User:TStein/Magnetic field



In physics, a magnetic field is a field that permeates space and which can exert a magnetic force on moving electric charges and on magnetic dipoles (such as permanent magnets). When placed in a magnetic field, magnetic dipoles tend to align their axes to be parallel with the magnetic field, as can be seen when iron filings are in the presence of a magnet (see picture at right). In addition, a changing magnetic field can induce an electric field. Magnetic fields surround and are created by electric currents, magnetic dipoles, and changing electric fields. Magnetic fields also have their own energy, with an energy density proportional to the square of the field intensity.

There are some notable specific instances of the magnetic field. 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 stationary dipoles and steady currents, see magnetostatics. For magnetic fields created by changing electric fields, see electromagnetism.

The electric field and the magnetic field are tightly interlinked, in two senses. First, changes in either of these fields can cause ("induce") changes in the other, according to Maxwell's equations. Second according to Einstein's theory of special relativity, a magnetic force in one inertial frame of reference may be an electric force in another, or vice-versa (see relativistic electromagnetism for examples). Together, these two fields make up the electromagnetic field, which is best known for underlying light and other electromagnetic waves.

B and H
There are two quantities that physicists may refer to as the magnetic field, notated $$\mathbf{H}$$ and $$\mathbf{B}$$. Although the term "magnetic field" was historically reserved for $$\mathbf{H}$$, with $$\mathbf{B}$$ being termed the "magnetic induction", $$\mathbf{B}$$ is now understood to be the more fundamental entity. Most modern writers refer to $$\mathbf{B}$$ as the magnetic field. See: This article will follow that convention and will discuss the more fundamental $$\mathbf{B}$$ magnetic field, before treating the $\mathbf{H}$ field.

See History of B and H below for further discussion.

Alternative Names for B and H
The vector field $$\mathbf{H}$$ is known among electrical engineers as the magnetic field intensity or magnetic field strength also known as auxiliary magnetic field or magnetizing field. The vector field $$\mathbf{B}$$ is known as magnetic flux density or magnetic induction or simply magnetic field, as used by physicists, and has the SI units of teslas (T), equivalent to webers (Wb) per square meter or volt seconds per square meter. Magnetic flux has the SI units of webers so the $$\mathbf{B}$$ field is that of its areal density. The vector field $$\mathbf{H}$$ is measured in Amperes/meter (SI units) or oersted (cgs units).

Permanent Magnets and Magnetic Poles


Permanent magnets are objects that produce their own persistent magnetic fields. All permanent magnets have both a north and a south pole. (Magnetic poles always come in north-south pairs.)   Like poles repel and opposite poles attract. (See Force on a magnetic dipole due to a non-uniform B below.)  Permanent magnets may be thought of being made of many tiny magnets called magnetic dipoles that line up on average to create the magnetic field. Ferromagnetic materials such as iron or nickel can be used to form permanent magnets after these materials have been magnetized.

If allowed to twist a magnet will turn to point in the direction of the magnetic field at its location. (See Torque on a magnetic dipole below.)   A compass is a small magnet that uses this effect to point in the direction the local magnetic field. By definition, the direction of the magnetic field at a point is the direction that the north pole of a magnet wants to point.

If a compass is placed near the north pole of a magnet then it will point away from that pole---like poles repel. In other words, the magnetic field leaves a magnet near its north pole. The opposite occurs if we place the compass near a magnet's south pole; the magnetic field enters the magnet near its south pole. Not all magnetic fields are describable in terms of poles, though. A straight current carrying wire, for instance, produces a magnetic field that neither enters nor exits the wire but encircles it instead.

Visualizing the Magnetic Field
The strength and direction of the magnetic field due to an object varies from position to position. If one wanted to map out a magnetic field, they could in principle measure the strength and direction of the magnetic field at a large number of points. They could then mark each location with an arrow (called [[vector (spatial)|vector]s whose direction points in the direction of the magnetic field and whose length is proportional to the strength of the magnetic field. This is a valid and useful way of marking out and visualizing the magnetic field of an object.  It has the unfortunate consequence, though, of cluttering up a graph even when using a small number of points.  An alternative method of visualizing the magnetic field is to use magnetic field lines.

