Magnetic field

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Magnetic field lines shown by iron filings. The high permeability of individual iron filings causes the magnetic field to be larger at the ends of the filings. This causes individual filings to attract each other, forming elongated clusters that trace out the appearance of lines. It would not be expected that these "lines" be precisely accurate field lines for this magnet; rather, the magnetization of the iron itself would be expected to alter the field somewhat.

A magnetic field is the term used in physics to describe the invisible forces in the space around a magnet, which pull on iron and other magnets. Mathematically, it is a vector field which surrounds magnets and electric currents, and is detected by the force it exerts on moving electric charges and on magnetic materials. When placed in a magnetic field, magnetic dipoles tend to align their axes parallel to the magnetic field. Magnetic fields also have their own energy with an energy density proportional to the square of the field intensity.

For the physics of magnetic materials, see magnetism and magnet, and more specifically ferromagnetism, paramagnetism, and diamagnetism. For constant magnetic fields, such as are generated by magnetic materials and steady currents, see magnetostatics. A changing electric field results in a magnetic field, and a changing magnetic field also generates a electric field (see electromagnetism).

In special relativity, the electric field and magnetic field are two interrelated aspects of a single object, called the electromagnetic field. A pure electric field in one reference frame is observed as a combination of both an electric field and a magnetic field in a moving reference frame.

[edit] B and H

The term magnetic field is used for two different vector fields, denoted B and H,^[1] although there are many alternative names for both (see sidebar). To avoid confusion, this article uses B-field and H-field for these fields, and uses magnetic field where either or both fields apply.

The B-field can be defined in many equivalent ways based on the effects it has on its environment. For instance, a particle having an electric charge, q, and moving in a B-field with a velocity, v, experiences a force, F, called the Lorentz force (see below). In SI units, the Lorentz force equation is

where × is the vector cross product. The B-field is measured in teslas in SI units and in gauss in cgs units.

Technically, B is a pseudovector (also called an axial vector). (This is a technical statement about how the magnetic field behaves when you reflect the world in a mirror.) This fact is apparent from the above definition of B.

Although views have shifted over the years, B is now understood as being the fundamental quantity, while H is a derived field. It is defined as a modification of B due to material media such that (in SI):

(definition of H )

where M is the magnetization of the material and μ₀ is the magnetic constant.^[2] The H-field is measured in amperes per meter (A/m) in SI units and in oersteds (Oe) in cgs units.^[3]

In materials for which M is proportional to B the relationship between B and H can be cast into the simpler form: H = B ⁄ μ, where μ is a material dependent parameter called the permeability. In free space, there is no magnetization M so that H = B ⁄ μ₀ (free space). For many materials, though, there is no simple relationship between B and M. For example, ferromagnetic materials and superconductors have a magnetization that is a multiple-valued function of B due to hysteresis.^[4]

See History of B and H below for further discussion.

[edit] The magnetic field and permanent magnets

Permanent magnets are objects that produce their own persistent magnetic fields. All permanent magnets have both a north and a south pole. Like poles repel and opposite poles attract. Permanent magnets are made of ferromagnetic materials such as iron and nickel that have been magnetized. The strength of a magnet is represented by its magnetic moment, m; for simple magnets, m points in the direction of a line drawn from the south to the north pole of the magnet. For more details about magnets see magnetization below and the article ferromagnetism.

[edit] Force on a magnet due to a non-uniform B

Like magnetic poles brought near each other repel while opposite poles attract. This is a specific example of a general rule that magnets are attracted (or repulsed depending on the orientation of the magnet) to regions of higher magnetic field. For example, opposite poles attract because the magnetic moment m of each magnet is in the same direction as the magnetic field B of the other, pulling each into the larger magnetic field near the poles.

Mathematically, the force on a magnet having a magnetic moment m is:^[5]

where the gradient ∇ is the change of the quantity m·B per unit distance and the direction is that of maximum increase of m·B. (The dot product m·B = |m||B|cos(θ), where | | represent the magnitude of the vector and θ is the angle between them.) This equation is strictly only valid for magnets of zero size, but it can often be used as an approximation for not too large magnets. The magnetic force on larger magnets is determined by dividing them into smaller regions having their own m then summing up the forces on each of these regions.

Reversing the direction of m reverses the resultant force. Magnets with m opposite to B are pushed into regions of lower magnetic field, provided that the magnet, and therefore, m does not flip due to magnetic torque. The ability of a nonuniform magnetic field to sort differently oriented dipoles is the basis of the Stern-Gerlach experiment, which established the quantum mechanical nature of the magnetic dipoles associated with atoms and electrons.^[6][7]

The force between two magnets is quite complicated and depends on the orientation of both magnets and the distance of the magnets relative to each other. The force is particularly sensitive to rotations of the magnets due to magnetic torque.

