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Magnetic Field

Magnetic field

:For other senses of this term, see magnetic field (disambiguation). In physics, a magnetic field is an entity produced by moving electric charges (electric currents) that exerts a force on other moving charges. (The quantum-mechanical spin of a particle produces magnetic fields and is acted on by them as though it were a current; this accounts for the fields produced by "permanent" ferromagnets.) A magnetic field is a vector field: it associates with every point in space a (pseudo-)vector that may vary in time. The direction of the field is the equilibrium direction of a compass needle placed in the field.

Symbols and terminology

Magnetic field is usually denoted by the symbol

\mathbf \

. Historically,

\mathbf \

was called the magnetic flux density, magnetic induction, or magnetic field strength.

\mathbf

was called the magnetic field (or magnetic field intensity), and this terminology is still often used to distinguish the two in the context of magnetic materials (non-trivial permeability μ). Otherwise, however, this distinction is often ignored, and both symbols are frequently referred to as the magnetic field. (Some authors call H the auxiliary field, instead.) In linear materials, such as air or free space, the two quantities are linearly related: :

\mathbf = \mu \mathbf \

where

\ \mu

is the magnetic permeability (in henries per meter) of the medium. In SI units,

\mathbf \

and

\mathbf \

are measured in teslas (T) and amperes per meter (A/m), respectively; or, in cgs units, in gauss (G) and oersteds (Oe), respectively. Two parallel wires carrying an electric current in the same sense will generate a magnetic field which will cause a force of attraction to each other. This fact is used to generate the value of an ampere of electric current. Note that while like charges repel and unlike ones attract, the opposite holds for currents: if the current in one of the two parallel wires is reversed, the two will repel.

Definition

Like the electric field, the magnetic field can be defined by the force it produces. In SI units, this is: :

\mathbf = q \mathbf \times \mathbf

where :F is the force produced, measured in newtons :

\times \

indicates a vector cross product :

q \

is electric charge that the magnetic field is acting on, measured in coulombs :

\mathbf \

is velocity of the electric charge

q \

, measured in metres per second :B is magnetic flux density, measured in teslas This law is called the Lorentz force law. (More precisely, it is the special case of that law when there is no electric field. It holds in any reference frame, although the force due to the magnetic field may be different in different frames because magnetic fields transform into electric fields under Lorentz transformations. The total force due to the electric and magnetic fields is the same in any frame.)

Current loop

A simpler form of the force equation in a wire current loop is: Force = BLi = (Tesla)x(meter length of wire)x(ampere current of wire). A more complex explanation is that if the moving charge is part of a current in a wire, then an equivalent form of the law is :

\frac   = \mathbf \times \mathbf

In words, this equation says that the force per unit length of wire is the cross product of the current vector and the magnetic field. In the equation above, the current vector,

\mathbf

, is a vector with magnitude equal to the usual scalar current,

i

, and direction pointing along the wire that the current is flowing.

Point charge generating magnetic field

The field can be computed as the sum of the contributions from individual charged particles. The magnetic flux density from a point charge is: :

\mathbf = \frac\mathbf \times \mathbf

which, for constant velocities, can be expanded into the Biot-Savart law: :

\mathbf = \frac\frac\mathbf \times \mathbf

q \

is electric charge generating the magnetic field, measured in coulombs :

\mathbf \

is velocity of the electric charge

q \

that is generating B, measured in metres per second :B is magnetic flux density, measured in teslas

Vector calculus

The most compact and elegant mathematical statements describing how magnetic fields are produced makes use of vector calculus. In free space: :

\nabla \times \mathbf = \mu_0 \mathbf + \mu_0 \epsilon_0 \frac

\nabla \cdot \mathbf = 0

where :

\nabla \times

is the curl operator :

\nabla \cdot

is the divergence operator :

\mu_0 \

is permeability :

\mathbf \

is current density :

\partial \

is the partial derivative :

\epsilon_0 \

is the free-space permittivity :

\mathbf \

is the electric field :

t \

is time The first equation is known as Ampère's law with James Clerk Maxwell's correction. The second term of this equation (Maxwell's correction) disappears in static or quasi-static systems. The second equation is a statement of the observed non-existence of magnetic monopoles. These are two of four Maxwell's equations; the notation is due to Oliver Heaviside.

Energy in the magnetic field

The general relation for nonlinear materials, the differential energy is: :

dU_H = \int_^ H \cdot dB \, dV

Where V is the volume and dV is the differential volume. For linear materials, H is proportional to B, so the above equation can be simplified: :

U_H = \frac\int_^ B \cdot H \, dV

For linear materials and a constant volume: :

U_H = \frac

Energy can produce a force, so :

F = \frac

F = \frac

Where dl is differential distance and A is the surface area. Force per unit area (pressure) is :

P = \frac

In the case of free space (air),

\mu_o = 4 \pi \cdot 10^ \frac

: :

P \approx 398 \, \mbox \, \approx 57.7 \, \frac

at B = 1 tesla :

P \approx 1592 \, \mbox \, \approx 231 \, \frac

at B = 2 teslas This is the force observed when a high permeability, ferromagnetic materials, such as iron and steel alloys, are in the proximity of magnetic fields.

Properties

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 observer may perceive a magnetic force where a moving observer perceives only an electrostatic force. Thus, using special relativity, magnetic forces are a manifestation of electrostatic forces of charges in motion and may be predicted from knowledge of the electrostatic forces and the 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 more slowly than that which the "moving" observer sees. The reduction of repulsive acceleration can be thought of as an attractive force, in a classical physics context, that reduces the electrostatic repulsive force and also that is increasing with increasing velocity. This pseudo-force is precisely the same as the electromagnetic force in a classical context. Changing magnetic fields, according to Faraday's law of induction, can induce an electric field and thus an electric current; similar currents can be induced by conductors moving in a fixed magnetic field. These phenomena are the basis for many electric generators and electric motors.

Magnetic field lines

electric motor Formally, the magnetic field is not a vector, it is a pseudovector. That is, it gains an extra sign flip under improper rotations of the coordinate system. (The distinction is important when using symmetry to analyze magnetic-field problems.) This is a consequence of the fact that B is related to two true vectors by a cross product (e.g. in the Lorentz force law). To simplify the study of magnets an arbitrary (but valid) description of magnetic field lines was created. 1 magnetic field line = 1 gauss line. 10,000 gauss lines per square meter is equal to 1 tesla. The total number of lines emanating from a magnet pole is the magnetic flux. Count only north or only south pole lines, i.e. monopole or one sided value. Although the field line orientation is typically indicated in diagrams with an arrow, the arrow should not be interpreted to indicate any actual movement or flow of the field line.

Pole labeling confusions

It is necessary to note that the labeling of north and south on a compass is in opposition to the labeling of the north and south pole of the Earth. If you have two labeled magnets, it is clear that like poles repel, while opposing poles attract. However, this is clearly wrong when using a compass to find the North Pole of the Earth, because the "north" end of the compass points to the "North" Pole. By convention, the pole of a magnet is labelled according to the direction it points, hence when we speak of the "north pole" of a magnet, we really mean the "north-seeking pole". Magnetic field lines point from north to south of a magnet, and hence the natural magnetic field lines run from south to north along the Earth's surface. This choice, along with the choice of sign convention in the Biot-Savart law, is equivalent to choosing a sign convention for electric charge.

