circuit is closed, a current is induced, which is called induced current.

Changing the Magnetic Field Intensity in a Closed Loop

Magnetic field intensity in a closed loop

Mentioned here are a few ways to change the magnetic field intensity in a closed loop:

• By rotating the coil relative to the magnet.
• By moving the coil into or out of the magnetic field.
• By changing the area of a coil placed in the magnetic field.
• By moving a magnet towards or away from the coil.

Faraday’s Second Law of Electromagnetic Induction

Faraday’s second law of electromagnetic induction states that

The induced emf in a coil is equal to the rate of change of flux linkage.

The flux linkage is the product of the number of turns in the coil and the flux associated with the coil. The formula of Faraday’s law is given below:

Where ε is the electromotive force, Φ is the magnetic flux, and N is the number of turns.

Learn more about Faraday’s Law of induction and the relationship between the electric circuit and magnetic field by watching this engaging video from BYJU’S.

The German physicist Heinrich Friedrich Lenz deduced a rule known as Lenz’s law that describes the polarity of the induced emf.

Lenz’s law states that “The polarity of induced emf is such that it tends to produce a current which opposes the change in magnetic flux that produced it.”

The negative sign in the formula represents this effect. Thus, the negative sign indicates that the direction of the induced emf and the change in the direction of magnetic fields have opposite signs.

Read more: Lenz’s law

Faraday’s Law Derivation

Consider a magnet approaching a coil. Consider two-time instances T 1 and T 2 .

Flux linkage with the coil at the time T 1 is given by NΦ 1 .

Flux linkage with the coil at the time T 2 is given by NΦ 2

Change in the flux linkage is given by

N(Φ 2 – Φ 1 )

Let us consider this change in flux linkage as

Φ = Φ 2 – Φ 1

Hence, the change in flux linkage is given by

The rate of change of flux linkage is given by

Taking the derivative of the above equation, we get

According to Faraday’s second law of electromagnetic induction, we know that the induced emf in a coil is equal to the rate of change of flux linkage. Therefore,

Considering Lenz’s law,

From the above equation, we can conclude the following

• Increase in the number of turns in the coil increases the induced emf
• Increasing the magnetic field strength increases the induced emf
• Increasing the speed of the relative motion between the coil and the magnet, results in the increased emf

Faraday’s Experiment: Relationship Between Induced EMF and Flux

• In the first experiment, he proved that when the strength of the magnetic field is varied, only then current is induced. An ammeter was connected to a loop of wire; the ammeter deflected when a magnet was moved towards the wire.

In the second experiment, he proved that passing a current through an iron rod would make it electromagnetic. He observed that when a relative motion exists between the magnet and the coil, an electromotive force will be induced. When the magnet was held stationary about its axis, no electromotive force was observed, but when the magnet was rotated about its own axis then the induced electromotive force was produced. Thus, there was no deflection in the ammeter when the magnet was held stationary.

While conducting the third experiment, he recorded that the galvanometer did not show any deflection and no induced current was produced in the coil when the coil was kept away in a stationary magnetic field. The ammeter deflected in the opposite direction when the magnet was kept away from the loop.

Summarising the above points in a table, we have mapped out the relationship between the position of the magnet and the deflection in the Galvanometer.

Conclusion:

After conducting all the experiments, Faraday finally concluded that if relative motion existed between a conductor and a magnetic field, the flux linkage with a coil changed and this change in flux produced a voltage across a coil.

Faraday law basically states, “when the magnetic flux or the magnetic field changes with time, the electromotive force is produced”. Additionally, Michael Faraday also formulated two laws on the basis of the above experiments.

The below videos help to revise the chapter Magnetic Effects of Electric Current Class 10

Applications of Faraday’s Law

Following are the fields where Faraday’s law finds applications:

• Electrical equipment like transformers works on the basis of Faraday’s law.
• Induction cooker works on the basis of mutual induction, which is based on the principle of Faraday’s law.
• By inducing an electromotive force into an electromagnetic flowmeter, the velocity of the fluids is recorded.
• Electric guitar and electric violin are musical instruments that find an application of Faraday’s law.
• Maxwell’s equation is based on the converse of Faraday’s laws which states that a change in the magnetic field brings a change in the electric field.

Frequently Asked Questions – FAQs

What does faraday’s first law of electromagnetic induction state.

Faraday’s first law of electromagnetic induction states, “Whenever a conductor is placed in a varying magnetic field, an electromotive force is induced. Likewise, if the conductor circuit is closed, a current is induced, which is called induced current.”

