The field around a long straight wire is found to be in circular loops. The right hand rule 2 RHR-2 emerges from this exploration and is valid for any current segment— point the thumb in the direction of the current, and the fingers curl in the direction of the magnetic field loops created by it.

The magnetic field strength magnitude produced by a long straight current-carrying wire is found by experiment to be. So a moderately large current produces a significant magnetic field at a distance of 5.

The magnetic field of a long straight wire has more implications than you might at first suspect. Each segment of current produces a magnetic field like that of a long straight wire, and the total field of any shape current is the vector sum of the fields due to each segment.

The formal statement of the direction and magnitude of the field due to each segment is called the Biot-Savart law. Integral calculus is needed to sum the field for an arbitrary shape current.

Most of this is beyond the scope of this text in both mathematical level, requiring calculus, and in the amount of space that can be devoted to it.

But for the interested student, and particularly for those who continue in physics, engineering, or similar pursuits, delving into these matters further will reveal descriptions of nature that are elegant as well as profound.

In this text, we shall keep the general features in mind, such as RHR-2 and the rules for magnetic field lines listed in Chapter Hearing all we do about Einstein, we sometimes get the impression that he invented relativity out of nothing.

The magnetic field near a current-carrying loop of wire is shown in Figure 2. Both the direction and the magnitude of the magnetic field produced by a current-carrying loop are complex.

RHR-2 can be used to give the direction of the field near the loop, but mapping with compasses and the rules about field lines given in Chapter There is a simple formula for the magnetic field strength at the center of a circular loop. It is. This equation is very similar to that for a straight wire, but it is valid only at the center of a circular loop of wire.

The similarity of the equations does indicate that similar field strength can be obtained at the center of a loop. Note that the larger the loop, the smaller the field at its center, because the current is farther away. A solenoid is a long coil of wire with many turns or loops, as opposed to a flat loop.

Because of its shape, the field inside a solenoid can be very uniform, and also very strong. The field just outside the coils is nearly zero. Figure 3 shows how the field looks and how its direction is given by RHR The magnetic field inside of a current-carrying solenoid is very uniform in direction and magnitude.

Only near the ends does it begin to weaken and change direction. The field outside has similar complexities to flat loops and bar magnets, but the magnetic field strength inside a solenoid is simply. Large uniform fields spread over a large volume are possible with solenoids, as Example 2 implies. What is the field inside a 2.

First, we note the number of loops per unit length is. This is a large field strength that could be established over a large-diameter solenoid, such as in medical uses of magnetic resonance imaging MRI. The very large current is an indication that the fields of this strength are not easily achieved, however.

Higher currents can be achieved by using superconducting wires, although this is expensive. An instrument used to measure the local magnetic field is known as a magnetometer. Important classes of magnetometers include using induction magnetometers or search-coil magnetometers which measure only varying magnetic fields, rotating coil magnetometers , Hall effect magnetometers, NMR magnetometers , SQUID magnetometers , and fluxgate magnetometers.

The magnetic fields of distant astronomical objects are measured through their effects on local charged particles.

For instance, electrons spiraling around a field line produce synchrotron radiation that is detectable in radio waves. The field can be visualized by a set of magnetic field lines , that follow the direction of the field at each point. The lines can be constructed by measuring the strength and direction of the magnetic field at a large number of points or at every point in space.

Then, mark each location with an arrow called a vector pointing in the direction of the local magnetic field with its magnitude proportional to the strength of the magnetic field. Connecting these arrows then forms a set of magnetic field lines.

The direction of the magnetic field at any point is parallel to the direction of nearby field lines, and the local density of field lines can be made proportional to its strength. Magnetic field lines are like streamlines in fluid flow , in that they represent a continuous distribution, and a different resolution would show more or fewer lines.

An advantage of using magnetic field lines as a representation is that many laws of magnetism and electromagnetism can be stated completely and concisely using simple concepts such as the "number" of field lines through a surface. These concepts can be quickly "translated" to their mathematical form.