Magnetic B field lines
Various physical phenomena have the effect of displaying magnetic field line. For example, iron filings placed in a magnetic field will 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, the direction of the magnetic field is revealed by the direction that a magnetic dipole (such as a small magnet) will orient itself in that magnetic field (see definition below).

These field lines provide us with a way to depict or draw the magnetic field (or any other vector field). Technically, field lines are a set of lines through space whose direction at any point is the direction of the local magnetic field, and whose density is proportional to the magnitude of the local magnetic field. Note that when a magnetic field is depicted with field lines, it is not meant to imply that the field is only nonzero along the drawn-in field lines. The field is typically smooth and continuous everywhere, and 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. The choice of which field lines to draw in such a depiction is arbitrary, apart from the requirement that they be spaced out so that their density approximates the magnitude of the local field. The level of detail at which the magnetic field is depicted can be increased by increasing the number of lines.

This visualization is particularly helpful for the magnetic field, as it makes certain aspects of it more transparent. For example, consider the field lines around the magnet in the above figure. Notice that all the magnetic field lines start at one end of the magnet and end at the other. If we were able to trace out the entire field lines we would note that they extend through the magnet to form a complete loop. Examining the magnetic field around a current carrying wire we notice magnetic field lines the formed complete loops around the wire. In fact, this is a general rule about magnetic fields; they either make a complete circle or extend out to infinity. Magnetic field lines cannot have starting or ending points. This is mathematically equivalent to Gauss's law for magnetism which states that the magnetic field is solenoidal (has zero divergence). Further when we examine the magnetic field loops we will see that within the loops there is always some source that is creating the magnetic field. This source could be a current, a dipole or magnet, or a changing electric field but it is always within the loops of magnetic field they create.

Since magnetic field lines always come in loops, magnetic poles always come in N and S pairs. If a magnetic field line enters a magnet somewhere it has to leave the magnet somewhere else, since it is not allowed to end. For this reason as well, cutting a magnet in half will result in two separate magnets each with both a north and a south pole.

Pole labeling confusions


See also North Magnetic Pole and South Magnetic Pole.

Placed anywhere on Earth the north pole of a compass will point roughly North toward Earth's North Magnetic Pole. This may be considered the definition of the north pole of a magnet. The fact that the compass turn in this manner is proof that Earth has a magnetic field deep in its interior. This definition of the north pole of a magnet is unfortunate since the north pole of a compass points in a direction that leads to the 'south pole of a magnet. (Opposites attract.)  Earth's magnetic field is such that its south pole is directly beneath the north magnetic pole deep in its interior. The north magnetic pole is so named not because of the polarity of the field there but because of its geological location.

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

The 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. The Earth's magnetic field is currently getting weaker and maybe about to start this reversal process.

Definition of B
In classical physics, the magnetic field $$\mathbf{B}$$ is a vector field (that is, some vector at every point of space and time), with SI units of teslas (one tesla is one newton-second per coulomb-meter) and cgs units of gauss. It has the property of being a solenoidal vector field. (The term solenoidal means that the field lines always formed closed loops, as described above.)

The field $$\mathbf{B}$$ can be both defined and measured by means of a small magnetic dipole (i.e., bar magnet). The magnetic field exerts a torque on magnetic dipoles that tends to make them point in the same direction as the magnetic field (as in a compass), and moreover the magnitude of that torque is proportional to the magnitude of the magnetic field. Therefore, in order to measure the magnetic field at a particular point in space, you can put a small freely-rotating bar magnet (such as a compass) there: the direction it winds up pointing is the direction of $$\mathbf{B}$$; and the ratio of the maximum magnitude of the torque to the dipole moment of the bar magnet is the magnitude $$|\mathbf{B}|$$.