What does Faraday’s Second Law of Electromagnetic Induction state?

Why are faraday’s laws important, what does the negative sign indicate in faraday’s law of electromagnetic induction formula, what is meant by emf.

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• Faraday Electromagnetic Induction Experiment

What is the Electromagnetic Induction Experiment?

Michael Faraday was an English physicist and chemist who lived from September 22, 1791, in Newington, Surrey, England, until August 25, 1867, at Hampton Court, Surrey. Many of his experiments have had a profound effect on electromagnetic knowledge.

Faraday began his career as a pharmacist before becoming one of the leading scientists of the nineteenth century. He discovered a biological novel combination, including benzene, and became the first to 'immerse in gas' permanently. He also published a workbook on practical chemical science showing his strengths in the technical aspects of his business. He invented the first electric motor and dynamo, demonstrated the link between electricity and chemical bonding, identified the effect of magnetism on light, and named diamagnetism, the distinctive behaviour of other things in strong magnetic fields. He laid the experimental and some theoretical groundwork for James Clerk Maxwell's construction of classical electromagnetic field theory.

Electromagnetic Induction

Michael Faraday was the first to discover electromagnetic induction in the 1830s. When Faraday removed a permanent magnet from a coil or single telephone loop, he discovered that ElectroMotive Force or emf, or voltage, had been created, so a stream was generated.

The Galvanometer needle, which is actually the most sensitive center ammeter of a zero-moving coil, will move from its center to one side only if the magnet shown below is pushed "towards" the coil. Because there is no real movement of the magnetic field when the magnet stops moving and is kept upright toward the coil, the galvanometer needle returns to zero.

If the magnet shown below is pulled "towards" the coil, point or needle of the Galvanometer, which is simply the most sensitive center of the zero-moving moving ammeter, it will deviate from its center in only one direction.

The galvanometer needle returns to zero as there is no real movement of the magnetic field when the magnet stops rotating and is kept upright relative to the coil.

The galvanometer needle will also deviate in any direction if the magnet is now held in place and only the coil is moved in or out of the magnet. Moving a coil or wire loop in a magnetic field produces a voltage in the coil, its magnitude relative to the speed or speed of movement. To be sure, Faraday's law requires "related movement" or movement between the coil and the magnetic field, whether magnetic, coil, or both.

Michael Faraday's basic law of electromagnetic induction states that there is a link between electrical energy and a flexible magnetic field. In other words, Electromagnetic Induction is a method of generating electricity and still using magnetic fields in a closed circuit.

So, with magnetism alone, how much voltage (emf) can the coil produce? This is governed by the three conditions listed below.

Increasing the number of coils in the coil - By increasing the number of single conductors across the magnetic field, the amount of emf produced will be the sum of all the coils of the coil, so if the coil is 20 curves, the total number of emf produced will be 20 times more than one wire.

Increase the relative movement between the coil and the magnet - If the same telephone coil moves in the same magnetic field, but the speed or speed is increased, the wire would cut through the flow lines at a faster speed, resulting in more. idud emf.

Increasing the magnetic field - When the same telephone coil is moved at the same speed as a large magnetic field, more emf is produced as more power lines must be cut.

A small endless magnet is rotated by the movement of a bicycle wheel inside a coil that does not turn on small generators like a bicycle dynamo. The electromagnetic voltage provided by the fixed DC voltage can also be made to rotate inside a constant coil, as in large generators generating alternating power in both cases.

The permanent magnet surrounds the middle shaft in a simple dynamo-type generator, and a telephone coil is placed near the rotating magnetic field. The magnetic field surrounding the top and bottom of the coil constantly shifts between the north and south poles as the magnet rotates. According to Faraday's law of electromagnetic induction, this rotating motion of the magnetic field causes an alternating emf in the coil.

Faraday's law states that generating voltage in a conductor can be achieved by transmitting it to the magnetic field or by transmitting the magnetic field past the conductor, and that electrical energy will flow if the conductor is part of a closed circuit. Because it is fitted to the conductor by a magnetic field that changes as a result of the magnetic field, this voltage is known as the inserted emf, which has a negative signal in Faraday's calculations that indicates the direction of the available force.