For example, the number of field lines through a given surface is the surface integral of the magnetic field. Various phenomena "display" magnetic field lines as though the field lines were physical phenomena.

For example, iron filings placed in a magnetic field form lines that correspond to "field lines". Field lines can be used as a qualitative tool to visualize magnetic forces.

In ferromagnetic substances like iron and in plasmas, magnetic forces can be understood by imagining that the field lines exert a tension , like a rubber band along their length, and a pressure perpendicular to their length on neighboring field lines.

Permanent magnets are objects that produce their own persistent magnetic fields. They are made of ferromagnetic materials, such as iron and nickel , that have been magnetized, and they have both a north and a south pole.

The magnetic field of permanent magnets can be quite complicated, especially near the magnet. The magnetic field of a small [note 6] straight magnet is proportional to the magnet's strength called its magnetic dipole moment m. The equations are non-trivial and depend on the distance from the magnet and the orientation of the magnet.

For simple magnets, m points in the direction of a line drawn from the south to the north pole of the magnet. Flipping a bar magnet is equivalent to rotating its m by degrees.

The magnetic field of larger magnets can be obtained by modeling them as a collection of a large number of small magnets called dipoles each having their own m. The magnetic field produced by the magnet then is the net magnetic field of these dipoles; any net force on the magnet is a result of adding up the forces on the individual dipoles.

There are two simplified models for the nature of these dipoles: the magnetic pole model and the Amperian loop model. These two models produce two different magnetic fields, H and B.

Outside a material, though, the two are identical to a multiplicative constant so that in many cases the distinction can be ignored.

This is particularly true for magnetic fields, such as those due to electric currents, that are not generated by magnetic materials.

A realistic model of magnetism is more complicated than either of these models; neither model fully explains why materials are magnetic.

The monopole model has no experimental support. The Amperian loop model explains some, but not all of a material's magnetic moment. The model predicts that the motion of electrons within an atom are connected to those electrons' orbital magnetic dipole moment , and these orbital moments do contribute to the magnetism seen at the macroscopic level.

However, the motion of electrons is not classical, and the spin magnetic moment of electrons which is not explained by either model is also a significant contribution to the total moment of magnets.

Historically, early physics textbooks would model the force and torques between two magnets as due to magnetic poles repelling or attracting each other in the same manner as the Coulomb force between electric charges.

At the microscopic level, this model contradicts the experimental evidence, and the pole model of magnetism is no longer the typical way to introduce the concept. In this model, a magnetic H -field is produced by fictitious magnetic charges that are spread over the surface of each pole.

These magnetic charges are in fact related to the magnetization field M. The H -field, therefore, is analogous to the electric field E , which starts at a positive electric charge and ends at a negative electric charge. Near the north pole, therefore, all H -field lines point away from the north pole whether inside the magnet or out while near the south pole all H -field lines point toward the south pole whether inside the magnet or out.

Too, a north pole feels a force in the direction of the H -field while the force on the south pole is opposite to the H -field. The magnetic pole model predicts correctly the field H both inside and outside magnetic materials, in particular the fact that H is opposite to the magnetization field M inside a permanent magnet.

Since it is based on the fictitious idea of a magnetic charge density , the pole model has limitations. Magnetic poles cannot exist apart from each other as electric charges can, but always come in north—south pairs. If a magnetized object is divided in half, a new pole appears on the surface of each piece, so each has a pair of complementary poles.

The magnetic pole model does not account for magnetism that is produced by electric currents, nor the inherent connection between angular momentum and magnetism.

The pole model usually treats magnetic charge as a mathematical abstraction, rather than a physical property of particles.

However, a magnetic monopole is a hypothetical particle or class of particles that physically has only one magnetic pole either a north pole or a south pole.

In other words, it would possess a "magnetic charge" analogous to an electric charge. Magnetic field lines would start or end on magnetic monopoles, so if they exist, they would give exceptions to the rule that magnetic field lines neither start nor end.

