Magnetic field

In physics, a magnetic field is a solenoidal vector field in the space surrounding moving electric charges and magnetic dipoles, such as those in electric currents and magnets. Where such a field is present, magnetic force acts. Michael Faraday talked about the magnetic field in terms of "lines of force". The relationship between magnetic fields, electric fields, and electric sources (stationary charges and current densities) was later quantified by James Clerk Maxwell in Maxwell's equations.

Magnetic field lines shown by iron filings

The intensity of the magnetic field has SI units of teslas. Unlike the electric field, the force exerted by a magnetic field is perpendicular to both the field and the velocity of a moving test charge. The direction of a magnetic field can be demonstrated with magnetic dipoles; magnetic dipoles in a magnetic field align themselves to be parallel with the field lines, as can be seen when iron filings are in the presence of a magnet. Magnetic fields also have an energy density proportional to the square of the field intensity.

For a closer look at the effects of a magnetic field, see under ferromagnetism, paramagnetism, diamagnetism, electromagnetism, and electromagnetic induction.

Definition

The magnetic field ${\displaystyle \mathbf {B} }$  of a moving charge, as described by the Biot-Savart law, is:

${\displaystyle \mathbf {B} =\mathbf {v} \times {\frac {1}{c^{2}}}\mathbf {E} }$ 

or

${\displaystyle \mathbf {B} =\mathbf {v} \times \mu \mathbf {D} }$ 

where

${\displaystyle \mathbf {v} }$  is velocity vector of the electric charge, measured in metres per second

${\displaystyle \times }$  indicates a vector cross product

${\displaystyle c\ }$  is the speed of light in a vacuum measured in metres per second

${\displaystyle \mathbf {E} }$  is the electric field vector measured in newtons per coulomb or volts per metre

${\displaystyle \mathbf {D} }$  is the electric displacement vector

${\displaystyle \mu }$  is the magnetic permeability

This same expression for the magnetic field can also be derived from the Lorentz transformation of an electric field exerted on an electric charge from the charge's proper frame to any other inertial frame.

As seen from the definition, the unit of magnetic field is newton-second per coulomb-metre (or newton per ampere-metre) and is called the tesla, and the direction of magnetic field vector is perpendicular to both source electric field and velocity of motion of source's reference frame v. Also it follows from the definition that the magnetic field vector being a vector product is a pseudovector (axial vector).

Like the electric field, the magnetic field exerts force on electric charge — but unlike an electric field, it exerts force only on a moving charge, and the direction of the force is orthogonal to both magnetic field and charge's velocity:

${\displaystyle \mathbf {F} =q\mathbf {v} \times \mathbf {B} }$ 

where

${\displaystyle \mathbf {F} }$  is the force vector produced, measured in newtons

${\displaystyle q\ }$  is electric charge that the magnetic field is acting on, measured in coulombs

${\displaystyle \mathbf {v} \,}$  is velocity vector of the electric charge ${\displaystyle q\ }$ , measured in metres per second

Intuitively ${\displaystyle \mathbf {B} }$  can be seen as a vector whose direction gives the axis of the possible directions of the force on a charged particle due to the magnetic field; the possible directions being at right angles to the axis ${\displaystyle \mathbf {B} }$ , and the exact direction being at right angles to both the velocity of the particle and ${\displaystyle \mathbf {B} }$ . The magnitude of ${\displaystyle \mathbf {B} }$  is the amount of force the magnetic field causes on the particle, per unit of particle charge by particle speed. Another intuitive way to view ${\displaystyle \mathbf {B} }$  is as a bundle of lines of force that pull two unlike magnetic poles together.

The Difference between B and H

There are two quantities that physicists may refer to as the magnetic field, notated ${\displaystyle \mathbf {H} }$  and ${\displaystyle \mathbf {B} }$ . The vector H is properly known as the magnetic field intensity. The vector field ${\displaystyle \mathbf {B} }$  represents the quantity known as Magnetic Flux Density and has the SI units of Tesla (T), equivalent to webers (Wb) per square metre. Magnetic Flux has the SI units of webers so the ${\displaystyle \mathbf {B} }$  field is that of its density (an areal density). The vector field ${\displaystyle \mathbf {H} }$  is known as Magnetic Field Strength [1][2][3][4]^[1] and has the SI units of amperes per metre. The Magnetic Field Strength vector field (${\displaystyle \mathbf {H} }$ ) is something of the magnetic analog to the Electric Field Strength vector field represented by ${\displaystyle \mathbf {E} }$ , with the SI units of the latter being volts per metre -- note the analogy with the units for ${\displaystyle \mathbf {H} }$ . Although the term "magnetic field" was historically reserved for ${\displaystyle \mathbf {H} }$ , with ${\displaystyle \mathbf {B} }$  being termed the "magnetic induction," ${\displaystyle \mathbf {B} }$  is now understood to be the more fundamental entity, and most modern writers refer to ${\displaystyle \mathbf {B} }$  as the magnetic field, except when context fails to make it clear whether the quantity being discussed is ${\displaystyle \mathbf {H} }$  or ${\displaystyle \mathbf {B} }$ . See^[2]