FAQs on Faraday Electromagnetic Induction Experiment

1. Explain Faraday's Law

Michael Faraday was an English scientist of the 19th century, credited with many great discoveries in the field of physics and chemistry, specifically on the relationship between current and magnets, and electrochemistry. Law of Faraday, by the 19th-century physicist Michael Faraday. This relates the rate of magnetic flux shift through the loop to the magnitude of the electromotive force E caused by the loop. There's a relationship which is stated as - 

E =  dΦ​ / dtE

The electromotive force or EMF refers to the potential difference between the unloaded loop (i.e. when the resistance in the circuit is high). In practice, it is always necessary to regard EMF as a voltage, because both the voltage and the EMF are calculated using the same unit, the volt.

2.  A small 10 mm diameter permanent magnet produces a field of 100 mT. The field drops away rapidly with distance and is negligible more than 1 mm from the surface. If this magnet moves at a speed of 1 m/s through a 100-turn coil of length 1 mm and diameter just larger than the magnet, what is the EMF Induced?

We can use Faraday 's induction law to find the induced EMF. This needs us to know the change in the flow through the coil and how quickly the change is going to happen.

We can start by looking at the cases where the magnet is outside and inside the coil separately. Since we are told that the field decays quickly, we can assume that the flux is zero when the magnet is outside the coil. Since the coil is a close fit around the magnet, we can assume that the field is always orthogonal to the coil and that the flux is orthogonal.

Since the magnet is considered to be traveling at 1000 mm / s, we know that it will be inside a 1 mm long coil for just 1/1000 s (1 ms). So by applying Faraday 's law,

E = -N dΦ / dt 

= - (100 turns) (100. 10 -3 T) (5. 10 -3 m ) 2 / 1. 10 -3 s

20.3 Electromagnetic Induction

Section learning objectives.

By the end of this section, you will be able to do the following:

• Explain how a changing magnetic field produces a current in a wire
• Calculate induced electromotive force and current

Teacher Support

The learning objectives in this section will help your students master the following standards:

• (G) investigate and describe the relationship between electric and magnetic fields in applications such as generators, motors, and transformers.

In addition, the OSX High School Physics Laboratory Manual addresses content in this section in the lab titled: Magnetism, as well as the following standards:

Section Key Terms

Changing Magnetic Fields

In the preceding section, we learned that a current creates a magnetic field. If nature is symmetrical, then perhaps a magnetic field can create a current. In 1831, some 12 years after the discovery that an electric current generates a magnetic field, English scientist Michael Faraday (1791–1862) and American scientist Joseph Henry (1797–1878) independently demonstrated that magnetic fields can produce currents. The basic process of generating currents with magnetic fields is called induction ; this process is also called magnetic induction to distinguish it from charging by induction, which uses the electrostatic Coulomb force.

When Faraday discovered what is now called Faraday’s law of induction, Queen Victoria asked him what possible use was electricity. “Madam,” he replied, “What good is a baby?” Today, currents induced by magnetic fields are essential to our technological society. The electric generator—found in everything from automobiles to bicycles to nuclear power plants—uses magnetism to generate electric current. Other devices that use magnetism to induce currents include pickup coils in electric guitars, transformers of every size, certain microphones, airport security gates, and damping mechanisms on sensitive chemical balances.

One experiment Faraday did to demonstrate magnetic induction was to move a bar magnet through a wire coil and measure the resulting electric current through the wire. A schematic of this experiment is shown in Figure 20.33 . He found that current is induced only when the magnet moves with respect to the coil. When the magnet is motionless with respect to the coil, no current is induced in the coil, as in Figure 20.33 . In addition, moving the magnet in the opposite direction (compare Figure 20.33 with Figure 20.33 ) or reversing the poles of the magnet (compare Figure 20.33 with Figure 20.33 ) results in a current in the opposite direction.

Virtual Physics

Faraday’s law.

Try this simulation to see how moving a magnet creates a current in a circuit. A light bulb lights up to show when current is flowing, and a voltmeter shows the voltage drop across the light bulb. Try moving the magnet through a four-turn coil and through a two-turn coil. For the same magnet speed, which coil produces a higher voltage?

• The sign of voltage will change because the direction of current flow will change by moving south pole of the magnet to the left.
• The sign of voltage will remain same because the direction of current flow will not change by moving south pole of the magnet to the left.
• The sign of voltage will change because the magnitude of current flow will change by moving south pole of the magnet to the left.
• The sign of voltage will remain same because the magnitude of current flow will not change by moving south pole of the magnet to the left.