Some theories such as Grand Unified Theories have predicted the existence of magnetic monopoles, but so far, none have been observed.

In the model developed by Ampere , the elementary magnetic dipole that makes up all magnets is a sufficiently small Amperian loop with current I and loop area A.

The magnetic field of a magnetic dipole is depicted in the figure. From outside, the ideal magnetic dipole is identical to that of an ideal electric dipole of the same strength. Unlike the electric dipole, a magnetic dipole is properly modeled as a current loop having a current I and an area a.

This model clarifies the connection between angular momentum and magnetic moment, which is the basis of the Einstein—de Haas effect rotation by magnetization and its inverse, the Barnett effect or magnetization by rotation. Specifying the force between two small magnets is quite complicated because it depends on the strength and orientation of both magnets and their distance and direction relative to each other.

The force is particularly sensitive to rotations of the magnets due to magnetic torque. The force on each magnet depends on its magnetic moment and the magnetic field [note 7] of the other. To understand the force between magnets, it is useful to examine the magnetic pole model given above.

In this model, the H -field of one magnet pushes and pulls on both poles of a second magnet. If this H -field is the same at both poles of the second magnet then there is no net force on that magnet since the force is opposite for opposite poles.

If, however, the magnetic field of the first magnet is nonuniform such as the H near one of its poles , each pole of the second magnet sees a different field and is subject to a different force.

This difference in the two forces moves the magnet in the direction of increasing magnetic field and may also cause a net torque. This is a specific example of a general rule that magnets are attracted or repulsed depending on the orientation of the magnet into regions of higher magnetic field.

Any non-uniform magnetic field, whether caused by permanent magnets or electric currents, exerts a force on a small magnet in this way.

If m is in the same direction as B then the dot product is positive and the gradient points "uphill" pulling the magnet into regions of higher B -field more strictly larger m · B. This equation is strictly only valid for magnets of zero size, but is often a good approximation for not too large magnets.

The magnetic force on larger magnets is determined by dividing them into smaller regions each having their own m then summing up the forces on each of these very small regions. If two like poles of two separate magnets are brought near each other, and one of the magnets is allowed to turn, it promptly rotates to align itself with the first.

In this example, the magnetic field of the stationary magnet creates a magnetic torque on the magnet that is free to rotate. This magnetic torque τ tends to align a magnet's poles with the magnetic field lines. A compass, therefore, turns to align itself with Earth's magnetic field.

In terms of the pole model, two equal and opposite magnetic charges experiencing the same H also experience equal and opposite forces. Since these equal and opposite forces are in different locations, this produces a torque proportional to the distance perpendicular to the force between them.

Mathematically, the torque τ on a small magnet is proportional both to the applied magnetic field and to the magnetic moment m of the magnet:. where × represents the vector cross product. This equation includes all of the qualitative information included above.

There is no torque on a magnet if m is in the same direction as the magnetic field, since the cross product is zero for two vectors that are in the same direction. Further, all other orientations feel a torque that twists them toward the direction of magnetic field.

Currents of electric charges both generate a magnetic field and feel a force due to magnetic B-fields. All moving charged particles produce magnetic fields. Moving point charges, such as electrons , produce complicated but well known magnetic fields that depend on the charge, velocity, and acceleration of the particles.

Magnetic field lines form in concentric circles around a cylindrical current-carrying conductor, such as a length of wire.

The direction of such a magnetic field can be determined by using the " right-hand grip rule " see figure at right. The strength of the magnetic field decreases with distance from the wire.

For an infinite length wire the strength is inversely proportional to the distance. Bending a current-carrying wire into a loop concentrates the magnetic field inside the loop while weakening it outside.

Bending a wire into multiple closely spaced loops to form a coil or " solenoid " enhances this effect. A device so formed around an iron core may act as an electromagnet , generating a strong, well-controlled magnetic field.