In many cases, both the force and the torque on a magnet can be modeled quite well by assuming a 'magnetic charge' at the poles of each magnet and using a magnetic equivalent to Coulomb's law. In this useful but physically incorrect and sometimes misleading model each magnetic pole is a source of a magnetic field that is stronger near the pole. An external magnetic field exerts a force in the direction of the magnetic field for a north pole and in the opposite direction for the south pole. In a nonuniform magnetic field, each pole sees a different field and is subject to a different force. The difference in the two forces moves the magnet in the direction of increasing magnetic field and may also cause a net torque.

Unfortunately, the idea of "poles" does not accurately reflect what happens inside a magnet (see ferromagnetism). For instance, a small magnet placed inside of a larger magnet feels a force in the opposite direction. The more physically correct description of magnetism involves atomic sized loops of current distributed throughout the magnet.^[8]

[edit] Torque on a magnet due to a B-field

A magnet placed in a magnetic field feels a torque that tries to align the magnet with the magnetic field. The torque on a magnet due to an external magnetic field is easy to observe by placing two magnets near each other while allowing one to rotate. The torque N on a small magnet is proportional both to the applied B-field and to the magnetic moment m of the magnet:

where × represents the vector cross product.

The alignment of a magnet with the magnetic field of the Earth is how compasses work. It is used to determine the direction of a local magnetic field as well (see below). A small magnet is mounted such that it is free to turn (in a given plane) and its north pole is marked. By definition, the direction of the local magnetic field is the direction that the north pole of a compass (or of any magnet) tends to point.

Magnetic torque is used to drive simple electric motors. In one simple motor design, a magnet is fixed to a freely rotating shaft (forming a rotor) and subjected to a magnetic field from an array of electromagnets —called the stator. By continuously switching the electrical current through each of the electromagnets and thereby flipping the polarity of their magnetic field the stator keeps like poles next to the rotor; The resultant magnetic torque is then transferred to the shaft. The inverse process, changing mechanical motion to electrical energy, is accomplished by the inverse of the above mechanism in the electric generator.

See Rotating magnetic fields below for an example using this effect with electromagnets.

[edit] Visualizing the magnetic field using field lines

Magnetic field lines shown by iron filings. The field lines are not precisely the same as the isolated magnet; the magnetization of the filings alter the field somewhat.

Mapping out the strength and direction of the magnetic field is simple in principle. First, measure the strength and direction of the magnetic field at a large number of locations. Then mark each location with an arrow (called a vector) pointing in the direction of the local magnetic field with a length proportional to the strength of the magnetic field. An alternative method of visualizing the magnetic field which greatly simplifies the diagram while containing the same information is to 'connect' the arrows to form "magnetic field lines".

Various physical phenomena have the effect of displaying magnetic field lines. For example, iron filings placed in a magnetic field line up in such a way as to visually show the orientation of the magnetic field (see figure at top). Another place where magnetic fields are visually displayed is in the polar auroras, in which visible streaks of light line up with the local direction of Earth's magnetic field (due to plasma particle dipole interactions). In these phenomena, lines or curves appear that follow along the direction of the local magnetic field.

These field lines provide a simple way to depict or draw the magnetic field (or any other vector field). ^[9] The magnetic field can be estimated at any point (whether on a field line or not) by looking at the direction and density of the field lines nearby.

Field lines are also a good qualitative tool for visualizing 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. 'Unlike' poles of magnets attract because they are linked by many field lines; 'like' poles repel because their field lines do not meet, but run parallel, pushing on each other.

The direction of the magnetic field near the poles of a magnet is revealed by placing compasses nearby. As seen here, the magnetic field points towards a magnet's south pole and away from its north pole.

Magnetic field lines also have a direction that can be revealed using a compass. A compass placed near the north pole of a magnet will point away from that pole—like poles repel. The opposite occurs for a compass placed near a magnet's south pole. The magnetic field points away from a magnet near its north pole and towards a magnet near its south pole. Magnetic field lines outside of a magnet point from the north pole to the south. Not all magnetic fields are describable in terms of poles, though. A straight current-carrying wire, for instance, produces a magnetic field that points neither towards nor away from the wire, but encircles it instead.

[edit] B-field lines always form closed loops

Field lines are a useful way to represent any vector field and often reveal sophisticated properties of fields quite simply. One important property of the B-field that can be verified with field lines is that magnetic field lines always make complete loops. Magnetic field lines neither start nor end (although they can extend to or from infinity). To date no exception to this rule has been found. (See magnetic monopole below.)