The difference between the ${\displaystyle \mathbf {B} }$  and the ${\displaystyle \mathbf {H} }$  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, ${\displaystyle \mathbf {H} }$  represented pure vorticity (spin), whereas ${\displaystyle \mathbf {B} }$  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

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

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

(2) Electric Convection Current

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

where ${\displaystyle \rho }$  is electric charge density. ${\displaystyle \mathbf {B} }$  was seen as a kind of magnetic current of vortices aligned in their axial planes, with ${\displaystyle \mathbf {H} }$  being the circumferential velocity of the vortices.

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

In SI units, ${\displaystyle \mathbf {B} \ }$  and ${\displaystyle \mathbf {H} \ }$  are measured in teslas (T) and amperes per metre (A/m), respectively; or, in cgs units, in gauss (G) and oersteds (Oe), respectively. Two parallel wires carrying an electric current in the same direction will generate a magnetic field that will cause a force of attraction between them. This fact is used to define the value of an ampere of electric current.

Magnetic field of current of charged particles

 

Current (I) through a wire produces a magnetic field (${\displaystyle \mathbf {B} }$ ) around the wire. The field is oriented according to the right hand grip rule.

 

Charged particle drifts in a homogenous magnetic field. (A) No disturbing force (B) With an electric field, E (C) With an independent force, F (eg. gravity) (D) In an inhomgeneous magnetic field, grad H

Substituting into the definition of magnetic field

${\displaystyle \mathbf {B} =\mathbf {v} \times {\frac {1}{c^{2}}}\mathbf {E} }$ 

the proper electric field of point-like charge (see Coulomb's law)

${\displaystyle \mathbf {E} ={1 \over 4\pi \epsilon _{0}}{q \over r^{2}}{\hat {r}}={10^{-7}}{c^{2}}{q \over \ {r}^{2}}{\hat {r}}}$ 

results in the equation of magnetic field of moving charge, which is usually called the Biot-Savart law:

${\displaystyle \mathbf {B} =\mathbf {v} \times {\frac {\mu _{0}}{4\pi }}{\frac {q}{r^{2}}}{\hat {r}}}$ 

where

${\displaystyle q}$  is electric charge, whose motion creates the magnetic field, measured in coulombs

${\displaystyle \mathbf {v} }$  is velocity of the electric charge ${\displaystyle q}$  that is generating ${\displaystyle \mathbf {B} }$ , measured in metres per second

${\displaystyle \mathbf {B} }$  is the magnetic field (measured in teslas)

Integrating the above expression around closed loop results in Ampere's law which is one of four Maxwell's equations.

Lorentz force on wire segment

Integrating the Lorentz force on an individual charged particle over a flow (current) of charged particles results in the Lorentz force on a stationary wire carrying electric current:

${\displaystyle F=Il\times B\,}$ 

where

F = forces, measured in newtons

I = current in wire, measured in amperes

B = magnetic field, measured in teslas

${\displaystyle \times }$  = vector cross-product

l = length of wire, measured in metres

In the equation above, the current vector I is a vector with magnitude equal to the scalar current, I, and direction pointing along the wire in which the current is flowing.

Alternatively, instead of current, the wire segment l can be considered a vector.

The Lorentz force on a macroscopic current carrier is often referred to as the Laplace force.

Properties

Magnetic field lines

 

Magnetic field lines shown by iron filings

The direction of the magnetic field vector follows from the definition above. It coincides with the direction of orientation of a magnetic dipole, such as a small magnet, or a small loop of current in the magnetic field. So, a cluster of small particles of ferromagnetic material being brought in the magnetic field can be used to show the direction of magnetic field lines (see figure). A trajectory of charged particle (electron Such motion of Solar wind plasma in the magnetic field of Earth results in Northern Lights (and Southern Lights) - spots of glow in upper atmosphere above magnetic poles of Earth where energetic electrons and protons can reach air and ionize nitrogen and oxygen molecules.