An infinitely long cylindrical electromagnet has a uniform magnetic field inside, and no magnetic field outside. A finite length electromagnet produces a magnetic field that looks similar to that produced by a uniform permanent magnet, with its strength and polarity determined by the current flowing through the coil.

The direction is tangent to a circle perpendicular to the wire according to the right hand rule. Ampère's law is always valid for steady currents and can be used to calculate the B -field for certain highly symmetric situations such as an infinite wire or an infinite solenoid.

In a modified form that accounts for time varying electric fields, Ampère's law is one of four Maxwell's equations that describe electricity and magnetism. A charged particle moving in a B -field experiences a sideways force that is proportional to the strength of the magnetic field, the component of the velocity that is perpendicular to the magnetic field and the charge of the particle.

The Lorentz force is always perpendicular to both the velocity of the particle and the magnetic field that created it. When a charged particle moves in a static magnetic field, it traces a helical path in which the helix axis is parallel to the magnetic field, and in which the speed of the particle remains constant.

Because the magnetic force is always perpendicular to the motion, the magnetic field can do no work on an isolated charge. It is often claimed that the magnetic force can do work to a non-elementary magnetic dipole , or to charged particles whose motion is constrained by other forces, but this is incorrect [24] because the work in those cases is performed by the electric forces of the charges deflected by the magnetic field.

The force on a current carrying wire is similar to that of a moving charge as expected since a current carrying wire is a collection of moving charges. A current-carrying wire feels a force in the presence of a magnetic field. The Lorentz force on a macroscopic current is often referred to as the Laplace force.

The formulas derived for the magnetic field above are correct when dealing with the entire current. A magnetic material placed inside a magnetic field, though, generates its own bound current , which can be a challenge to calculate.

This bound current is due to the sum of atomic sized current loops and the spin of the subatomic particles such as electrons that make up the material. The H -field as defined above helps factor out this bound current; but to see how, it helps to introduce the concept of magnetization first.

The magnetization vector field M represents how strongly a region of material is magnetized. It is defined as the net magnetic dipole moment per unit volume of that region. The magnetization of a uniform magnet is therefore a material constant, equal to the magnetic moment m of the magnet divided by its volume.

The magnetization M field of a region points in the direction of the average magnetic dipole moment in that region. Magnetization field lines, therefore, begin near the magnetic south pole and ends near the magnetic north pole.

Magnetization does not exist outside the magnet. In the Amperian loop model, the magnetization is due to combining many tiny Amperian loops to form a resultant current called bound current. This bound current, then, is the source of the magnetic B field due to the magnet.

In the magnetic pole model, magnetization begins at and ends at magnetic poles. If a given region, therefore, has a net positive "magnetic pole strength" corresponding to a north pole then it has more magnetization field lines entering it than leaving it.

A closed surface completely surrounds a region with no holes to let any field lines escape. The negative sign occurs because the magnetization field moves from south to north. For the differential equivalent of this equation see Maxwell's equations.

where H 0 is the applied magnetic field due only to the free currents and H d is the demagnetizing field due only to the bound currents. The magnetic H -field, therefore, re-factors the bound current in terms of "magnetic charges". The H field lines loop only around "free current" and, unlike the magnetic B field, begins and ends near magnetic poles as well.

Most materials respond to an applied B -field by producing their own magnetization M and therefore their own B -fields. Typically, the response is weak and exists only when the magnetic field is applied. The term magnetism describes how materials respond on the microscopic level to an applied magnetic field and is used to categorize the magnetic phase of a material.

Materials are divided into groups based upon their magnetic behavior:. In some cases the permeability may be a second rank tensor so that H may not point in the same direction as B.

These relations between B and H are examples of constitutive equations. However, superconductors and ferromagnets have a more complex B -to- H relation; see magnetic hysteresis. Energy is needed to generate a magnetic field both to work against the electric field that a changing magnetic field creates and to change the magnetization of any material within the magnetic field.