Since magnetic field lines always come in loops, magnetic poles always come in N and S pairs. Magnetic field leaves a magnet near its north pole and enters the magnet near its south pole but inside the magnet the magnetic field continues from the south pole back to the north.^[10] If a magnetic field line enters a magnet somewhere it has to leave the magnet somewhere else; it is not allowed to have an end point. For this reason as well, cutting a magnet in half results in two separate magnets each with both a north and a south pole.

[edit] Magnetic monopole (hypothetical)

A magnetic monopole is a hypothetical particle (or class of particles) that has, as its name suggests, only one magnetic pole (either a north pole or a south pole). In other words, it would possess a "magnetic charge" analogous to electric charge.

Modern interest in this concept stems from particle theories, notably Grand Unified Theories and superstring theories, that predict either the existence or the possibility of magnetic monopoles. These theories and others have inspired extensive efforts to search for monopoles. Despite these efforts, no magnetic monopole has been observed to date.^[11]

[edit] The magnetic field and electrical currents

Currents of electrical charges both generate a magnetic field and feel a force due to magnetic B-fields.

[edit] Magnetic field due to moving charges and electrical currents

All moving charges produce a magnetic field.^[12] For instance a moving point charge produces a complicated but well known magnetic field that is a function of the charge, velocity, and acceleration of the particle.^[13] It forms closed loops around a line pointing in the direction the charge is moving.

Current (I) through a wire produces a magnetic field (B) around the wire. The field is oriented according to the right hand grip rule.

Current carrying wires generate magnetic field lines that form concentric circles around the wire. The direction of the magnetic field in these loops is determined by the right hand grip rule. When moving along the current, to the left the magnetic field points up while to the right it points down. (See figure to the right.) The strength of the magnetic field decreases with distance from the wire.

Bending a current carrying wire into a loop concentrates the magnetic field inside the loop and weakens it outside. Stacking many such loops to form a solenoid (or long coil) enhances this effect. Such devices, called electromagnets, are important because they generate strong well controlled magnetic fields. An infinitely long electromagnet has a uniform magnetic field inside and no magnetic field outside. A finite length electromagnet produces essentially the same magnetic field as a uniform permanent magnet of the same shape and size with a strength (and polarity) that is controlled by the input current. For example, one class of electric motors continually switches the polarity of stationary electromagnets to apply torque to a rotating permanent magnet using the fact that like poles repel.

The magnetic field generated by a steady current I (a constant flow of charges in which charge is neither accumulating nor depleting at any point) is described by^[14] the Biot-Savart law:

where the integral sums over the entire loop of a wire with dl a particular infinitesimal piece of that loop, μ₀ is the magnetic constant, r is the distance between the location of dl and the location at which the magnetic field is being calculated, and is a unit vector in the direction of r.

A slightly more general^[15][16] way of relating the current I to the B-field is through Ampère's law:

where the integral is over any arbitrary loop and I_enc is the current enclosed by that loop. 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.

[edit] Force due to a B-field on a moving charge

[edit] Force on a charged particle

Beam of electrons moving in a circle. Lighting is caused by excitation of atoms of gas in a bulb.

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. This force is known as the Lorentz force, and is given by

where

F is the force (in newtons)

q is the electric charge of the particle (in coulombs)

v is the instantaneous velocity of the particle (in meters per second)

B is the magnetic field (in teslas).

The force is always perpendicular to both the velocity of the particle and the magnetic field that created it. Neither a stationary particle nor one moving in the direction of the magnetic field lines experiences a force. For that reason, charged particles move in a circle (or more generally, in a helix) around magnetic field lines; this is called cyclotron motion. Because the magnetic force is always perpendicular to the motion, the magnetic fields can do no work on an isolated charge. It can and does, however, change the particle's direction, even to the extent that a force applied in one direction can cause the particle to drift in a perpendicular direction. (See figure.) The magnetic force can do work to a magnetic dipole, or to a charged particle whose motion is constrained by other forces^{[citation needed]}.

[edit] Force on current-carrying wire

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

[edit] Direction of force

The right-hand rule: With the thumb of the right hand pointing in the direction of the conventional current or moving positive charge and the fingers pointing in the direction of the magnetic field the force on the current is in a direction out of the palm. The direction of the force is reversed for a negative charge.

The direction of force on a positive charge or a current is determined by the right-hand rule. See the figure on the right. Using the right hand and pointing the thumb in the direction of the moving positive charge or positive current and the fingers in the direction of the magnetic field the resulting force on the charge points outwards from the palm. The force on a negative charged particle is in the opposite direction. If both the speed and the charge are reversed then the direction of the force remains the same. For that reason a magnetic field measurement (by itself) cannot distinguish whether there is a positive charge moving to the right or a negative charge moving to the left. (Both of these cases produce the same current.) On the other hand, a magnetic field combined with an electric field can distinguish between these, see Hall effect below.