Rotating magnetic fields

A rotating magnetic field is a magnetic field which rotates in polarity at non-relativistic speeds. This is a key principle to the operation of alternating-current motor. A permanent magnet in such a field will rotate so as to maintain its alignment with the external field. This effect is utilised in alternating current electric motors. A good rotating magnetic field can be constructed using three phase alternating currents (or even with higher order polyphase systems). Synchronous motors and induction motors use a stator's rotating magnetic fields to turn rotors. In 1882, Nikola Tesla identified the concept of the rotary 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.

References

Books
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External articles

Information
- Nave, R., "[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magfie.html Magnetic Field]". HyperPhysics.
- "Magnetism", [http://theory.uwinnipeg.ca/physics/mag/node2.html#SECTION00110000000000000000 The Magnetic Field]. theory.uwinnipeg.ca.
- Hoadley, Rick, "[http://my.execpc.com/~rhoadley/magfield.htm What do magnetic fields look like]?" 17 July 2005. Rotating magnetic fields
- "[http://www.tpub.com/neets/book5/18a.htm Rotating magnetic fields]". Integrated Publishing.
- "Introduction to Generators and Motors", [http://www.tpub.com/content/neets/14177/css/14177_87.htm rotating magnetic field]. Integrated Publishing.
- "[http://www.egr.msu.edu/~jurkovi4/Experiment4.pdf Induction Motor-Rotating Fields]". Diagrams
- McCulloch, Malcolm,"A2: Electrical Power and Machines", [http://www.eng.ox.ac.uk/~epgmdm/A2/img89.htm Rotating magnetic field]. eng.ox.ac.uk.
- "AC Motor Theory" [http://www.tpub.com/content/doe/h1011v4/css/h1011v4_23.htm Figure 2 Rotating Magnetic Field]. Integrated Publishing. Journal Articles
- Yaakov Kraftmakher, "[http://www.iop.org/EJ/abstract/0143-0807/22/5/302 Two experiments with rotating magnetic field]". 2001 Eur. J. Phys. 22 477-482.
- Bogdan Mielnik and David J. Fernández C., "[http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JMAPAQ000030000002000537000001&idtype=cvips&gifs=yes 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 "[http://prola.aps.org/abstract/PRE/v61/i4/p4111_1 Structure and dynamics of magnetorheological fluids in rotating magnetic fields]". Phys. Rev. E 61, 4111–4117 (2000). Category:Magnetism Category:Physical quantity Category:Introductory physics ja:磁場 th:สนามแม่เหล็ก

Magnetic field (disambiguation)

Magnetic field can refer to:
- A computer game developer; see Magnetic Fields (computer game developer)
- A magnetic field, the physical phenomenon produced by moving electric charges and exhibited by ferrous materials.
- Les Champs Magnétiques, a surrealist novel by André Breton.
- Magnetic Fields, the English name of the album Les Chants Magnétiques, by French composer Jean-Michel Jarre.
- The Magnetic Fields, an American synth-pop band led by Stephin Merritt.

Electric current

In electricity, current refers to electric current, which is the flow of electric charge. Lightning is an example of an electric current, as is the solar wind, the source of the polar aurora. Probably the most familiar form of electric current is the flow of conduction electrons in a metallic wire. This is how utility companies deliver electricity. In electronics, electric current is most often the flow of electrons through conductors and devices such as resistors, but it is also the flow of ions inside a battery or the flow of holes within a semiconductor.

Relation between current and charge

The symbol typically used for the amount of current (the amount of charge Q flowing per unit of time t) is I, from the German word Intensität, which means 'intensity'. :

I =

Formally this is written as :

i(t) =

or inversely as

q(t) = \int_^ i(x)\, dx

Conventional current

Conventional current was defined early in the history of electrical science as a flow of positive charge. In solid metals, like wires, the positive charges are immobile, and only the negatively charged electrons flow in the direction opposite conventional current, but this is not the case in most non-metallic conductors. In other materials, charged particles flow in both directions at the same time. Electric currents in electrolytes are flows of electrically charged atoms (ions), which exist in both positive and negative varieties. For example, an electrochemical cell may be constructed with salt water (a solution of sodium chloride) on one side of a membrane and pure water on the other. The membrane lets the positive sodium ions pass, but not the negative chlorine ions, so a net current results. Electric currents in plasma are flows of electrons as well as positive and negative ions. In ice and in certain solid electrolytes, flowing protons constitute the electric current. To simplify this situation, the original definition of conventional current still stands. There are also instances where the electrons are the charge that is physically moving, but where it makes more sense to think of the current as the movement of positive "holes" (the spots that should have an electron to make the conductor neutral). This is the case in a p-type semiconductor. The SI unit of electrical current is the ampere. Electric current is therefore sometimes informally referred to as amperage or ampage, by analogy with the term voltage. Though this is a valid term, some engineers frown on it.

The speed of an electric current

The charged particles whose movement causes an electric current do not always move in straight lines. In metals, for example, they follow an erratic path, bouncing from atom to atom, but generally drifting in the direction of the electric field. The speed at which they drift can be calculated from the equation: :

I=nAvQ \!\

where :I is the current :n is number of charged particles per unit volume :A is the cross-sectional area of the conductor :v is the drift velocity, and :Q is the charge on each particle. For example, in a copper wire of cross-section 0.5 mm², carrying a current of 5 A, the drift velocity of the electrons is of the order of a millimetre per second. To take a different example, in the near-vacuum inside a cathode ray tube, the electrons travel in near-straight lines ("ballistically") at about a tenth of the speed of light. However, we know that an electric signal travels much faster than this; usually close to the speed of light. These results show that the speed of the charged particles is not necessarily related to the speed of the electric signal. To understand how signals travel faster than the particles that carry them, it is necessary to understand the properties of electromagnetic waves (see article).

Current density

Current density is the current per unit (cross-sectional) area. Mathematically, current is defined as the net flux through an area. Thus: :

I = j \cdot A

where, in the MKS or SI system of measurement, :I is the current, measured in amperes :j is the "current density" measured in amperes per square metre :A is the area through which the current is flowing, measured in square metres The current density is defined as: :

j=\int_i n_i \cdot x_i \cdot \mathbf

where :n is the particle density (number of particles per unit volume) :x is the mass, charge, or any other characteristic whose flow one would like to measure. :u is the average velocity of the particles in each volume Current density is an important consideration in the design of electrical and electronic systems. Most electrical conductors have a finite, positive resistance, making them dissipate power in the form of heat. The current density must be kept sufficiently low to prevent the conductor from melting or burning up, or the insulating material failing. In superconductors, excessive current density may generate a strong enough magnetic field to cause spontaneous loss of the superconductive property.

Electromagnetism

Every electric current produces a magnetic field. The magnetic field can be visualized as a pattern of circular field lines surrounding the wire. Electric current can be directly measured with a galvanometer, but this method involves breaking the circuit, which is sometimes inconvenient. Current can also be measured without breaking the circuit by detecting the magnetic field it creates. Devices used for this include Hall effect sensors, current clamps and Rogowski coils.