For non-dispersive materials, this same energy is released when the magnetic field is destroyed so that the energy can be modeled as being stored in the magnetic field. If there are no magnetic materials around then μ can be replaced by μ 0. The above equation cannot be used for nonlinear materials, though; a more general expression given below must be used.

In general, the incremental amount of work per unit volume δW needed to cause a small change of magnetic field δ B is:. Once the relationship between H and B is known this equation is used to determine the work needed to reach a given magnetic state.

For hysteretic materials such as ferromagnets and superconductors, the work needed also depends on how the magnetic field is created.

For linear non-dispersive materials, though, the general equation leads directly to the simpler energy density equation given above. Like all vector fields, a magnetic field has two important mathematical properties that relates it to its sources. For B the sources are currents and changing electric fields.

These two properties, along with the two corresponding properties of the electric field, make up Maxwell's Equations. Maxwell's Equations together with the Lorentz force law form a complete description of classical electrodynamics including both electricity and magnetism. As discussed above, a B -field line never starts or ends at a point but instead forms a complete loop.

This is mathematically equivalent to saying that the divergence of B is zero. Such vector fields are called solenoidal vector fields.

This property is called Gauss's law for magnetism and is equivalent to the statement that there are no isolated magnetic poles or magnetic monopoles.

The result of the curl is called a "circulation source". The equations for the curl of B and of E are called the Ampère—Maxwell equation and Faraday's law respectively.

One important property of the B -field produced this way is that magnetic B -field lines neither start nor end mathematically, B is a solenoidal vector field ; a field line may only extend to infinity, or wrap around to form a closed curve, or follow a never-ending possibly chaotic path.

More formally, since all the magnetic field lines that enter any given region must also leave that region, subtracting the "number" [note 12] of field lines that enter the region from the number that exit gives identically zero.

Since d A points outward, the dot product in the integral is positive for B -field pointing out and negative for B -field pointing in. A changing magnetic field, such as a magnet moving through a conducting coil, generates an electric field and therefore tends to drive a current in such a coil.

This is known as Faraday's law and forms the basis of many electrical generators and electric motors. This definition of magnetic flux is why B is often referred to as magnetic flux density. This phenomenon is known as Lenz's law.

This integral formulation of Faraday's law can be converted [note 13] into a differential form, which applies under slightly different conditions. Similar to the way that a changing magnetic field generates an electric field, a changing electric field generates a magnetic field.

This fact is known as Maxwell's correction to Ampère's law and is applied as an additive term to Ampere's law as given above. This additional term is proportional to the time rate of change of the electric flux and is similar to Faraday's law above but with a different and positive constant out front.

The electric flux through an area is proportional to the area times the perpendicular part of the electric field. The full law including the correction term is known as the Maxwell—Ampère equation. It is not commonly given in integral form because the effect is so small that it can typically be ignored in most cases where the integral form is used.

The Maxwell term is critically important in the creation and propagation of electromagnetic waves. Maxwell's correction to Ampère's Law together with Faraday's law of induction describes how mutually changing electric and magnetic fields interact to sustain each other and thus to form electromagnetic waves , such as light: a changing electric field generates a changing magnetic field, which generates a changing electric field again.

These, though, are usually described using the differential form of this equation given below. where J is the complete microscopic current density , and ε 0 is the vacuum permittivity. As discussed above, materials respond to an applied electric E field and an applied magnetic B field by producing their own internal "bound" charge and current distributions that contribute to E and B but are difficult to calculate.

To circumvent this problem, H and D fields are used to re-factor Maxwell's equations in terms of the free current density J f :. These equations are not any more general than the original equations if the "bound" charges and currents in the material are known.

They also must be supplemented by the relationship between B and H as well as that between E and D. On the other hand, for simple relationships between these quantities this form of Maxwell's equations can circumvent the need to calculate the bound charges and currents.