An alternative, similar trick to the right hand rule is Fleming's left hand rule.

[edit] The H-field

In the formulation of Maxwell's equations at a microscopic level where all charges and currents are treated explicitly, only the E- and B-fields occur. On the other hand, when charges and currents are divided into "free" and "bound" categories, D- and H-fields are used, with the H-field determined by the "free" current and time rate of change of D.^[17] Thus, when the "free" and "bound" division of currents and charges is introduced, the H-field appears and simplifies the equations for the magnetic field because microscopic details of the B- and E-fields inside materials can be treated separately as problems of condensed-matter physics. The H-field is defined as:

(definition of H in SI units)

(definition of H in cgs units)

where M is magnetization density of any magnetic material. H is measured in amperes per meter (A/m) in SI and in oersteds (Oe) for cgs. In SI units, μ₀ is a defined constant called the magnetic constant (μ₀ = 4π × 10⁻⁷ Tm/A).

[edit] Magnetization

Hierarchy of types of magnetism. See Myers.^[18]

Materials placed in a magnetic field can become magnetized. Magnetization is due to the accumulated effect of many tiny magnetic dipole moments that occur on the atomic level. In non-magnetized materials, the magnetic dipoles align randomly such that the net magnetic moment cancels producing no net magnetic field. But, if the magnetic dipoles of the material becomes aligned a net magnetization and magnetic field is produced. The magnetization field M represents how strongly a region is magnetized and is defined as the volume density of the net magnetic dipole moment in that region of material.

An equivalent way to represent magnetization is to add all of the currents of the dipole moments that produce the magnetization. The resultant current is called bound current and is the source of the magnetic field due to the magnet. Mathematically, the curl of M equals the bound current. Unlike B, though, magnetization must begin and end near the poles. (There is no magnetization outside of the material.) Therefore, the divergence of M must be non-zero near the poles of a magnet.

Most materials produce a magnetization in response to an applied B-field. Typically, the response is very weak and exists only when the magnetic field is applied. Materials are divided into groups based upon their magnetic behavior:

• Diamagnetic materials^[19] produce a magnetization that opposes the magnetic field.
• Paramagnetic materials^[19] produce a magnetization in the same direction as the applied magnetic field.
• Ferromagnetic materials and the closely related ferrimagnetic materials and antferromagnetic materials^[20][21] can have a magnetization independent of an applied B-field with a complex relationship between the two fields.
• Superconductors (and ferromagnetic superconductors)^[22][23] are materials that are characterized by perfect conductivity below a critical temperature and magnetic field. They also are highly magnetic and can be perfect diamagnets below a lower critical magnetic field. Superconductors often have a broad range of temperatures and magnetic fields (the so named mixed state) for which they exhibit a complex hysteretic dependence of M on B.

In the case of paramagnetism, and diamagnetism the B-field often is proportional to the H-field such that:

where μ is a material dependent parameter called the permeability (see constitutive equations). 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 hysteresis. In all cases the original definitions of H in terms of B and M still are valid.

The advantage of the H-field is that its bound sources are treated so differently that they can often be isolated from the free sources. For example, a line integral of the H-field in a closed loop will yield the total free current in the loop (not including the bound current). Similarly, a surface integral of H over any closed surface picks out the 'magnetic charges' within that closed surface.

Magnetic field lines around a ”magnetostatic dipole” the magnetic dipole itself is in the center and is seen from the side.

[edit] Magnetic dipoles

The magnetic field of an ideal magnetic dipole is depicted on the right. As discussed in magnetic moment, however, due to the inherent connection between angular momentum and magnetism, magnetic dipoles in actual materials are not ideal magnetic dipoles. The connection between angular momentum and magnetism is the basis of the Einstein-de Haas effect "rotation by magnetization" and its inverse, the Barnett effect or "magnetization by rotation".^[24]

The magnetic field of permanent magnets and of all magnetic material originate at the atomic level.

[edit] Energy stored in magnetic fields

In asking how much energy does it take to create a specific magnetic field using a particular current it is important to distinguish between free and bound currents. It is the free current that we directly 'push' on to create the magnetic field. The bound currents create a magnetic field that the free current has to work against without doing any of the work.

It is not surprising, therefore, that the H-field is important in magnetic energy calculations since it treats the two sources differently. In general the incremental amount of work per unit volume δW needed to cause a small change of magnetic field δB is:

If there are no magnetic materials around then we can replace H with B ⁄ μ₀,

For linear materials (such that B = μH ), the energy density can be expressed as:

(Valid only for linear materials)

Nonlinear materials cannot use the above equation but must return to the first equation which is always valid. In particular, the energy density stored in the fields of hysteretic materials such as ferromagnets and superconductors depends on how the magnetic field was created.