Ohm's law

Ohm's law predicts the current in an (ideal) resistor (or other ohmic device) to be the quotient of applied voltage over electrical resistance: :

I = \frac

where :I is the current, measured in amperes :V is the potential difference measured in volts :R is the resistance measured in ohms

Electrical safety

The danger of an electric shock depends on the current (in milliamperes), duration and the current's path in the body:
- 1 mA causes a tingle
- 5 mA causes a slight shock
- 50 to 150 mA may result in death, e.g. through rhabdomyolysis (muscle breakdown) and resultant acute renal failure
- 1-4 A causes ventricular fibrillation
- 10 A causes cardiac arrest (only at this current will a typical home fuse break the circuit) Currents through the heart and the nervous system are the most dangerous. As most dangerous sources are voltage sources, the current present depends on the resistance of the body between the points of contact and any current limiting built into the source. The comparison between the dangers of alternating current and direct current has been a subject of debate ever since the War of Currents in the 1880s. DC tends to cause continuous muscular contractions that make the victim hold on to a live conductor, thereby increasing the risk of deep tissue burns. On the other hand, mains-frequency AC tends to interfere more with the heart's electrical pacemaker, leading to an increased risk of fibrillation. AC at higher frequencies holds a different mixture of hazards, such as RF burns and the possibility of tissue damage with no immediate sensation of pain.

External links

- [http://www.unitconversion.org/unit_converter/current.html Online Current Converter] - convert between various units of current, such as ampere, biot, abampere, statampere, and so on
- [http://www.unitconversion.org/unit_converter/current-v.html Interactive Current Conversion Table] - convert selected unit to all other units of current
- [http://amasci.com/amateur/elecdir.html Which direction does electricity really flow?] Category:Electromagnetism Category:Magnetism ko:전류 ja:電流 th:กระแสไฟฟ้า

Force (physics)

:For other senses of this word, see force (disambiguation). In physics, a force is an external cause responsible for any change of a physical system. For instance, a person holding a dog by a rope is experiencing the force applied by the rope on their hand, and the cause for its pulling forward is the force exercised by the rope. The kinetic expression of this change is, according to Newton's second law, acceleration, non kinetic expressions such as deformation can also occur. The SI unit for force is the newton.

Elementary concepts

Force in its most primitive definition can be thought of as that which when acting alone causes an object to accelerate. In a practical sense forces can be divided into two groups: contact forces and field forces. Contact forces require the physical contact of one object with other such as a hammer striking a nail or the force exerted by a gas under pressure - gas produced by exploding gunpowder forces a heavy ball out of a cannon. Field forces on the other hand need no physical medium of contact. Gravity and magnetism are examples of such forces. It should be noted however, that fundamentally all forces are in fact field forces. The force of hammer striking the nail in the previous example turns out to be a clash of the electric forces in both hammer and nail. Nevertheless it is appropriate in some cases to maintain these two classifications for ease of understanding.

Quantitative definition

In physics models, the point-like system is used, where objects are represented as one-dimensional points at their centre of mass. The only change the system can experience is a change of its momentum (its speed). Since the rise of the atomic theory, any physical system has been considered in classical physics as composed of point-like systems called atoms or molecules. Therefore, all forces can be defined by their effect; that is, by the change of movement they induce on point-like systems. This change of movement can be quantified by the acceleration (the derivative of velocity). The discovery by Isaac Newton that a given force will induce an acceleration in inverse proportion to a quantity called the mass of inertia or inertial mass which is independent of the speed of the system is Newton's second law. This law allows us to predict the effect of a force on any point-like system whose mass is known. It is usually written as: :F = dp/dt = d(m·v)/dt = m·a (in the case where m does not depend on t) where :F is the force (a vector quantity), :p is the momentum, :t is the time, :v is the velocity, :m is the mass, and :a=d²x/dt² is the acceleration, the second derivative with respect to t of the position vector x. If the mass m is measured in kilograms and the acceleration a is measured in metres per second squared, then the unit of force is kilogram × metre/second squared. This unit is called the newton: 1 N = 1 kg x 1 m/s². This equation is a system of three second-order differential equations with respect to the three-dimensional position vector which is an unknown function of time. This equation can be solved if F is a known function of x and some of its derivatives and if the mass m is known. Morevover the boundary conditions are required; for example, the values of the position vector and x and the velocity v at the starting time, say t=0. Of course, this formula is only useful if one knows the numerical values of F and m. The definition above is an implicit definition, arrived at as follows. One defines a reference system (one litre of water) and a reference force (the gravitational force applied by the Earth on it at the altitude of Paris). One takes Newton's second law for granted (one postulates that it is true) and measures the acceleration induced by the reference force on the reference system. This gives us a mass unit (1 kg) and a force unit (the older unit of 1 kilogram-force = 9.81 N). Once this is done, one can measure any force by the acceleration it induces on the reference system and measure the inertial mass of any system by measuring the acceleration induced on this system by the reference force. Force is often considered a fundamental quantity in physics, but there are more fundamental quantities, such as momentum (p = mass m x velocity v). Energy, measured in joules, is still less fundamental than force and momentum, because it is defined as work, and work is defined in terms of force. The two most fundamental theories of nature - quantum electrodynamics and general relativity - do not contain the concept of force at all. Although not the most fundamental quantity in physics, force is an important basic mathematical concept from which other concepts, such work and pressure (measured in pascals), are derived. Force is sometimes confused with stress.

Types of force

There are four known fundamental forces in nature.
- Nuclear forces acting between subatomic particles
- Electromagnetic forces between electric charges
- Weak forces arising from radioactive decay
- Gravitational forces between masses Quantum field theory accurately models the first three fundamental forces, but does not model quantum gravity. Quantum gravity on a large scale can, however, be described by general relativity. The four fundamental forces describe every observable phenomenon including the many other forces observed such as: Coulomb's force (the force between electrical charges), gravitational force (force between masses), magnetic force, frictional forces, centripetal, centrifugal, impact force, and spring force, to name a few. Forces can also be classified into conservative forces and nonconservative forces. Conservative forces are equivalent to the gradient of a potential, and include gravity, electromagnetic force, and spring force. Nonconservative forces include friction and drag.

Properties of force

Because momentum is a vector, then force, being its time derivative, is also a vector - it has magnitude and direction. Forces can be added together using the parallelogram of force. When two forces act on an object, the resulting force, the resultant, is the vector sum of the original forces. This is called the principle of superposition. The magnitude of the resultant varies from zero to the sum of the magnitudes of the two forces, depending on the angle between their lines of action. If the two forces are equal, but opposite, the resultant is zero. This condition is called static equilibrium, with the result that the object remains at rest or moves with a constant velocity. As well as being added, forces can be can also be broken down (or 'resolved'). For example, a horizontal force pointing northeast can be split into two forces, one pointing north, and one pointing east. Summing these component forces using vector addition yields the original force. Force vectors can also be three-dimensional, with the third (vertical) component at right-angles to the two horizontal components.