Induced Electromotive Force

If a current is induced in the coil, Faraday reasoned that there must be what he called an electromotive force pushing the charges through the coil. This interpretation turned out to be incorrect; instead, the external source doing the work of moving the magnet adds energy to the charges in the coil. The energy added per unit charge has units of volts, so the electromotive force is actually a potential. Unfortunately, the name electromotive force stuck and with it the potential for confusing it with a real force. For this reason, we avoid the term electromotive force and just use the abbreviation emf , which has the mathematical symbol ε . ε . The emf may be defined as the rate at which energy is drawn from a source per unit current flowing through a circuit. Thus, emf is the energy per unit charge added by a source, which contrasts with voltage, which is the energy per unit charge released as the charges flow through a circuit.

To understand why an emf is generated in a coil due to a moving magnet, consider Figure 20.34 , which shows a bar magnet moving downward with respect to a wire loop. Initially, seven magnetic field lines are going through the loop (see left-hand image). Because the magnet is moving away from the coil, only five magnetic field lines are going through the loop after a short time Δ t Δ t (see right-hand image). Thus, when a change occurs in the number of magnetic field lines going through the area defined by the wire loop, an emf is induced in the wire loop. Experiments such as this show that the induced emf is proportional to the rate of change of the magnetic field. Mathematically, we express this as

where Δ B Δ B is the change in the magnitude in the magnetic field during time Δ t Δ t and A is the area of the loop.

Note that magnetic field lines that lie in the plane of the wire loop do not actually pass through the loop, as shown by the left-most loop in Figure 20.35 . In this figure, the arrow coming out of the loop is a vector whose magnitude is the area of the loop and whose direction is perpendicular to the plane of the loop. In Figure 20.35 , as the loop is rotated from θ = 90° θ = 90° to θ = 0° , θ = 0° , the contribution of the magnetic field lines to the emf increases. Thus, what is important in generating an emf in the wire loop is the component of the magnetic field that is perpendicular to the plane of the loop, which is B cos θ . B cos θ .

This is analogous to a sail in the wind. Think of the conducting loop as the sail and the magnetic field as the wind. To maximize the force of the wind on the sail, the sail is oriented so that its surface vector points in the same direction as the winds, as in the right-most loop in Figure 20.35 . When the sail is aligned so that its surface vector is perpendicular to the wind, as in the left-most loop in Figure 20.35 , then the wind exerts no force on the sail.

Thus, taking into account the angle of the magnetic field with respect to the area, the proportionality E ∝ Δ B / Δ t E ∝ Δ B / Δ t becomes

Another way to reduce the number of magnetic field lines that go through the conducting loop in Figure 20.35 is not to move the magnet but to make the loop smaller. Experiments show that changing the area of a conducting loop in a stable magnetic field induces an emf in the loop. Thus, the emf produced in a conducting loop is proportional to the rate of change of the product of the perpendicular magnetic field and the loop area

where B cos θ B cos θ is the perpendicular magnetic field and A is the area of the loop. The product B A cos θ B A cos θ is very important. It is proportional to the number of magnetic field lines that pass perpendicularly through a surface of area A . Going back to our sail analogy, it would be proportional to the force of the wind on the sail. It is called the magnetic flux and is represented by Φ Φ .

The unit of magnetic flux is the weber (Wb), which is magnetic field per unit area, or T/m 2 . The weber is also a volt second (Vs).

The induced emf is in fact proportional to the rate of change of the magnetic flux through a conducting loop.

Finally, for a coil made from N loops, the emf is N times stronger than for a single loop. Thus, the emf induced by a changing magnetic field in a coil of N loops is

The last question to answer before we can change the proportionality into an equation is “In what direction does the current flow?” The Russian scientist Heinrich Lenz (1804–1865) explained that the current flows in the direction that creates a magnetic field that tries to keep the flux constant in the loop. For example, consider again Figure 20.34 . The motion of the bar magnet causes the number of upward-pointing magnetic field lines that go through the loop to decrease. Therefore, an emf is generated in the loop that drives a current in the direction that creates more upward-pointing magnetic field lines. By using the right-hand rule, we see that this current must flow in the direction shown in the figure. To express the fact that the induced emf acts to counter the change in the magnetic flux through a wire loop, a minus sign is introduced into the proportionality ε ∝ Δ Φ / Δ t . ε ∝ Δ Φ / Δ t . , which gives Faraday’s law of induction.