(There are, in addition, several other different but physically equivalent ways to define the magnetic field, for example via the Lorentz force law (see below), or as the solution to Maxwell's equations.)

It follows from any of these definitions that the magnetic field vector (being a vector product) is a pseudovector (also called an axial vector).

Effects of the Magnetic Field, B
The magnetic field at a point can reveal itself in 4 different ways.



Sideways force on a moving charge or current

 Torque on a magnetic dipole

 Force on a magnetic dipole due to a non-uniform B

 force due to a changing B





Force on a charged particle
A charged particle moving in a magnetic field will feel a sideways force that is proportional to the strength of the magnetic field, the component of the speed that is perpendicular to the magnetic field and the charge of the particle. This force is known as the Lorentz Force. The force is always perpendicular to both the speed of the particle and the magnetic field that created it. Neither a stationary particle nor one moving in the direction of the magnetic field line $$\mathbf{B}$$ will experience a force. For that reason, charged particles move in a helix along magnetic field lines. (The continuous sideways force due to the magnetic field will cause it to change direction continuously but not speed up nor slow down.) Because the magnetic field is always perpendicular to the motion magnetic fields can do no work on a moving charge. Therefore magnetic fields cannot change the energy of a simple charged particle. It can, however, change the direction of a particle, so that a force applied in one direction will cause the particle to drift in a perpendicular direction. (See above figure.)

Force on current-carrying wire
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 will feel 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.



Direction of force
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 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 will point 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 (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 will produce the same current.)  A magnetic field combined with an electric field can distinguish between current made of positive charge and that made of negative charge, but only by measuring the sign of the (sideways to the current) charge build up. (See the above figure part B, where the direction of the electric field corresponds to the direction of a moving current.) This phenomenon is known as the  Hall effect. Another similar trick to the right hand rule is the right hand grip rule.

Torque on a magnetic dipole
A magnet placed in a magnetic field will feel a torque that will try 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. This magnetic torque is the basis for how compasses work. It is used to define the direction of the magnetic field (see above).

The magnetic torque also provides the driving torque for simple electric motors. By attaching a magnet to a shaft we can cause the magnet to rotate by placing an external magnetic field that causes it to twist. The simplest way to do this is to place the north pole of an external magnet near the north pole of the rotating magnet (called the rotor) and a south pole of the external magnet near the south pole of the rotor. If we use an electromagnet we can continually flip the poles of the external stationary magnet (called the stator) causing the rotor to turn as long as the poles of the stator are being flipped.

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

Force on a magnetic dipole due to a non-uniform B
The most commonly experienced effect of the magnetic field is the force between two magnets. Like poles repel and opposites attract. One, in principle, could express this force in terms of the poles having a positive (north pole) or negative (south pole) charge. This model is called the Gilbert model and produces the correct $$\mathbf{B}$$ field outside of the magnet. Unfortunately, this model predicts the wrong magnetic field inside a magnet. (Due to its relative simplicity the Gilbert model is still useful as long as nothing depends on the magnetic $$\mathbf{B}$$ field inside a magnet.)

Expressing the same force between two magnets using the magnetic $$\mathbf{B}$$ field is more complicated. For example, a uniform magnetic field will not cause a force on an external magnet. The south pole of a magnet is attracted to the north pole of the magnet because the field is stronger closer to the magnet. The force on a magnetic dipole depends not on the strength nor direction of the magnetic $$\mathbf{B}$$ field but how it varies with location. A magnet will move to maximize the magnetic field in the direction of its magnetic moment.

Care should be taken to distinguish the magnetic force on a magnetic dipole from the magnetic force on a moving charge. The magnetic force on a charge only occurs when the charge is moving and is in a sideways direction. It is felt for uniform and non-uniform magnetic fields. The magnetic force on a dipole, on the other hand, is present only in non-uniform (in space) fields and is in the direction that increases magnetic field strength in the same direction as the magnetic moment of the magnet.. Neither does the force on a magnetic dipole depend on its speed.