[edit] Electromagnetism: the relationship between magnetic and electric fields

[edit] The magnetic field due to a changing electric field

A changing electric field generates a magnetic field proportional to the time rate of the change of the electric field. This fact is known as Maxwell's correction to Ampere's Law. Therefore the full Ampere's Law is:

where J is the current density, and partial derivatives indicate spatial location is fixed when the time derivative is taken. The last term is Maxwell's correction. This equation is valid even when magnetic materials are involved, but in practice it is often easier to use an alternate equation.

[edit] Electric force due to a changing B-field

Above is a discussion of how a changing E-field creates a B-field. The inverse process also occurs: a changing magnetic field, such as a magnet moving through a stationary coil, generates an electric field (and therefore tends to drive a current in the coil). (These two effects bootstrap together to form electromagnetic waves, such as light.) This is known as Faraday's Law and forms the basis of many electric generators and electrical motors.

Faraday's law is commonly represented as:

where is the electromotive force or EMF (the voltage generated around a closed loop) and Φ_m is the magnetic flux—the product of the area times the magnetic field normal to that area. (This definition of magnetic flux is why engineers often refer to B as "magnetic flux density".) This law includes both flux changes because of the magnetic field generated by a time varying E-field (transformer EMF) and flux changes because of movement through a magnetic field (motional EMF).

A form of Faraday's law of induction that does not include motional EMF is the Maxwell-Faraday equation:

one of Maxwell's equations. This equation is valid even in the presence of magnetic material.^[25]

[edit] Maxwell's equations

Like all vector fields the B-field has two important mathematical properties that relates it to its sources. 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.

The first property is the divergence of a vector field A, ∇ · A which represents how A 'flows' outward from a given point. As discussed above a B-field line never starts nor 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' law for magnetism and is equivalent to the statement that there are no magnetic charges or magnetic monopoles. The electric field on the other hand begins and ends at electrical charges so that its divergence is non-zero and proportional to the charge density (See Gauss' law).

The second mathematical property is called the curl, ∇ × such that ∇ × A represents how A curls or 'circulates' around a given point. The result of the curl is called a 'circulation source' The curl of B and of E are given above and are called the Ampère-Maxwell equation and Faraday's law respectively.

The complete set of Maxwell's equations then are:

where J = complete microscopic current density and ρ is the charge density.

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 the auxillary H and D fields are defined so that Maxwell's equations can be re-factored in terms of the free current density J_f and free charge density ρ_f:

These equations are not any more general then the original equations (if the 'bound' charges and currents in the material are known'). They also need to 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.

[edit] Electric and magnetic fields: different aspects of the same phenomenon

According to special 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 is perceived by another (in a different frame of reference) as a mixture of electric and magnetic forces. (Too, a magnetic force in one reference frame is perceived as a mixture of electric and magnetic forces in another.)

More specifically, 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.

[edit] Measuring the B-field

Devices used to measure the local magnetic field are called magnetometers. Important classes of magnetometers include using a rotating coil, Hall effect magnetometers, NMR magnetometer, SQUID magnetometer, and a fluxgate magnetometer. The magnetic fields of distant astronomical objects can be determined by noting their effects on local charged particles. For instance, electrons spiraling around a field line produce synchotron radiation which is detectable in radio waves.

The smallest magnetic field measured^[26] is on the order of attoteslas (10^-18 tesla); the largest magnetic field produced in a laboratory is 2,800 T (VNIIEF in Sarov, Russia, 1998)^[27] The magnetic field of some astronomical objects such as magnetars are much higher; magnetars range from 0.1 to 100 GT (10⁸ to 10¹¹ T).^[28] See orders of magnitude (magnetic field).

[edit] History

The modern understanding that the B-field is the more fundamental field with the H-field being an auxiliary field was not easy to arrive at. Indeed, largely because of mathematical similarities to the electric field, the H-field was developed first and was thought at first to be the more fundamental of the two. A brief history of this important transition in thought is instructional in giving insight into the nature of both H and B.

Perhaps the earliest description of a magnetic field was performed by Petrus Peregrinus and published in his “Epistola Petri Peregrini de Maricourt ad Sygerum de Foucaucourt Militem de Magnete” and is dated 1269 A.D. Petrus Peregrinus mapped out the magnetic field on the surface of a spherical magnet. Noting that the resulting field lines crossed at two points he named those points 'poles' in analogy to Earth's poles. Almost three centuries later, near the end of the sixteenth century, William Gilbert of Colchester replicated Petrus Peregrinus work and was the first to state explicitly that Earth itself was a magnet. William Gilbert's great work De Magnete was published in 1600 A.D. and helped to establish the study of magnetism as a science.