Lenz’s law is very important. To better understand it, consider Figure 20.36 , which shows a magnet moving with respect to a wire coil and the direction of the resulting current in the coil. In the top row, the north pole of the magnet approaches the coil, so the magnetic field lines from the magnet point toward the coil. Thus, the magnetic field B → mag = B mag ( x ^ ) B → mag = B mag ( x ^ ) pointing to the right increases in the coil. According to Lenz’s law, the emf produced in the coil will drive a current in the direction that creates a magnetic field B → coil = B coil ( − x ^ ) B → coil = B coil ( − x ^ ) inside the coil pointing to the left. This will counter the increase in magnetic flux pointing to the right. To see which way the current must flow, point your right thumb in the desired direction of the magnetic field B → coil, B → coil, and the current will flow in the direction indicated by curling your right fingers. This is shown by the image of the right hand in the top row of Figure 20.36 . Thus, the current must flow in the direction shown in Figure 4(a) .

In Figure 4(b) , the direction in which the magnet moves is reversed. In the coil, the right-pointing magnetic field B → mag B → mag due to the moving magnet decreases. Lenz’s law says that, to counter this decrease, the emf will drive a current that creates an additional right-pointing magnetic field B → coil B → coil in the coil. Again, point your right thumb in the desired direction of the magnetic field, and the current will flow in the direction indicate by curling your right fingers ( Figure 4(b) ).

Finally, in Figure 4(c) , the magnet is reversed so that the south pole is nearest the coil. Now the magnetic field B → mag B → mag points toward the magnet instead of toward the coil. As the magnet approaches the coil, it causes the left-pointing magnetic field in the coil to increase. Lenz’s law tells us that the emf induced in the coil will drive a current in the direction that creates a magnetic field pointing to the right. This will counter the increasing magnetic flux pointing to the left due to the magnet. Using the right-hand rule again, as indicated in the figure, shows that the current must flow in the direction shown in Figure 4(c) .

Faraday’s Electromagnetic Lab

This simulation proposes several activities. For now, click on the tab Pickup Coil, which presents a bar magnet that you can move through a coil. As you do so, you can see the electrons move in the coil and a light bulb will light up or a voltmeter will indicate the voltage across a resistor. Note that the voltmeter allows you to see the sign of the voltage as you move the magnet about. You can also leave the bar magnet at rest and move the coil, although it is more difficult to observe the results.

• Yes, the current in the simulation flows as shown because the direction of current is opposite to the direction of flow of electrons.
• No, current in the simulation flows in the opposite direction because the direction of current is same to the direction of flow of electrons.

Watch Physics

Induced current in a wire.

This video explains how a current can be induced in a straight wire by moving it through a magnetic field. The lecturer uses the cross product , which a type of vector multiplication. Don’t worry if you are not familiar with this, it basically combines the right-hand rule for determining the force on the charges in the wire with the equation F = q v B sin θ . F = q v B sin θ .

Grasp Check

What emf is produced across a straight wire 0.50 m long moving at a velocity of (1.5 m/s) x ^ x ^ through a uniform magnetic field (0.30 T) ẑ ? The wire lies in the ŷ -direction. Also, which end of the wire is at the higher potential—let the lower end of the wire be at y = 0 and the upper end at y = 0.5 m)?

• 0.15 V and the lower end of the wire will be at higher potential
• 0.15 V and the upper end of the wire will be at higher potential
• 0.075 V and the lower end of the wire will be at higher potential
• 0.075 V and the upper end of the wire will be at higher potential

Worked Example

Emf induced in conducing coil by moving magnet.

Imagine a magnetic field goes through a coil in the direction indicated in Figure 20.37 . The coil diameter is 2.0 cm. If the magnetic field goes from 0.020 to 0.010 T in 34 s, what is the direction and magnitude of the induced current? Assume the coil has a resistance of 0.1 Ω. Ω.

Use the equation ε = − N Δ Φ / Δ t ε = − N Δ Φ / Δ t to find the induced emf in the coil, where Δ t = 34 s Δ t = 34 s . Counting the number of loops in the solenoid, we find it has 16 loops, so N = 16 . N = 16 . Use the equation Φ = B A cos θ Φ = B A cos θ to calculate the magnetic flux

where d is the diameter of the solenoid and we have used cos 0° = 1 . cos 0° = 1 . Because the area of the solenoid does not vary, the change in the magnetic of the flux through the solenoid is

Once we find the emf, we can use Ohm’s law, ε = I R , ε = I R , to find the current.

Finally, Lenz’s law tells us that the current should produce a magnetic field that acts to oppose the decrease in the applied magnetic field. Thus, the current should produce a magnetic field to the right.