Electric force due to a changing B
If the magnetic field in an area is varying with time it generates an electric field that forms closed loops around that area. A conducting wire that forms a closed loop around the area will have an induced voltage generated by this changing magnetic field. This effect is represented mathematically as Faraday's Law and forms the basis of generators. Indeed, Faraday's Law forms the basis of modern civilization in that it allows the conversion of mechanical energy to electrical energy. Care must be taken to understand that the changing magnetic field is a source for an extended electric field. The changing magnetic field does not only create an electric field at that location;  rather it generates an electric field that forms closed loops around the location where the magnetic field is changing.

Mathematically, Faraday's law is most often represented in terms of the change of magnetic flux with time. The magnetic flux is the property of a closed loop (say of a coil of wire) and is the product of the area times the magnetic field that is normal to that area. Engineers and physicists often use magnetic flux as a convenient physical property of a loop(s). They then express the magnetic $$\mathbf{B}$$ field as the magnetic flux per unit area. It is for this reason that the $$\mathbf{B}$$ field is often referred to as the magnetic flux density. This approach has the benefit of making certain calculations easier such as in magnetic circuits. It is typically not used outside of electrical circuits, though, because the magnetic $$\mathbf{B}$$ field truly is the more 'fundamental' quantity in that it directly connects all of electrodynamics in the simplest manner.

Sources of Magnetic Fields
Magnetic $$\mathbf{B}$$ fields, essentially, are generated by currents of electrical charge. We say that currents are sources for magnetic fields. Altogether there are 4 sources for magnetic fields—the last is only hypothetical:



 Electrical currents (moving charges)

 Magnetic dipoles

 Changing electric field </li>

<li> Magnetic monopole (hypothetical)</li>

</ol>

Electrical currents (moving charges)
All moving charges produce a magnetic field. The magnetic field of a moving charge is very complicated but is well known. (See Jefimenko's equations.) It forms closed loops around a line that is pointing in the direction the charge is moving. The magnetic field of a current on the other hand is much easier to calculate.

Magnetic field of a steady current


The magnetic field generated by a steady current (a continual flow of charges, for example through a wire, which is constant in time and in which charge is neither building up nor depleting at any point), is described by the Biot-Savart law. This is a consequence of Ampere's law, one of the four Maxwell's equations that describe electricity and magnetism. The magnetic field lines generated by a current carrying wire form concentric circles around the wire. The direction of the magnetic field of the loops is determined by the right hand grip rule. (See figure to the right.)  The strength of the magnetic field decreases with distance from the wire.

A current carrying wire can be bent in a loop such that the field is concentrated (and in the same direction) inside of the loop. The field will be weaker outside of the loop. Stacking many such loops to form a solenoid (or long coil) can greatly increase the magnetic field in the center and decrease the magnetic field outside of the solenoid. Such devices are called electromagnets and are extremely important in generating strong and well controlled magnetic fields. An infinitely long solenoid will have a uniform magnetic field inside of the loops and no magnetic field outside. A finite length electromagnet will produce essentially the same magnetic field as a uniform permanent magnet of the same shape and size. An electromagnet has the advantage, though, that you can easily vary the strength (even creating a field in the opposite direction) simply by controlling the input current. One important use is to continually switch the polarity of a stationary electromagnet to force a rotating permanent magnet to continually rotate using the fact that opposite poles attract and like poles repel. This can be used to create an important type of electrical motor.

Magnetic dipoles


The magnetic field due to a permanent magnet is well known. (See the first figure of article.)  But, what causes the magnetic field of a permanent magnet? The answer again is that the magnetic field is essentially created due to currents. But this time it is due to the cumulative effect of many small 'currents' of electrons 'orbiting' the nuclei of the magnetic material. Alternatively it is due to the structure of the electron itself which, in some sense, we can think of as forming a tiny loop of current. (The true nature of the electron's magnetic field is relativistic in nature, but this model often works.) Both of these tiny loops are modeled in terms of what we call the magnetic dipole. We can define the dipole moment of that dipole as the current times the area of the loop, then derive an equation for the magnetic field due to that magnetic dipole. (See the above image for what that magnetic field looks like.) By adding up the magnetic fields of many magnetic dipoles we can calculate the magnetic field of a larger magnet.