The modern distinction between the B- and H- fields does not become important until Siméon-Denis Poisson (1781–1840) developed one of the first mathematical theories of magnetism. Poisson's model, developed in 1824, assumed that magnetism was due to magnetic charges. In analogy to electric charges, these magnetic charges produce a H-field. In modern notation, Poisson's model was exactly analogous to electrostatics with the H-field replacing the electric field E-field and the B-field replacing the auxiliary D-field.

Poisson's model was, unfortunately, incorrect. Magnetism is not due to magnetic charges. Nor is magnetism created by the H-field polarizing magnetic charge in a material. The model, however, was remarkably successful for being fundamentally wrong. It predicts the correct relationship between the H-field and the B-field, even though it wrongly places H as the fundamental field with B as the auxiliary field. It predicts the correct forces between magnets.

It even predicts the correct energy stored in the magnetic fields. By the definition of magnetization, in this model, and in analogy to the physics of springs, the work done per unit volume, in stretching and twisting the bonds between magnetic charge to increment the magnetization by μ₀δM is W = H · μ₀δM. In this model, B = μ₀ (H + M ) is an effective magnetization which includes the H-field term to account for the energy of setting up the magnetic field in a vacuum. Therefore the total energy density increment needed to increment the magnetic field is W = H · δB. This is the correct result, but it is derived from an incorrect model.

In retrospect the success of this model is due largely to the remarkable coincidence that from the 'outside' the field of an electric dipole has the exact same form as that of a magnetic dipole. It is therefore only for the physics of magnetism 'inside' of magnetic material where the simpler model of magnetic charges fails. It is also important to note that this model is still useful in many situations dealing with magnetic material. One example of its utility is the concept of magnetic circuits.

The formation of the correct theory of magnetism begins with a series of revolutionary discoveries in 1820, four years before Poisson's model was developed. (The first clue that something was amiss, though, was that unlike electrical charges magnetic poles cannot be separated from each other or form magnetic currents.) The revolution began when Hans Christian Oersted discovered that an electrical current generates a magnetic field that encircles the wire. In a quick succession that discovery was followed by Andre Marie Ampere showing that parallel wires having currents in the same direction attract, and by Jean-Baptiste Biot and Felix Savart developing the correct equation, the Biot-Savart Law, for the magnetic field of a current carrying wire. In 1825, Ampere extended this revolution by publishing his Ampere's Law which provided a more mathematically subtle and correct description of the magnetic field generated by a current than the Biot-Savart Law.

Subsequent development in the nineteenth century interlinked magnetic and electric phenomena even tighter, until the concept of magnetic charge was not needed. Magnetism became an electric phenomenon with even the magnetism of permanent magnets being due to small loops of current in their interior. This development was aided greatly by Michael Faraday, who in 1831 showed that a changing magnetic field generates an encircling electric field.

In 1861, James Clerk-Maxwell wrote a paper entitled 'On Physical Lines of Force' [2] in which he attempted to explain Faraday's magnetic lines of force in terms of a sea of tiny molecular vortices. These molecular vortices occupied all space and they were aligned in a solenoidal fashion such that their rotation axes traced out the magnetic lines of force. When two like magnetic poles repel each other, the magnetic lines of force spread outwards from each other in the space between the two poles. Maxwell considered that magnetic repulsion was the consequence of a lateral pressure between adjacent lines of force, due to centrifugal force in the equatorial plane of the molecular vortices. When deriving the equation for magnetic force in part I of his 1861 paper, Maxwell used a quantity which was closely related to the circumferential speed of the vortices. This quantity was therefore a measure of the vorticity in the magnetic lines of force, and Maxwell referred to it as the intensity of the magnetic force. In the 1861 paper, the magnetic intensity which we denote as v, was always multiplied by the term μ as a weighting for the cross sectional density of the lines of force. The quantity v corresponds reasonably closely to the modern magnetic field vector H, and the product μv corresponds very closely to the modern magnetic flux density B, where μ is referred to as the magnetic permeability.

Although the classical theory of electrodynamics was essentially complete with Maxwell's equations, the twentieth century saw a number of improvements and extensions to the theory. Albert Einstein, in his great paper of 1905 that established relativity, showed that both the electric and magnetic fields were part of the same phenomena viewed from different reference frames. Finally, the emergent field of quantum mechanics was merged with electrodynamics to form quantum electrodynamics or QED.