Forces in theory

The total (Newtonian) force, in newtons, on an object at any given time is defined as the rate of change of the object's velocity multiplied by the object's mass: :

\mathbf = \lim_ \frac

where :m is the inertial mass of the particle (measured in kilograms) :v_o is its initial velocity (measured in metres per second) :v is its final velocity (measured in metres per second) :T is the time from the initial state to the final state (measured in seconds); :Lim T→0 is the limit as T tends towards zero. Force was so defined to explain the effects of superimposing situations: if in one situation, a force is experienced by a particle, and if in another situation another force is experienced by that particle, then in a third situation, which (according to standard physical practice) is taken to be a combination of the two individual situations, the force experienced by the particle will be the vector sum of the individual forces experienced in the first two situations. This superposition of forces, and the definition of inertial frames and inertial mass, are the empirical content of Newton's laws of motion. There are other ways to look at the above definition of force. First, the mass of a body multiplied by its velocity is called its momentum, p, so the above definition is equivalent to: :

\textbf=

If F is not constant over Δt, then this is the definition of average force over the time interval. To apply it at an instant we apply an idea from calculus. If we graph p as a function of time, the average force will be the slope of the line connecting the momentum at two times. Taking the limit as the two times get closer together gives the slope at an instant, which is called the derivative: :

\textbf=

Many forces are associated with a potential energy field. For instance, the gravitational force acting upon a body can be seen as the action of the gravitational field that is present at the body's location. The potential field is defined as that field whose gradient is equal and opposite to the force produced at every point: :

\textbf=-\nabla U

The derivative of force with respect to time is called yank. Higher order derivatives are sometimes used, but they lack names because of their rarity. In most explanations of mechanics, force is usually defined only implicitly, in terms of the equations that work with it. Some physicists, philosophers and mathematicians, such as Ernst Mach, Clifford Truesdell and Walter Noll, have found this problematic and sought a more explicit definition of force. According to the Special theory of relativity the mass of an object increases as it's velocity and therefore it's energy increases. The law of force must then be modified to the following: :

\textbf=

where :v is the mass's velocity :c is the speed of light. Note that the equation is undefined if the mass's speed is equal to c because one then has to divide by zero. This is one reason most physicists believe an object with nonzero rest mass can not be accelerated to the speed of light, as this would require an infinite force.

Units of measurement

The SI unit used to measure force is the newton (symbol N), which is equivalent to kg·m·s⁻².

Non-SI units of force and mass

The F=m·a relationship can be used with any consistent units (SI or CGS). If these units are not consistent, a more general form, F=k·m·a, can be used, where the constant k is a conversion factor dependent upon the units being used. For example, in imperial engineering units, F is measured in "pounds force" or "lbf", m in "pounds mass" or "lb", and a in feet per second squared. In this particular system, one needs to use the more general form above, usually written F=m·a/g_c with the constant normally used for this purpose g_c = 32.174 lb·ft/(lbf·s²) equal to the reciprocal of the k above. As with the kilogram, the pound is colloquially used as both a unit of mass and a unit of force. 1 lbf is the force required to accelerate 1 lb at 32.174 ft per second squared, since 32.174 ft per second squared is the standard acceleration due to terrestrial gravity. Another imperial unit of mass is the slug, defined as 32.174 lb. It is the mass that accelerates by one foot per second squared when a force of one lbf is exerted on it. When the standard gee (an acceleration of 9.80665 m/s²) is used to define pounds force, the mass in pounds is numerically equal to the weight in pounds force. However, even at sea level on Earth, the actual acceleration of free fall is quite variable, over 0.53% more at the poles than at the equator. Thus, a mass of 1.0000 lb at sea level at the equator exerts a force due to gravity of 0.9973 lbf, whereas a mass of 1.000 lb at sea level at the poles exerts a force due to gravity of 1.0026 lbf. The normal average sea level acceleration on Earth (World Gravity Formula 1980) is 9.79764 m/s², so on average at sea level on Earth, 1.0000 lb will exerts a force of 0.9991 lbf. The equivalence 1 lb = 0.453 592 37 kg is always true, by definition, anywhere in the universe. If you use the standard gee which is official for defining kilograms force to define pounds force as well, then the same relationship will hold between pounds-force and kilograms-force (an old non-SI unit is still used). If a different value is used to define pounds force, then the relationship to kilograms force will be slightly different—but in any case, that relationship is also a constant anywhere in the universe. What is not constant throughout the universe is the amount of force in terms of pounds-force (or any other force units) which 1 lb will exert due to gravity. By analogy with the slug, there is a rarely used unit of mass called the "metric slug". This is the mass that accelerates at one metre per second squared when pushed by a force of one kgf. An item with a mass of 10 kg has a mass of 1.01972661 metric slugs (= 10 kg divided by 9.80665 kg per metric slug). This unit is also known by various other names such as the hyl, TME (from a German acronym), and mug (from metric slug). Another unit of force called the poundal (pdl) is defined as the force that accelerates 1 lbm at 1 foot per second squared. Given that 1 lbf = 32.174 lb times one foot per second squared, we have 1 lbf = 32.174 pdl. The kilogram-force is a unit of force that was used in various fields of science and technology. In 1901, the CGPM improved the definition of the kilogram-force, adopting a standard acceleration of gravity for the purpose, and making the kilogram-force equal to the force exerted by a mass of 1 kg when accelerated by 9.80665 m/s². The kilogram-force is not a part of the modern SI system, but is still used in applications such as:
- Thrust of jet and rocket engines
- Spoke tension of bicycles
- Draw weight of bows
- Torque wrenches in units such as "meter kilograms" or "kilogram centimetres" (the kilograms are rarely identified as units of force)
- Engine torque output (kgf·m expressed in various word orders, spellings, and symbols)
- Pressure gauges in "kg/cm²" or "kgf/cm²" In colloquial, non-scientific usage, the "kilograms" used for "weight" are almost always the proper SI units for this purpose. They are units of mass, not units of force. The symbol "kgm" for kilograms is also sometimes encountered. This might occasionally be an attempt to disintinguish kilograms as units of mass from the "kgf" symbol for the units of force. It might also be used as a symbol for those obsolete torque units (kilogram-force metres) mentioned above, used without properly separating the units for kilogram and metre with either a space or a centered dot.

Conversions

Below are several coversion factors between various mesurements of force:
- 1 kgf (kilopond kp) = 9.80665 newtons
- 1 metric slug = 9.80665 kg
- 1 lbf = 32.174 poundals
- 1 slug = 32.174 lb
- 1 kgf = 2.2046 lbf

Forces in everyday life

Forces are part of everyday life, with examples such as:
- gravity: objects fall, even after being thrown upwards, or slide and roll down
- friction: floors and objects that are not extremely slippery
- spring force, objects resist tensile stress, compressive stress and/or shear stress, objects bounce back.
- electromagnetic force: attraction of magnets
- movement created by force: the movement of objects when force is applied.

Forces in the laboratory

Founding experiments

- Galileo Galilei used rolling balls to disprove the Aristotelian theory of motion (1602 - 1607)
- Henry Cavendish's torsion bar experiment measured the force of gravity between two masses (1798)

Instruments to measure forces

- spring balance
- pivot balance
- forcemeter

History

Force was first described by Archimedes.