Changing electric field
The final known source of magnetic fields is a changing electric field. Just as a changing magnetic field generates an electric field so does a changing electric field generate a magnetic field. (Together these two effects bootstrap together to form and electromagnetic wave, light.) Similar to the way magnetic field lines formed close loops around current and to the way that electric field lines form closed loops around areas where the magnetic $$\mathbf{B}$$ field is changing, a changing (with time) electric field generates a magnetic field that forms closed loops around the region where the electric field is changing. Since the changing electric field causes the same indistinguishable magnetic field as a current in that region (and because of the prevailing yet incorrect ether model of electrodynamics at that time) this source of magnetic $$\mathbf{B}$$ field is known as 'displacement current'. The fact that a changing electric field creates a magnetic field is known as Maxwell's correction to Ampere's Law.

Magnetic monopole (hypothetical)
The magnetic monopole is a hypothetical particle that has not been found and may or may not exist. A magnetic monopole has, as it name suggests, only one pole. It is essentially a magnetic charge. It would either be a magnetic north pole corresponding to a positive magnetic charge or a magnetic south pole corresponding to a negative magnetic charge. To date, no one has found an isolated magnetic pole despite effort to discover it. Magnetic poles always come in pairs due to currents of electrical charge. The reason for the search is that if just one magnetic monopole was found then it would provide a reason due to quantum mechanics for the quantization of electrical charge.

The H Field
The term 'magnetic field' can also be used to describe the magnetic $$\mathbf{H}$$ field. The magnetic $$\mathbf{H}$$ field is a vector field similar to the magnetic $$\mathbf{B}$$ field except with different units and is completely different than the magnetic $$\mathbf{B}$$ field in the interior of a magnetic material. In SI units, $$\mathbf{B}$$ and $$\mathbf{H}$$ are measured in teslas (T) and amperes per meter (A/m), respectively; or, in cgs units, in gauss (G) and oersteds (Oe), respectively. The magnetic $$\mathbf{H}$$ field is determined from the

magnetic $$\mathbf{B}$$ field using the following equation:

The fields $$\mathbf{B}$$ and $$\mathbf{H}$$ are also related by the equation


 * $$\mathbf{B}=\mu_0(\mathbf{H}+\mathbf{M})$$ (SI units)


 * $$\mathbf{B}=\mathbf{H}+4\pi\mathbf{M}$$ (cgs units),

where $$\mathbf{M}$$ is magnetization of any magnetic material. Since magnetization exists only where there is magnetic material the form of $$\mathbf{H}$$ varies from $$\mathbf{B}$$ only inside a magnetic material. Outside of magnetic material $$\mathbf{B}$$ and $$\mathbf{H}$$ differ only by a multiplicative constant.

Physical Interpretation of the H field
It is often important to distinguish between two different types of currents: free currents and bound current. Free currents are currents that can be directly controlled and easily measured, for instance by changing the voltage applied to the wire. Bound currents are, as their name implies, bound to magnetic materials. As an example of bound current consider a uniform permanent bar magnet. A bar magnet is formed by many tiny magnets called magnetic dipoles each of which is essentially a tiny loop of current. By lining up a huge number of these dipoles we can create a large magnetic field. If we add up the currents of all these tiny loops we will find that the currents cancel in the interior of the bar magnet but add up along the outside edge of the bar magnet. (This current loops around the sides and not at the poles.) No one charge makes the complete trip around the magnet (each charge is bound to their tiny loop) but the net effect is a real current that flows on the outside of the magnet. (If the magnetization is not uniform then a bound current will flow through the bulk of the magnetic material as well.)