In the late nineteenth century the moving magnet and conductor problem developed as an important thought experiment that eventually helped Albert Einstein to develop special relativity. This thought experiment revolves around the interpretation of Faraday's law, as explained next:

Imagine a conducting loop moving relative to a magnet as seen by two different observers: one on the magnet the other on the loop. Both observers see the identical EMF generated in the coil using the flux form of Faraday's law, but explain the result using two different reasons. The observer on the magnet sees the magnet as stationary with an unchanging magnetic field, while the conducting loop moves. All of the charges within the loop move with the loop, and due to the B-field experience a sideways Lorentz force, which generates the EMF. On the other hand, an observer on the loop sees a changing magnetic field due to a moving magnet (relative to the loop's reference frame) and no Lorentz force (charges in the loop are not moving). This changing magnetic field means ∂B / ∂t ≠ 0, which creates an electric field that generates the current.

Prior to special relativity, it was customary to draw a sharp distinction between these two cases; a stationary magnet and a moving loop only produces motional EMF due to the Lorentz force from the B-field, while a moving magnet through a stationary loop produces only transformer EMF due to the electric field E generated by a changing B. See Faraday's law as two different phenomena. Einstein, on the other hand, proposed the equivalence of these two scenarios^[29] in the first postulate of relativity that the physics depends on only relative motion. Motional EMF and transformer EMF, therefore are the same phenomenon as seen in different reference frames. Likewise, the same is true of E and B, which are not separate, but are aspects of the same electromagnetic tensor.

[edit] Important uses and examples of magnetic field

[edit] Magnetic circuits

An important use of H is in magnetic circuits where inside a linear material B = μ H. Here, μ is the magnetic permeability of the material. This result is similar in form to Ohm's Law J = σ E, where J is the current density, σ is the conductance and E is the electric field. Extending this analogy we derive the counterpart to the macroscopic Ohm's law ( I = V ⁄ R ) as:

where is the magnetic flux in the circuit, is the magnetomotive force applied to the circuit, and R_m is the reluctance of the circuit. Here the reluctance R_m is a quantity similar in nature to resistance for the flux.

Using this analogy it is straight-forward to calculate the magnetic flux of complicated magnetic field geometries, by using all the available techniques of circuit theory.

[edit] Hall effect

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).

[edit] Magnetic field shape descriptions

Schematic quadrupole magnet ("four-pole") magnetic field. There are four steel pole tips, two opposing magnetic north poles and two opposing magnetic south poles.

• An azimuthal magnetic field is one that runs east-west.

• A meridional magnetic field is one that runs north-south. In the solar dynamo model of the Sun, differential rotation of the solar plasma causes the meridional magnetic field to stretch into an azimuthal magnetic field, a process called the omega-effect. The reverse process is called the alpha-effect.^[30]

• A dipole magnetic field is one seen around a bar magnet or around a charged elementary particle with nonzero spin.

• A quadrupole magnetic field is one seen, for example, between the poles of four bar magnets. The field strength grows linearly with the radial distance from its longitudinal axis.

• A solenoidal magnetic field is similar to a dipole magnetic field, except that a solid bar magnet is replaced by a hollow electromagnetic coil magnet.

• A toroidal magnetic field occurs in a doughnut-shaped coil, the electric current spiraling around the tube-like surface, and is found, for example, in a tokamak.

• A poloidal magnetic field is generated by a current flowing in a ring, and is found, for example, in a tokamak.

• A radial magnetic field is one in which the field lines are directed from the center outwards, similar to the spokes in a bicycle wheel. An example can be found in a loudspeaker transducers (driver).^[31]

• A helical magnetic field is corkscrew-shaped, and sometimes seen in space plasmas such as the Orion Molecular Cloud.^[32]

[edit] Earth's magnetic field

A sketch of Earth's magnetic field representing the source of Earth's magnetic field as a magnet. The north pole of earth is near the top of the diagram, the south pole near the bottom. Notice that the south pole of that magnet is deep in Earth's interior below Earth's North Magnetic Pole. Earth's magnetic field is produced in the outer liquid part of its core due to a dynamo that produce electrical currents there.

Because of Earth's magnetic field, a compass placed anywhere on Earth will turn so that the "north pole" of the magnet inside the compass points roughly north, toward Earth's north magnetic pole in northern Canada. This is the traditional definition of the "north pole" of a magnet, although other equivalent definitions are also possible. One confusion that arises from this definition is that if Earth itself is considered as a magnet, the south pole of that magnet would be the one nearer the north magnetic pole, and vice-versa. (Opposite poles attract and the north pole of the compass magnet is attracted to the north magnetic pole.) The north magnetic pole is so named not because of the polarity of the field there but because of its geographical location.