According to the special theory of relativity , the partition of the electromagnetic force into separate electric and magnetic components is not fundamental, but varies with the observational frame of reference : An electric force perceived by one observer may be perceived by another in a different frame of reference as a magnetic force, or a mixture of electric and magnetic forces.

The magnetic field existing as electric field in other frames can be shown by consistency of equations obtained from Lorentz transformation of four force from Coulomb's Law in particle's rest frame with Maxwell's laws considering definition of fields from Lorentz force and for non accelerating condition.

This form of magnetic field can be shown to satisfy maxwell's laws within the constraint of particle being non accelerating. Formally, special relativity combines the electric and magnetic fields into a rank-2 tensor , called the electromagnetic tensor.

Changing reference frames mixes these components. This is analogous to the way that special relativity mixes space and time into spacetime , and mass, momentum, and energy into four-momentum. In advanced topics such as quantum mechanics and relativity it is often easier to work with a potential formulation of electrodynamics rather than in terms of the electric and magnetic fields.

The vector potential, A given by this form may be interpreted as a generalized potential momentum per unit charge [38] just as φ is interpreted as a generalized potential energy per unit charge.

There are multiple choices one can make for the potential fields that satisfy the above condition. However, the choice of potentials is represented by its respective gauge condition. Maxwell's equations when expressed in terms of the potentials in Lorentz gauge can be cast into a form that agrees with special relativity.

Using the four potential instead of the electromagnetic tensor has the advantage of being much simpler—and it can be easily modified to work with quantum mechanics. Special theory of relativity imposes the condition for events related by cause and effect to be time-like separated, that is that causal efficacy propagates no faster than light.

Electric and magnetic fields from classical electrodynamics obey the principle of locality in physics and are expressed in terms of retarded time or the time at which the cause of a measured field originated given that the influence of field travelled at speed of light. The retarded time for a point particle is given as solution of:.

The equation subtracts the time taken for light to travel from particle to the point in space from the time of measurement to find time of origin of the fields. The solution of maxwell's equations for electric and magnetic field of a point charge is expressed in terms of retarded time or the time at which the particle in the past causes the field at the point, given that the influence travels across space at the speed of light.

Any arbitrary motion of point charge causes electric and magnetic fields found by solving maxwell's equations using green's function for retarded potentials and hence finding the fields to be as follows:.

Hence by the principle of superposition , the fields of a system of charges also obey principle of locality. The classical electromagnetic field incorporated into quantum mechanics forms what is known as the semi-classical theory of radiation.

However, it is not able to make experimentally observed predictions such as spontaneous emission process or Lamb shift implying the need for quantization of fields. In modern physics, the electromagnetic field is understood to be not a classical field , but rather a quantum field ; it is represented not as a vector of three numbers at each point, but as a vector of three quantum operators at each point.

The most accurate modern description of the electromagnetic interaction and much else is quantum electrodynamics QED , [42] which is incorporated into a more complete theory known as the Standard Model of particle physics. In QED, the magnitude of the electromagnetic interactions between charged particles and their antiparticles is computed using perturbation theory.

These rather complex formulas produce a remarkable pictorial representation as Feynman diagrams in which virtual photons are exchanged.

This makes QED one of the most accurate physical theories constructed thus far. All equations in this article are in the classical approximation , which is less accurate than the quantum description mentioned here.

However, under most everyday circumstances, the difference between the two theories is negligible. The Earth's magnetic field is produced by convection of a liquid iron alloy in the outer core.

In a dynamo process , the movements drive a feedback process in which electric currents create electric and magnetic fields that in turn act on the currents. The field at the surface of the Earth is approximately the same as if a giant bar magnet were positioned at the center of the Earth and tilted at an angle of about 11° off the rotational axis of the Earth see the figure.

However, because a magnetic pole is attracted to its opposite, the North Magnetic Pole is actually the south pole of the geomagnetic field.