The magnetic $$\mathbf{H}$$ is useful because it treats these two types of currents differently. The free currents it treats in the normal fashion and therefore has the same form as the magnetic $$\mathbf{B}$$ field it would generate. The magnetic $$\mathbf{H}$$ fields treats the field inside of a magnetic material (due to that magnetic material) in a manner similar to the Gilbert model. (By subtracting the magnetization from the B field we are essentially converting the bound current sources to Gilbert-like magnetic charges at the poles.) Unlike the magnetic $$\mathbf{B}$$ which always forms closed loops the field due to the magnetic charges flow outward (or inward depending on the sign of the magnetic charge) in both directions from the poles. And while the magnetic field is exactly the same on the outside of the magnetic material for both models the magnetic fields inside are quite different.

Putting both sources together we see that the magnetic $$\mathbf{H}$$ field is the same as the magnetic $$\mathbf{B}$$ field to a multiplicative constant outside of magnetic materials, but is completely different from the magnetic $$\mathbf{B}$$ field inside a magnetic material. The advantage of this hybrid $$\mathbf{H}$$ field is that these sources are treated so differently that we can often pick out one source from the other. For example a line integral of the magnetic $$\mathbf{H}$$ field in a closed loop will yield the total free current in the loop (and not the bound current). This is unlike the magnetic $$\mathbf{B}$$ field where a similar integral will yield the sum of both the free and the bound current. If one wants to isolate the contribution due to the bound currents then a surface integral of $$\mathbf{H}$$ over any closed surface will pick out the 'magnetic charges' at the poles.

Sources of the H field
Unlike the magnetic $$\mathbf{B}$$ field that only has a current source such that the magnetic $$\mathbf{B}$$ field loops around currents, the magnetic $$\mathbf{H}$$ field has two types of sources. The first source of magnetic $$\mathbf{H}$$ field are the free currents for which $$\mathbf{H}$$ loop around similar to the way $$\mathbf{B}$$ field loops around the total current. The second source of the magnetic $$\mathbf{H}$$ field are 'magnetic charges' near the poles of the magnetic material. More precisely, these 'magnetic charges' are calculated as $$-\mathbf{\nabla}\cdot\mathbf{M}$$.

Energy Stored in Magnetic Fields
In order to create a magnetic field we need to do work to establish a free current. If we are to ask how much energy does it take to create a specific magnetic field using a particular free current then we need to distinguish between the free and the bound currents. It is the free current that we are 'pushing' on. The bound currents are freeloaders. They create a magnetic field that the free current has to work against without doing any of the work. If we are to calculate the energy of creating a magnetic field $$\mathbf{B}$$ we need to have a way of separating out the free current. The magnetic $$\mathbf{B}$$ cannot be used to determine this free current since $$\mathbf{B}$$ does not distinguish between bound and free current.

The magnetic $$\mathbf{H}$$ field does treat the two sources differently. Therefore it is useful in calculating the energy needed to create a magnetic field with a free current in the presence of magnetic materials. In this case the energy density needed, assuming a linear relationship between $$\mathbf{H}$$ and $$\mathbf{B}$$, has the form of:



u = \frac{\mathbf{H}\cdot\mathbf{B}}{2}

$$

If there are no magnetic materials around then we can replace $$\mathbf{H}$$ with $$\frac{\mathbf{B}}{\mu_o}$$.