The figure to the right is a sketch of Earth's magnetic field represented by field lines. The magnetic field at any given point does not point straight toward (or away) from the poles and has a significant up/down component for most locations. (In addition, there is an East/West component as Earth's magnetic poles do not coincide exactly with Earth's geological pole.) The magnetic field is as if there were a magnet deep in Earth's interior.

Earth's magnetic field is probably due to a dynamo that produces electric currents in the outer liquid part of its core. Earth's magnetic field is not constant: Its strength and the location of its poles vary. The poles even periodically reverse direction, in a process called geomagnetic reversal.

[edit] Rotating magnetic fields

The rotating magnetic field is a key principle in the operation of alternating-current motors. A permanent magnet in such a field will rotate 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. 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, in order to overcome it, three-phase systems are used where the three currents are equal in magnitude and have 120 degrees phase difference. Three similar coils having mutual geometrical angles of 120 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.

Because magnets degrade with time, synchronous motors 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 1882, Nikola Tesla identified the concept of the rotating magnetic field. In 1885, Galileo Ferraris independently researched the concept. In 1888, Tesla gained U.S. Patent 381,968 for his work. Also in 1888, Ferraris published his research in a paper to the Royal Academy of Sciences in Turin.

[edit] See also

General

• Electric field — field produced by electric charges and changing magnetic fields that affects charged particles.
• Electromagnetic field — a field composed of the electric field and the magnetic field.
• Electromagnetism — the physics of the electromagnetic field.
• Faraday's law of induction — the connection between electric and magnetic fields as found in motors and generators
• Lorentz force — the connection between fields and forces
• Magnetism — phenomenon by which materials exert a magnetic force on other materials.
• Magnetohydrodynamics — the study of the dynamics of electrically conducting fluids.
• Magnetic flux — amount of 'magnetic field' through a given loop.
• Magnetic monopole — hypothetical particle which causes nonzero divergence of magnetic field.
• Magnetic nanoparticles — extremely small magnetic particles that are tens of atoms wide
• Magnetic reconnection — an effect which causes solar flares and auroras.
• Magnetic potential — the vector and scalar potential representation of magnetism.
• SI electromagnetism units — common units used in electromagnetism.
• Orders of magnitude (magnetic field) — list of magnetic field sources and measurement devices from smallest magnetic fields to largest detected.

Mathematics

• Ampère's law — law describing how currents act as circulation sources for magnetic fields.
• Biot-Savart law — the magnetic field set up by a steadily flowing line current.
• Magnetic helicity — extent to which a magnetic field "wraps around itself".
• Maxwell's equations — four equations describing the behavior of electric and magnetic fields and their interaction with matter.

Applications

• Dynamo theory — a proposed mechanism for the creation of the Earth's magnetic field.
• Earth's magnetic field — a discussion of the magnetic field of the Earth.
• Electric motor — AC motors used magnetic fields.
• Helmholtz coil — a device for producing a region of nearly uniform magnetic field.
• Magnetic field viewing film — Film used to view the magnetic field of an area.
• Maxwell coil — a device for producing a large volume of almost constant magnetic field.
• Stellar magnetic field — a discussion of the magnetic field of stars.
• Teltron Tube — device used to display an electron beam and demonstrates effect of electric and magnetic fields on moving charges.

[edit] Notes

[edit] Further reading

[edit] External links

Information

• Crowell, B., "Electromagnetism".

• Nave, R., "Magnetic Field". HyperPhysics.

• "Magnetism", The Magnetic Field. theory.uwinnipeg.ca.

• Hoadley, Rick, "What do magnetic fields look like?" 17 July 2005.

Field density

• Jiles, David (1994). Introduction to Electronic Properties of Materials (1st ed.). Springer. ISBN 0-412-49580-5.

Rotating magnetic fields

• "Rotating magnetic fields". Integrated Publishing.

• "Introduction to Generators and Motors", rotating magnetic field. Integrated Publishing.

• "Induction Motor-Rotating Fields".

Diagrams

• McCulloch, Malcolm,"A2: Electrical Power and Machines", Rotating magnetic field. eng.ox.ac.uk.

• "AC Motor Theory" Figure 2 Rotating Magnetic Field. Integrated Publishing.

Journal Articles

• Yaakov Kraftmakher, "Two experiments with rotating magnetic field". 2001 Eur. J. Phys. 22 477-482.

• Bogdan Mielnik and David J. Fernández C., "An electron trapped in a rotating magnetic field". Journal of Mathematical Physics, February 1989, Volume 30, Issue 2, pp. 537–549.

• Sonia Melle, Miguel A. Rubio and Gerald G. Fuller "Structure and dynamics of magnetorheological fluids in rotating magnetic fields". Phys. Rev. E 61, 4111 – 4117 (2000).