Combining equations ε = − N Δ Φ / Δ t ε = − N Δ Φ / Δ t and Φ = B A cos θ Φ = B A cos θ gives

Solving Ohm’s law for the current and using this result gives

Lenz’s law tells us that the current must produce a magnetic field to the right. Thus, we point our right thumb to the right and curl our right fingers around the solenoid. The current must flow in the direction in which our fingers are pointing, so it enters at the left end of the solenoid and exits at the right end.

Let’s see if the minus sign makes sense in Faraday’s law of induction. Define the direction of the magnetic field to be the positive direction. This means the change in the magnetic field is negative, as we found above. The minus sign in Faraday’s law of induction negates the negative change in the magnetic field, leaving us with a positive current. Therefore, the current must flow in the direction of the magnetic field, which is what we found.

Now try defining the positive direction to be the direction opposite that of the magnetic field, that is positive is to the left in Figure 20.37 . In this case, you will find a negative current. But since the positive direction is to the left, a negative current must flow to the right, which again agrees with what we found by using Lenz’s law.

Magnetic Induction due to Changing Circuit Size

The circuit shown in Figure 20.38 consists of a U-shaped wire with a resistor and with the ends connected by a sliding conducting rod. The magnetic field filling the area enclosed by the circuit is constant at 0.01 T. If the rod is pulled to the right at speed v = 0.50 m/s, v = 0.50 m/s, what current is induced in the circuit and in what direction does the current flow?

We again use Faraday’s law of induction, E = − N Δ Φ Δ t , E = − N Δ Φ Δ t , although this time the magnetic field is constant and the area enclosed by the circuit changes. The circuit contains a single loop, so N = 1 . N = 1 . The rate of change of the area is Δ A Δ t = v ℓ . Δ A Δ t = v ℓ . Thus the rate of change of the magnetic flux is

where we have used the fact that the angle θ θ between the area vector and the magnetic field is 0°. Once we know the emf, we can find the current by using Ohm’s law. To find the direction of the current, we apply Lenz’s law.

Faraday’s law of induction gives

Solving Ohm’s law for the current and using the previous result for emf gives

As the rod slides to the right, the magnetic flux passing through the circuit increases. Lenz’s law tells us that the current induced will create a magnetic field that will counter this increase. Thus, the magnetic field created by the induced current must be into the page. Curling your right-hand fingers around the loop in the clockwise direction makes your right thumb point into the page, which is the desired direction of the magnetic field. Thus, the current must flow in the clockwise direction around the circuit.

Is energy conserved in this circuit? An external agent must pull on the rod with sufficient force to just balance the force on a current-carrying wire in a magnetic field—recall that F = I ℓ B sin θ . F = I ℓ B sin θ . The rate at which this force does work on the rod should be balanced by the rate at which the circuit dissipates power. Using F = I ℓ B sin θ , F = I ℓ B sin θ , the force required to pull the wire at a constant speed v is

where we used the fact that the angle θ θ between the current and the magnetic field is 90° . 90° . Inserting our expression above for the current into this equation gives

The power contributed by the agent pulling the rod is F pull v , or F pull v , or

The power dissipated by the circuit is

We thus see that P pull + P dissipated = 0 , P pull + P dissipated = 0 , which means that power is conserved in the system consisting of the circuit and the agent that pulls the rod. Thus, energy is conserved in this system.

Practice Problems

The magnetic flux through a single wire loop changes from 3.5 Wb to 1.5 Wb in 2.0 s. What emf is induced in the loop?

What is the emf for a 10-turn coil through which the flux changes at 10 Wb/s?

Check Your Understanding

• An electric current is induced if a bar magnet is placed near the wire loop.
• An electric current is induced if a wire loop is wound around the bar magnet.
• An electric current is induced if a bar magnet is moved through the wire loop.
• An electric current is induced if a bar magnet is placed in contact with the wire loop.
• Induced current can be created by changing the size of the wire loop only.
• Induced current can be created by changing the orientation of the wire loop only.
• Induced current can be created by changing the strength of the magnetic field only.
• Induced current can be created by changing the strength of the magnetic field, changing the size of the wire loop, or changing the orientation of the wire loop.

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Access for free at https://openstax.org/books/physics/pages/1-introduction

• Authors: Paul Peter Urone, Roger Hinrichs
• Publisher/website: OpenStax
• Book title: Physics
• Publication date: Mar 26, 2020
• Location: Houston, Texas
• Book URL: https://openstax.org/books/physics/pages/1-introduction
• Section URL: https://openstax.org/books/physics/pages/20-3-electromagnetic-induction

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