References

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External links

- [http://jumk.de/calc/force.shtml Calculation: force F - English and American units to metric units]
- [http://calc.skyrocket.de/en/ Online Unit Converter - Conversion of many different units]
- [http://www.patbelford.com/gallery/web3d/education/forceworkpower/index.html Interactive demonstration of Force-Work-Power Relationship] Category:Introductory physics ko:힘 ms:Daya (fizik) ja:力 simple:Force (physics)

Spin (physics)

In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is generated by the motion of its center of mass about an external point. In classical mechanics, the spin angular momentum of a body is associated with the rotation of the body around its own center of mass. For instance, the spin angular momentum of the Earth is associated with its 24-hourly rotation about the polar axis, which gives rise to the day-night cycle. On the other hand, the orbital angular momentum of the Earth is associated with its motion around the Sun. The orbital period of this motion defines the year. Spin angular momentum is particularly important for systems at atomic length scales or smaller, such as individual atoms, protons, or electrons. The effects of quantum mechanics are important when describing such particles. Quantum mechanical spin possesses several unusual features, which will be described in the remainder of this article. (We will use the term "particle" to refer to such quantum mechanical systems, with the understanding that they actually exhibit wave-particle duality, and thus display both particle-like and wave-like behaviors.)

Spin of elementary and composite particles

One of the most remarkable discoveries associated with quantum physics is the fact that elementary particles can possess non-zero spin. Elementary particles are particles that cannot be divided into any smaller units, such as the photon, the electron, and the various quarks. Theoretical and experimental studies have shown that the spin possessed by these particles cannot be explained by postulating that they are made up of even smaller particles rotating about a common center of mass (see classical electron radius); as far as we can tell, these elementary particles are true point particles. The spin that they carry is a truly intrinsic physical property. The concept of elementary particle spin was first proposed in 1925 by Ralph Kronig, George Uhlenbeck, and Samuel Goudsmit. A later section covers the history of this hypothesis and its subsequent developments. According to quantum mechanics, the angular momentum of any system is quantized. The magnitude of angular momentum can only take on the values :

\hbar \, \sqrt,

where

\hbar

is Planck's constant divided by 2π (sometimes called Dirac's constant), and s is a non-negative integer or half-integer (0, 1/2, 1, 3/2, 2, etc.). For instance, electrons (which are elementary particles) are called "spin-1/2" particles because their intrinsic spin angular momentum has s = 1/2. The spin carried by each elementary particle has a fixed s value that depends only by the type of particle, and cannot be altered in any known way (although, as we will see, it is possible to change the direction in which the spin "points".) Every electron in existence possesses s = 1/2. Other elementary spin-1/2 particles include neutrinos and quarks. On the other hand, photons are spin-1 particles, whereas the hypothetical graviton is a spin-2 particle. The spin of composite particles, such as protons, neutrons, atomic nuclei, and atoms, is made up of the spins of the constituent particles, plus the orbital angular momentum of their motions around one another. The angular momentum quantization condition applies to both elementary and composite particles. Composite particles are often referred to as having a definite spin, just like elementary particles; for example, the proton is a spin-1/2 particle. This is understood to refer to the spin of the lowest-energy internal state of the composite particle (i.e., a given spin and orbital configuration of the constituents). It is not always easy to deduce the spin of a composite particle from first principles; for example, even though we know that the proton is a spin-1/2 particle, the question of how this spin is distributed among the three internal quarks and the surrounding gluons is an active area of research. [http://www.cerncourier.com/main/article/42/1/16]

Spin direction

In classical mechanics, the angular momentum of a particle possesses not only a magnitude (how fast the body is rotating), but also a direction (the axis of rotation of the particle). Quantum mechanical spin also contains information about direction, albeit in a more subtle form. Quantum mechanics states that the component of angular momentum measured along any direction (say along the z-axis) can only take on the values :

\hbar s_z, \qquad s_z = - s, - s + 1, \cdots, s

where s is the principal spin quantum number discussed in the previous section. One can see that there are 2s+1 possible values of s_z. For example, there are only two possible values for a spin-1/2 particle: s_z = +1/2 and s_z = -1/2. These correspond to quantum states in which the spin is pointing in the +z or -z directions respectively, and are often referred to as "spin up" and "spin down". See spin-1/2. For a given quantum state , it is possible to describe a spin vector ⟨S⟩ whose components are the expectation values of the spin components along each axis, i.e., ⟨S⟩ = [⟨s_x⟩, ⟨s_y⟩, ⟨s_z⟩]. This vector describes the "direction" in which the spin is pointing, corresponding to the classical concept of the axis of rotation. It turns out that the spin vector is not very useful in actual quantum mechanical calculations, because it cannot be measured directly — s_x, s_y and s_z cannot possess simultaneous definite values, because of a quantum uncertainty relation between them. As a qualitative concept, however, the spin vector is often handy because it is easy to picture classically. For instance, quantum mechanical spin can exhibit phenomena analogous to classical gyroscopic effects. For example, one can exert a kind of "torque" on an electron by putting it in a magnetic field (the field acts upon the electron's intrinsic magnetic dipole moment — see the following section). The result is that the spin vector undergoes precession, just like a classical gyroscope. Mathematically, quantum mechanical spin is not described by a vector as in classical angular momentum. It is described using a family of objects known as spinors. There are subtle differences between the behavior of spinors and vectors under coordinate rotations. For example, rotating a spin-1/2 particle by 360 degrees does not bring it back to the same quantum state, but to the state with the opposite quantum phase; this is detectable, in principle, with interference experiments. To return the particle to its exact original state, one needs a 720 degree rotation!

Spin and magnetic moment

Particles with spin can possess a magnetic dipole moment, just like a rotating electrically charged body in classical electrodynamics. These magnetic moments can be experimentally observed in several ways, e.g. by the deflection of particles by inhomogeneous magnetic fields in a Stern-Gerlach experiment, or by measuring the magnetic fields generated by the particles themselves. The intrinsic magnetic moment μ of a particle with charge q, mass m, and spin S, is :

\mu = g \, \frac\, S

where the dimensionless quantity g is called the gyromagnetic ratio or g-factor. The electron, despite being an elementary particle, possesses a nonzero magnetic moment. One of the triumphs of the theory of quantum electrodynamics is its accurate prediction of the electron g-factor, which has been experimentally determined to have the value 2.002319... The value of 2 arises from the Dirac equation, a fundamental equation connecting the electron's spin with its electromagnetic properties, and the correction of 0.002319... arises from the electron's interaction with the surrounding electromagnetic field. Composite particles also possess magnetic moments associated with their spin. In particular, the neutron possesses a non-zero magnetic moment despite being electrically neutral. This fact was an early indication that the neutron is not an elementary particle. In fact, it is made up of quarks, which are charged particles. The magnetic moment of the neutron comes from the moments of the individual quarks and their orbital motions. The neutrinos are both elementary and electrically neutral, and theory indicates that they have zero magnetic moment. The measurement of neutrino magnetic moments is an active area of research. As of 2003, the latest experimental results have put the neutrino magnetic moment at less than 1.3 × 10^-10 times the electron's magnetic moment. In ordinary materials, the magnetic dipole moments of individual atoms produce magnetic fields that cancel one another, because each dipole points in a random direction. In ferromagnetic materials, however, the dipole moments are all lined up with one another, producing a macroscopic, non-zero magnetic field. These are the ordinary "magnets" with which we are all familiar. The study of the behavior of such "spin models" is a thriving cottage industry in condensed matter physics. For instance, the Ising model describes spins (dipoles) that have only two possible states, up and down, whereas in the Heisenberg model the spin vector is allowed to point in any direction. These models have many interesting properties, which have led to many interesting results in the theory of phase transitions. [http://www.hermetic.ch/compsci/thesis/chap1.htm] [http://www.hermetic.ch/compsci/thesis/chap1.htm#s1.3]