Pole labelling confusions

See also North Magnetic Pole and South Magnetic Pole.

The end of a compass needle that points north was historically called the "north" magnetic pole of the needle. Since dipoles are vectors and align "head to tail" with each other to minimize their magnetic potential energy, the magnetic pole located near the geographic North Pole is actually the "south" pole.

The "north" and "south" poles of a magnet or a magnetic dipole are labelled similarly to north and south poles of a compass needle. Near the north pole of a bar or a cylinder magnet, the magnetic field vector is directed out of the magnet; near the south pole, into the magnet. This magnetic field continues inside the magnet (so there are no actual "poles" anywhere inside or outside of a magnet where the field stops or starts). Breaking a magnet in half does not separate the poles but produces two magnets with two poles each.

Earth's magnetic field is probably produced by electric currents in its liquid core.

It can be more easily explained if one works backwards from the equation:

${\displaystyle B={\frac {F}{IL}}\,}$ 

where

B is the magnitude of flux density, measured in SI as teslas

F is the force experienced by a wire, measured in Newtons

I is the current, measured in amperes

L is the length of the wire, measured in metres

 

Demonstration of Fleming's left hand rule

For a magnetic flux density to equal 1 tesla, a force of 1 newton must act on a wire of length 1 metre carrying 1 ampere of current.

1 newton of force is not easily accomplished. For example: the most powerful superconducting electromagnets in the world have flux densities of 'only' 20 T. This is true obviously for both electromagnets and natural magnets, but a magnetic field can only act on moving charge — hence the current, I, in the equation.

The equation can be adjusted to incorporate moving single charges, ie protons, electrons, and so on via

${\displaystyle F=BQv\,}$ 

where

Q is the charge in coulombs, and

v is the velocity of that charge in metres per second.

Fleming's left hand rule for motion, current and polarity can be used to determine the direction of any one of those from the other two, as seen in the example. It can also be remembered in the following way. The digits from the thumb to second finger indicate 'Force', 'B-field', and 'I(Current)' respectively, or F-B-I in short. For professional use, the right hand grip rule is used instead which originated from the definition of cross product in the right hand system of coordinates.

Other units of magnetic flux density are

1 gauss = 10^-4 teslas = 100 microteslas (µT)

1 gamma = 10^-9 teslas = 1 nanotesla (nT)

Rotating magnetic fields

Main article: Alternator

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 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.

Hall effect

Main article: 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).

Extension to the Theory of Relativity

According to special relativity, electric and magnetic forces are part of a single physical phenomenon; an electric field perceived by one observer will be perceived by another observer in a different frame of reference as a mixture of electric and magnetic fields. A magnetic field can be considered as simply the relativistic part of an electric field when the latter is seen by a moving observer.^[3]

When an electric charge is moving from the perspective of an observer, the electric field of this charge due to space contraction is no longer seen by the observer as spherically symmetric due to non-radial time dilation, and it must be computed using the Lorentz transformations. One of the products of these transformations is the part of the electric field which only acts on moving charges — and we call it the "magnetic field". It is a relativistic manifestation of the more fundamental electric field.

The quantum-mechanical motion of electrons in atoms produces the magnetic fields of permanent ferromagnets. Spinning charged particles also have magnetic moment. Some electrically neutral particles (like the neutron) with non-zero spin also have magnetic moment due to the charge distribution in their inner structure. Particles with zero spin never have magnetic moment which is the consequence that a magnetic field is the result of motion of electric field.

A magnetic field is a vector field: it associates with every point in space a (pseudo) vector that may vary through time. The direction of the field is the equilibrium direction of a magnetic dipole (like a compass needle) placed in the field.

The Lorentz transformation of a spherically-symmetric proper electric field E of a moving electric charge (for example, the electric field of an electron moving in a conducting wire) from the charge's reference frame to the reference frame of a non-moving observer results in the following term which we can define or label as "magnetic field".

Maxwell did much to unify static electricity and magnetism, producing a set of four equations relating the two fields. However, under Maxwell's formulation, there were still two distinct fields describing different phenomena. It was Albert Einstein who showed, using special relativity, that electric and magnetic fields are two aspects of the same thing (a rank-2 tensor), and that one stationary observer may perceive a magnetic force where a moving observer perceives only an electric Field. Thus, using special relativity, magnetic forces are a manifestation of electric fields of charges in motion and may be predicted from knowledge of the electric fields and the velocity of movement (relative to some observer) of the charges.