Magnetic Circuits
The second main use for $$\mathbf{H}$$ is in magnetic circuits where inside a linear material $$\mathbf{B} = \mu \mathbf{H}$$. Here, $$\mu$$ is the permeability of the material. This is similar in form to Ohm's Law $$\mathbf{J} = \sigma \mathbf{E}$$, where $$\mathbf{J}$$ is the current density, $$\sigma$$ is the conductance and $$\mathbf{E}$$ is the Electric field. Extending this analogy we derive the counterpoint to the macroscopic Ohm's law ($$I = \frac{V}{R}$$) as:


 * $$\Phi = \frac F R_m,$$

where $$\Phi = \int \mathbf{B}\cdot d\mathbf{A}$$ is the magnetic flux in the circuit, $$F = \int \mathbf{H}\cdot d\mathbf{l}$$ is the magnetomotive force applied to the circuit, and $$R_m$$ is the reluctance of the circuit. Here the reluctance is defined as:


 * $$R_m = \frac{l}{\mu A},$$

where $$A$$ is the area, $$\mu$$ is the permeability of the material, and $$l$$ is the length.

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

History of B and H
The difference between the $$\mathbf{B}$$ and the $$\mathbf{H}$$ vectors can be traced back to Maxwell's 1855 paper entitled On Faraday's Lines of Force. It is later clarified in his concept of a sea of molecular vortices that appears in his 1861 paper On Physical Lines of Force - 1861. Within that context, $$\mathbf{H}$$ represented pure vorticity (spin), whereas $$\mathbf{B}$$ was a weighted vorticity that was weighted for the density of the vortex sea. Maxwell considered magnetic permeability µ to be a measure of the density of the vortex sea. Hence the relationship,

(1) Magnetic induction current causes a magnetic current density

$$\mathbf{B} = \mu \mathbf{H}$$

was essentially a rotational analogy to the linear electric current relationship,

(2) Electric convection current

$$\mathbf{J} = \rho \mathbf{v}$$

where $$\rho$$ is electric charge density. $$\mathbf{B}$$ was seen as a kind of magnetic current of vortices aligned in their axial planes, with $$\mathbf{H}$$ being the circumferential velocity of the vortices. With µ representing vortex density, we can now see how the product of µ with vorticity $$\mathbf{H}$$ leads to the term magnetic flux density which we denote as $$\mathbf{B}$$.

The electric current equation can be viewed as a convective current of electric charge that involves linear motion. By analogy, the magnetic equation is an inductive current involving spin. There is no linear motion in the inductive current along the direction of the $$\mathbf{B}$$ vector. The magnetic inductive current represents lines of force. In particular, it represents lines of inverse square law force.

The extension of the above considerations confirms that where $$\mathbf{B}$$ is to $$\mathbf{H}$$, and where $$\mathbf{J}$$ is to ρ, then it necessarily follows from Gauss's law and from the equation of continuity of charge that $$\mathbf{E}$$ is to $$\mathbf{D}$$. ie. $$\mathbf{B}$$ parallels with $$\mathbf{E}$$, whereas $$\mathbf{H}$$ parallels with $$\mathbf{D}$$.

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 utilised 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 will 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 for his work. Also in 1888, Ferraris published his research in a paper to the Royal Academy of Sciences in Turin.

Hall effect
Because the Lorentz force is charge-sign-dependent (see above), it results in charge separation when a conductor with current is placed in a transverse magnetic field, with a buildup of opposite charges on two opposite sides of conductor in the direction normal to the magnetic field, and the potential difference between these sides can be measured.

The Hall effect is often used to measure the magnitude of a magnetic field as well as to find the sign of the dominant charge carriers in semiconductors (negative electrons or positive holes).

Special relativity and electromagnetism
According to special relativity, electric and magnetic forces are part of a single physical phenomenon, electromagnetism; an electric force perceived by one observer will be perceived by another observer in a different frame of reference as a mixture of electric and magnetic forces. A magnetic force can be considered as simply the relativistic part of an electric force when the latter is seen by a moving observer.

More specifically, rather than treating the electric and magnetic fields as separate fields, special relativity shows that they naturally mix together into a rank-2 tensor, called the electromagnetic tensor. This is analogous to the way that special relativity "mixes" space and time into spacetime, and mass, momentum and energy into four-momentum.

Magnetic field shape descriptions



 * 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.


 * A dipole magnetic field is one seen around a bar magnet or around a 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.