The spin-statistics connection

It turns out that the spin of a particle is closely related to its properties in statistical mechanics. Particles with half-integer spin obey Fermi-Dirac statistics, and are known as fermions. They are subject to the Pauli exclusion principle, which forbids them from sharing quantum states, and are described in quantum theory by "antisymmetric states" (see the article on identical particles.) Particles with integer spin, on the other hand, obey Bose-Einstein statistics, and are known as bosons. These particles can share quantum states, and are described using "symmetric states". The proof of this is known as the spin-statistics theorem, which relies on both quantum mechanics and the theory of special relativity. In fact, the connection between spin and statistics is one of the most important and remarkable consequences of special relativity.

Applications

Well established applications of spin are Magnetic Resonance Imaging or MRI, and GMR drive head technology in modern hard disks. A possible application of spin is as a binary information carrier in spin transistors. Electronics based on spin transistors is called spintronics.

History

Wolfgang Pauli was possibly the most influential physicist in the theory of spin. Spin was first discovered in the context of the emission spectrum of alkali metals. In 1924 Pauli introduced what he called a "two-valued quantum degree of freedom" associated with the electron in the outermost shell. This allowed him to formulate the Pauli exclusion principle, stating that no two electrons can share the same quantum numbers. The physical interpretation of Pauli's "degree of freedom" was initially unknown. Ralph Kronig, one of Landé's assistants, suggested in early 1925 that it was produced by the self-rotation of the electron. When Pauli heard about the idea, he criticized it severely, noting that the electron's hypothetical surface would have to be moving faster than the speed of light in order for it to rotate quickly enough to produce the necessary angular momentum. This would violate the theory of relativity. Largely due to Pauli's criticism, Kronig decided not to publish his idea. In the fall of that year, the same thought came to two young Dutch physicists, George Uhlenbeck and Samuel Goudsmit. Under the advice of Paul Ehrenfest, they published their results in a small paper. It met a favorable response, especially after Llewellyn Thomas managed to resolve a factor of two discrepancy between experimental results and Uhlenbeck and Goudsmit's calculations (and Kronig's unpublished ones). This discrepancy was due to the necessity to take into account the orientation of the electron's tangent frame, in addition to its position; mathematically speaking, a fiber bundle description is needed. The tangent bundle effect is additive and relativistic (i.e. it vanishes if c goes to infinity); it is one half of the value obtained without regard for the tangent space orientation, but with opposite sign. Thus the combined effect differs from the latter by a factor two (Thomas precession). Despite his initial objections to the idea, Pauli formalized the theory of spin in 1927, using the modern theory of quantum mechanics discovered by Schrödinger and Heisenberg. He pioneered the use of Pauli matrices as a representation of the spin operators, and introduced a two-component spinor wave-function. Pauli's theory of spin was non-relativistic. However, in 1928, Paul Dirac published the Dirac equation, which described the relativistic electron. In the Dirac equation, a four-component spinor (known as a "Dirac spinor") was used for the electron wave-function. In 1940, Pauli proved the spin-statistics theorem, which states that fermions have half-integer spin and bosons integer spin.

References

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- "[http://www.sciam.com/article.cfm?articleID=0007A735-759A-1CDD-B4A8809EC588EEDF Spintronics. Feature Article]" in Scientific American, June 2002

External links

[http://www.lorentz.leidenuniv.nl/history/spin/goudsmit.html Goudsmit on the discovery of electron spin] Category:Rotational symmetry Category:Quantum field theory ko:스핀 ja:スピン角運動量

Vector field

In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. Vector fields are often used in physics to model for example the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from point to point. In the rigorous mathematical treatment, (tangent) vector fields are defined on manifolds as sections of the manifold's tangent bundle.

Definition

Given a subset S in Rⁿ a vector field is represented by a vector-valued function

V: S \to \mathbf^n

in standard Euclidean coordinates (x₁, ..., x_n). If there is another coordinate system y, then

V_y := \frac V

is the expression for the same vector field in the new coordinates. In particular a vector field is not a bunch of scalar fields. We say V is a C^k vector field if V is k times continuously differentiable. A point p in S is called stationary if the vector at that point is zero (

V(p) = 0

). A vector field can be visualized as a n-dimensional space with a n-dimensional vector attached to each point. Given two C^k-vector fields V, W defined on S and a real valued C^k-function f defined on S, the two operations scalar multiplication and vector addition :

(fV)(p) := f(p)V(p)

(V+W)(p) := V(p) + W(p)

define the module of C^k-vector fields over the ring of C^k-functions.

Notes

Vector fields should be compared to scalar fields, which associate a number or scalar to every point in space (or every point of some manifold). The divergence and curl are two operations on a vector field which result in a scalar field and another vector field respectively. The first of these operations is defined in any number of dimensions (that is, for any value of n). The curl however, is defined only for n=3, but it can be generalized to an arbitrary dimension using the exterior product and exterior derivative.

Examples

- A vector field for the movement of air on Earth will associate for every point on the surface of the Earth a vector with the wind speed and direction for that point. This can be drawn using arrows to represent the wind; the length (magnitude) of the arrow will be an indication of the wind speed. A "high" on the usual barometric pressure map would then act as a source (arrows pointing away), and a "low" would be a sink (arrows pointing towards), since air tends to move from high pressure areas to low pressure areas.
- Velocity field of a moving fluid. In this case, a velocity vector is associated to each point in the fluid.
- There are 3 types of lines that can be made from vector fields. They are : ::streaklines — as revealed in wind tunnels using smoke. ::fieldlines — as a line depicting the instantaneous field at a given time. ::pathlines — showing the path that a given particle (of zero mass) would follow.
- Magnetic fields. The fieldlines can be revealed using small iron filings.
- Maxwell's equations allow us to use a given set of initial conditions to deduce, for every point in Euclidean space, a magnitude and direction for the force experienced by a charged test particle at that point; the resulting vector field is the electromagnetic field.

Gradient field

Vector fields can be constructed out of scalar fields using the vector operator gradient which gives rise to the following definition. A vector field V over S is called a gradient field or a conservative field if there exists a real valued function f on X(a scalar field) such that

V = \nabla f.

The path integral along any closed curve γ (γ(0) = γ(1)) in a gradient field is zero. :

\int_\gamma \langle V(x), \mathrmx \rangle = \int_\gamma \langle \nabla f(x), \mathrmx \rangle = f((\gamma)(1)) - f((\gamma)(0))

Central field

A C^∞-vector field over Rⁿ \ is called a central field if :

V(T(p)) = T(V(p)) \qquad (T \in \mathrm(n, \mathbf))

where O(n, R) is the orthogonal group. We say central fields are invariant under orthogonal transformations around 0. The point 0 is called the center of the field. Since orthogonal transformations are actually rotations and reflections, the invariance conditions mean that vectors of a central field are always directed towards, or away from, 0; this is an alternate (and simpler) definition. A central field is always a gradient field, since defining it on one semiaxis and integrating gives an antigradient.