A thought experiment one can do to show this is with two identical infinite and parallel lines of charge having no motion relative to each other but moving together relative to an observer. Another observer is moving alongside the two lines of charge (at the same velocity) and observes only electrostatic repulsive force and acceleration. The first or "stationary" observer seeing the two lines (and second observer) moving past with some known velocity also observes that the "moving" observer's clock is ticking more slowly (due to time dilation) and thus observes the repulsive acceleration of the lines of charge more slowly than that which the "moving" observer sees. The reduction of repulsive acceleration can be thought of as an added attractive force, in a classical electrodynamics context, that reduces the electrostatic repulsive force and also increases in magnitude with increasing velocity. This pseudo-force is precisely the same as the electromagnetic force in a classical context.

A changing magnetic field is mathematically the same as a moving magnetic field (see relativity of motion). Thus, according to Einstein's field transformation equations (that is, the Lorentz transformation of the field from a proper reference frame to a non-moving reference frame), part of it is manifested as an electric field component. This is known as Faraday's law of induction and is the principle behind electric generators and electric motors.

See also

General

• Electric field — effect produced by an electric charge that exerts a force on charged objects in its vicinity.
• Electromagnetic field — a field composed of two related vector fields, the electric field and the magnetic field.
• Electromagnetism — the physics of the electromagnetic field: a field, encompassing all of space, composed of the electric field and the magnetic field.
• Magnetism — phenomenon by which materials exert an attractive or repulsive force on other materials.
• Magnetohydrodynamics — the academic discipline which studies the dynamics of electrically conducting fluids.
• Magnetic flux
• Magnetic monopole
• SI electromagnetism units

Mathematics

• Ampère's law — magnetic equivalent of Gauss's law.
• 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 the electric and magnetic fields, and their interaction with matter.

Applications

• Helmholtz coil — a device for producing a region of nearly uniform magnetic field.
• Maxwell coil — a device for producing a large volume of almost constant magnetic field.
• Earth's magnetic field — a discussion of the magnetic field of the Earth.
• Dynamo theory — a proposed mechanism for the creation of the Earth's magnetic field.
• Electric motor — AC motors used magnetic fields
• Rapid-decay theory - a creationist theory
• Stellar magnetic field — a discussion of the magnetic field of stars.
• Teltron Tube

References

Web

• Nave, R., Magnetic Field Strength H, retrieved 2007-06-04
• Keitch, Paul, Magnetic Field Strength and Magnetic Flux Density, retrieved 2007-06-04
• Oppelt, Arnulf (2006-11-02), magnetic field strength, retrieved 2007-06-04 {{citation}}: Check date values in: |date= (help)
• magnetic field strength converter, retrieved 2007-06-04

Books

• Durney, Carl H. and Johnson, Curtis C. (1969). Introduction to modern electromagnetics. McGraw-Hill. ISBN 0-07-018388-0.
• Rao, Nannapaneni N. (1994). Elements of engineering electromagnetics (4th ed.). Prentice Hall. ISBN 0-13-948746-8.
• Griffiths, David J. (1999). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 0-13-805326-X.
• Jackson, John D. (1999). Classical Electrodynamics (3rd ed.). Wiley. ISBN 0-471-30932-X.
• Tipler, Paul (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.). W. H. Freeman. ISBN 0-7167-0810-8.

Notes

1. Magnetic Field Strength is also sometimes called Magnetic Field Intensity. For more information reference the sources Durney and Johnson, and also Rao.
2. The standard graduate textbook by Jackson follows this usage. Edward Purcell, in Electricity and Magnetism, McGraw-Hill, 1963, writes, Even some modern writers who treat B as the primary field feel obliged to call it the magnetic induction because the name magnetic field was historically preempted by H. This seems clumsy and pedantic. If you go into the laboratory and ask a physicist what causes the pion trajectories in his bubble chamber to curve, he'll probably answer "magnetic field," not "magnetic induction." You will seldom hear a geophysicist refer to the earth's magnetic induction, or an astrophysicist talk about the magnetic induction of the galaxy. We propose to keep on calling B the magnetic field. As for H, although other names have been invented for it, we shall call it "the field H" or even "the magnetic field H".
3. http://www.iop.org/EJ/abstract/1402-4896/74/6/018 J H Field (2006) "Classical electromagnetism as a consequence of Coulomb's law, special relativity and Hamilton's principle and its relationship to quantum electrodynamics". Phys. Scr. 74 702-717

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).