This confusion in terminology arises because the pole of a magnet is defined by the geographical direction it points. Earth's magnetic field is not constant—the strength of the field and the location of its poles vary.

The most recent reversal occurred , years ago. The rotating magnetic field is a key principle in the operation of alternating-current motors.

A permanent magnet in such a field rotates so as to maintain its alignment with the external field. This effect was conceptualized by Nikola Tesla , and later utilized in his and others' early AC alternating current electric motors.

Magnetic torque is used to drive electric motors. In one simple motor design, a magnet is fixed to a freely rotating shaft and subjected to a magnetic field from an array of electromagnets. By continuously switching the electric current through each of the electromagnets, thereby flipping the polarity of their magnetic fields, like poles are kept next to the rotor; the resultant torque is transferred to the shaft.

A rotating magnetic field can be constructed using two orthogonal coils with 90 degrees phase difference in their AC currents. However, in practice such a system would be supplied through a three-wire arrangement with unequal currents. This inequality would cause serious problems in standardization of the conductor size and so, to overcome it, three-phase systems are used where the three currents are equal in magnitude and have degrees phase difference.

Three similar coils having mutual geometrical angles of degrees create the rotating magnetic field in this case. The ability of the three-phase system to create a rotating field, utilized in electric motors, is one of the main reasons why three-phase systems dominate the world's electrical power supply systems.

Synchronous motors use DC-voltage-fed rotor windings, which lets the excitation of the machine be controlled—and induction motors use short-circuited rotors instead of a magnet following the rotating magnetic field of a multicoiled stator.

The short-circuited turns of the rotor develop eddy currents in the rotating field of the stator, and these currents in turn move the rotor by the Lorentz force. In , Nikola Tesla identified the concept of the rotating magnetic field. In , Galileo Ferraris independently researched the concept.

In , Tesla gained U. patent , for his work. Also in , Ferraris published his research in a paper to the Royal Academy of Sciences in Turin. The charge carriers of a current-carrying conductor placed in a transverse magnetic field experience a sideways Lorentz force; this results in a charge separation in a direction perpendicular to the current and to the magnetic field.

The resultant voltage in that direction is proportional to the applied magnetic field. This is known as the Hall effect. The Hall effect is often used to measure the magnitude of a magnetic field. It is used as well to find the sign of the dominant charge carriers in materials such as semiconductors negative electrons or positive holes.

Here, μ is the magnetic permeability of the material. Here the reluctance R m is a quantity similar in nature to resistance for the flux. Using this analogy it is straightforward to calculate the magnetic flux of complicated magnetic field geometries, by using all the available techniques of circuit theory.

As of October [update] , the largest magnetic field produced over a macroscopic volume outside a lab setting is 2. While magnets and some properties of magnetism were known to ancient societies, the research of magnetic fields began in when French scholar Petrus Peregrinus de Maricourt mapped out the magnetic field on the surface of a spherical magnet using iron needles.

Noting the resulting field lines crossed at two points he named those points "poles" in analogy to Earth's poles. He also articulated the principle that magnets always have both a north and south pole, no matter how finely one slices them. Almost three centuries later, William Gilbert of Colchester replicated Petrus Peregrinus' work and was the first to state explicitly that Earth is a magnet.

Three discoveries in challenged this foundation of magnetism. Hans Christian Ørsted demonstrated that a current-carrying wire is surrounded by a circular magnetic field. Extending these experiments, Ampère published his own successful model of magnetism in Also in this work, Ampère introduced the term electrodynamics to describe the relationship between electricity and magnetism.

In , Michael Faraday discovered electromagnetic induction when he found that a changing magnetic field generates an encircling electric field, formulating what is now known as Faraday's law of induction.

In , Lord Kelvin , then known as William Thomson, distinguished between two magnetic fields now denoted H and B. The former applied to Poisson's model and the latter to Ampère's model and induction.

Between and , James Clerk Maxwell developed and published Maxwell's equations , which explained and united all of classical electricity and magnetism.