Curve integral

A common technique in physics is to integrate a vector field along a curve: a path integral. Given a particle in a gravitational vector field, where each vector represents the force acting on the particle at this point in space, the curve integral is the work done on the particle when it travels along a certain path. The curve integral is constructed analogously to the Riemann integral and it exists if the curve is rectifiable (has finite length) and the vector field is continuous. Given a vector field V and a curve γ parametrized by [0, 1] the curve integral is defined as :

\int_\gamma \langle V(x), \mathrmx \rangle = \int_0^1 \langle V(\gamma(t)), \gamma'(t)\;\mathrmt \rangle

Flow curves

Vector fields have a nice interpretation in terms of autonomous, first order ordinary differential equations. Given a vector field V defined on S, we can try to define curves γ on S such that for each t in an interval I :

\gamma'(t) = V(\gamma(t))

If V is Lipschitz continuous we can find a unique C¹-curve γ_x for each point x in X so that :

\gamma_x(0) = x

\gamma'_x(t) = V(\gamma_x(t)) \qquad ( t \in (-\epsilon, +\epsilon) \subset \mathbf)

The curves γ_x are called flow curves of the vector field V and partition S into equivalence classes. It is not always possible to extend the interval (-ε, +ε) to the whole real number line. The flow may for example reach the edge of S in a finite time. In two or three dimensions one can visualize the vector field as given rise to a flow on S. If we drop a particle into this flow at a point p it will move along the curve γ_p in the flow depending on the initial point p. If p is a stationary point of V then the particle will remain at p. Typical applications are streamline in fluid, geodesic flow, and one-parameter subgroups and the exponential map in Lie groups.

Difference between scalar and vector field

The difference between a scalar and vector field is not that a scalar is just one number while a vector is several numbers. The difference is in how their coordinates respond to coordinate transformations. A scalar is a coordinate whereas a vector can be described by coordinates, but it is not the collection of its coordinates.

Example 1

This example is about 2-dimensional Euclidean space (R²) where we examine Euclidean (x, y) and polar (r, θ) coordinates (which are undefined at the origin). Thus x = r cos θ and y = r sin θ and also r² = x² + y², cos θ = x/(x² + y²) and sin θ = y/(x² + y²). Suppose we have a scalar field which is given by the constant function 1, and a vector field which attaches a vector in the r-direction with length 1 to each point. More precisely, they are given by the functions :

s_:(r, \theta) \mapsto 1, \quad v_:(r, \theta) \mapsto (1, 0).

Let us convert these fields to Euclidean coordinates. The vector of length 1 in the r-direction has the x coordinate cos θ and the y coordinate sin θ. Thus in Euclidean coordinates the same fields are described by the functions :

s_:(x, y) \mapsto 1, \quad v_:(x, y) \mapsto (\cos \theta, \sin \theta) = \left(\frac, \frac\right).

We see that while the scalar field remains the same, the vector field now looks different. The same holds even in the 1-dimensional case, as illustrated by the next example.

Example 2

Consider the 1-dimensional Euclidean space R with its standard Euclidean coordinate x. Also consider the coordinate ξ := 2x. Suppose we have a scalar field and a vector field which are both given in the ξ coordinates by the constant function 1, :

s_:\xi \mapsto 1, \quad v_:\xi \mapsto 1.

Thus, we have a scalar field which has the value 1 everywhere and a vector field which attaches a vector in the ξ-direction with magnitude 1 unit of ξ to each point. But if ξ changes one unit then x changes 2 units. Then, this vector field has a magnitude of 2 in units of x. Therefore, in the x coordinate the scalar field and the vector field are described by the functions :

s_:x \mapsto 1, \quad v_:x \mapsto 2,

which are different.

Example 3

In 1D, an example of a scalar field is the electric potential V, which is e.g. 20 volt at a particular point. This is a scalar, not depending on the coordinate system. An electric field at that point of 5 volt/metre in some coordinate system is −5 volt/metre in an inverse coordinate system. Since a physical quantity is not just a number, but a number times a unit, there is no change of coordinate system that gives any other than one of these two values for the electric field at the point.

External links

- [http://mathworld.wolfram.com/VectorField.html Vector field] -- Mathworld
- [http://planetmath.org/encyclopedia/VectorField.html Vector field] -- PlanetMath
- [http://www.amasci.com/electrom/statbotl.html 3D Magnetic field viewer]
- [http://www-solar.mcs.st-and.ac.uk/~alan/MT3601/Fundamentals/node2.html Vector fields and field lines]
- [http://www.vias.org/simulations/simusoft_vectorfields.html Vector field simulation] An interactive application to show the effects of vector fields Category:Differential topology Category:Vector calculus Category:Infographics

Vector (spatial)

:This article discusses vectors that have a particular relation to the spatial coordinates. For a generalization, see vector space. In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). Although it is often described by a number of "components", each of which is dependent upon the particular coordinate system being used, a vector is an object with properties which do not depend on the coordinate system used to describe it. A common example of a vector is force — it has a magnitude and an orientation in three dimensions (or however many spatial dimensions one has), and multiple forces sum according to the parallelogram law. A spatial vector can be formally defined by its relationship to the spatial coordinate system under rotations. Alternatively, it can be defined in a coordinate-free fashion via a tangent space of a three-dimensional manifold in the language of differential geometry. These definitions are discussed in more detail below. A spatial vector is a special case of a tensor and is also analogous to a four-vector in relativity (and is sometimes therefore called a three-vector in reference to the three spatial dimensions, although this term also has another meaning for p-vectors of differential geometry). Vectors are the building blocks of vector fields and vector calculus. The word vector is also now used for more general concepts (see also vector and generalizations below), but this article describes the original spatial meaning except where otherwise noted.

Definitions

Informally, a vector is a quantity characterized by a magnitude (in mathematics a number, in physics a number times a unit) and a direction, often represented graphically by an arrow. Examples are "moving north at 90 km/h" or "pulling towards the center of Earth with a force of 70 newtons". The notion of having a "magnitude" and "direction" is formalized by saying that the vector has components that transform like the coordinates under rotations. That is, if the coordinate system undergoes a rotation described by a rotation matrix R, so that a coordinate vector x is transformed to x' = Rx, then any other vector v is similarly transformed via v' = Rv. This ensures the invariance of the operations dot product, Euclidean norm, cross product, gradient, divergence, curl, and scalar triple product, and trivially for vector addition and subtraction, and scalar multiplication. The terms scalar and vector as used here include pseudoscalars and pseudovectors or axial vectors (see also below). Accordingly, let, for example, each of two vectors be expressed as three space coordinates, and apply the formula for the cross product, resulting in three coordinates, which represent a third vector. If we rewrite the two vectors in rotated coordinates, and apply the formula for the cross product again, then the result is the original cross product in terms of rotated coordinates. Also, let, for example, a vector field be expressed as three space coordinate functions of three variables, and apply the formula for the curl based on these functions, resulting in three additional functions, which represent a second vector field. If we rewrite the original vector field in terms of rotated position coordinates and correspondingly rotated coordinates for the vector function values, and apply the formula for the curl based on these functions, then the result is the rewritten version of the original curl: also in terms of rotated position coordinates and correspondingly rotated coordinates for the vector function values. The same applies for dot product, gradient, divergence, vector addition and scalar multiplication. For these, also reflection in a plane can be applied. The scalars involved should not be transformed (e.g. in the case of a rotation by 180°, the scalar should not be multiplied by -1). Thus even in 1D we have to distinguish scalars and vectors: 2 × 3 = 6 can be interpreted as a scalar multiplication or a dot product, but not as a product of two vectors. Similarly differentiation in 1D can be interpreted as a gradient or a divergence: one of the two functions is scalar and one a vector, and the argument is a vector, ensuring invariance under inversion of the vectors without changing the scalars. Since rotation of the three Cartesian coordinate axes changes the formulas the same as an inverse rotation of the field itself, we can also conclude:
- if the same rotation is applied to two vectors, then the cross product is correspondingly rotated, but the dot product remains the same
- rotation of a scalar field results in a correspondingly rotated vector field for the gradient
- rotation of a vector field results in a correspondingly rotated scalar field for the divergence and a correspondingly rotated vector field for the curl where rotation of a scalar field involves only rotation of the position vectors, while rotation of a vector field involves also a corresponding rotation of the vector field values. Note that the concept of corresponding rotations applies even if different coordinate systems are used for field values and position vectors, so that e.g. for one we multiply by an orthogonal matrix and for the other we add an angle to an angle coordinate. In order to use the usual formulas, e.g. to compute mechanical work, the x-axis of forces should be in the same direction as the x-axis of position, etc. When, as described above, coordinate rotations of position are accompanied by corresponding coordinate rotations of forces, this property is preserved. On the other hand, the origin of forces is simply at the zero force (no force), while the origin of position can be chosen as desired. For example, work depends on displacement, which is the difference of positions and therefore does not depend on the origin. Position and function value of a vector field are often, but not necessarily, expressed in similar coordinate systems. For example gravitational field strength due to a particular point mass may be

g_r = -9.8 (r/r_0)^ m/s^2

, with both the function value and the position vector in spherical coordinates. For the position vector the origin is chosen here at the center of the point mass; for the field strength the origin is simply at "zero field strength" anyway. How the other two coordinates are chosen does not matter in this case, because the field does not depend on them, and the field has no components in their directions. More generally, a vector is a tensor of contravariant rank one. In differential geometry, the term vector usually refers to quantities that are closely related to tangent spaces of a differentiable manifold (assumed to be three-dimensional and equipped with a positive definite Riemannian metric). (A four-vector is a related concept when dealing with a 4 dimensional spacetime manifold in relativity.) Examples of vectors include displacement, velocity, electric field, momentum, force, and acceleration. Vectors can be contrasted with scalar quantities such as distance, speed, energy, time, temperature, charge, power, work, and mass, which have magnitude, but no direction (they are invariant under coordinate rotations). The magnitude of any vector is a scalar. A related concept is that of a pseudovector (or axial vector). This is a quantity that transforms like a vector under proper rotations, but gains an additional sign flip under improper rotations. Examples of pseudovectors include magnetic field, torque, and angular momentum. (This distinction between vectors and pseudovectors is often ignored, but it becomes important in studying symmetry properties.) To distinguish from pseudo/axial vectors, an ordinary vector is sometimes called a polar vector. See also parity (physics). Sometimes, one speaks informally of bound or fixed vectors, which are vectors additionally characterized by a "base point". Most often, this term is used for position vectors (relative to an origin point). More generally, however, the physical interpretation of a particular vector can be parameterized by any number of quantities.

Examples in one dimension

A force may be "15N to the right", with coordinate 15N if the basis vector is to the right, and −15N if the basis vector is to the left. The magnitude of the vector is 15N in both cases. A displacement may be "4m to the right", with coordinate 4m if the basis vector is to the right, and −4m if the basis vector is to the left. The magnitude of the vector is 4m in both cases. The work done by the force in the case of this displacement is 60J in both cases. The force and displacement are vectors, the magnitudes are scalars, and the coordinates are neither.

Generalizations

In mathematics, a vector is any element of a vector space over some field. The spatial vectors of this article are a very special case of this general definition (they are not simply any element of R^d in d dimensions), which includes a variety of mathematical objects (algebras, the set of all functions from a given domain to a given linear range, and linear transformations). Note that under this definition, a tensor is a special vector!

Representation of a vector

Symbols standing for vectors are usually printed in boldface as a; this is also the convention adopted in this encyclopedia. Other conventions include

\vec

or a, especially in handwriting. Alternately, some use a tilde (~) placed under the vector. The length or magnitude or norm of the vector a is denoted by |a|. Vectors are usually shown in graphs or other diagrams as arrows, as illustrated below: Image:vecab.png Here the point A is called the tail, base, start, or origin; point B is called the head, tip, endpoint, or destination. The length of the arrow represents the vector's magnitude, while the direction in which the arrow points represents the vector's direction. If a vector is itself spatial, the length of the arrow depends on a dimensionless scale. If it represents e.g. a force, the "scale" is of physical dimension length/force. Thus there is typically consistency in scale among quantities of the same dimension, but otherwise scale ratios may vary; for example, if "1 newton" and "5 m" are both represented with an arrow of 2cm, the scales are 1:250 and 1m:50N respectively. Equal length of vectors of different dimension has no particular significance unless there is some proportionality constant inherent in the system that the diagram represents. Also length of a unit vector (of dimension length, not length/force, etc.) has no coordinate-system-invariant significance. In the figure above, the arrow can also be written as

\overrightarrow

or AB In order to calculate with vectors, the graphical representation is too cumbersome. Vectors in a n-dimensional Euclidean space can be represented as a linear combination of n mutually perpendicular unit vectors. In this article, we will consider R³ as an example. In R³, we usually denote the unit vectors parallel to the x-, y- and z-axes by i, j and k respectively. Any vector a in R³ can be written as a = a₁i + a₂j + a₃k with real numbers a₁, a₂ and a₃ which are uniquely determined by a. Sometimes a is then also written as a 3-by-1 or 1-by-3 matrix: :

= \begin
 a_1\\
 a_2\\
 a_3\\
\end

= \begin
 a_1 & a_2 & a_3 \\
\end

even though this notation suppresses the dependence of the coordinates a₁, a₂ and a₃ on the specific choice of coordinate system i, j and k.

Length of a vector

The length of the vector a = a₁i + a₂j + a₃k can be computed with the Euclidian norm :

\left\|\mathbf\right\|=\sqrt

which is a consequence of the Pythagorean theorem.

Vector equality

Two vectors are said to be equal if they have the same magnitude and direction. However if we are talking about bound vector, then two bound vectors are equal if they have the same base point and end point. For example, the vector i + 2j + 3k with base point (1,0,0) and the vector i+2j+3k with base point (0,1,0) are different bound vectors, but the same (unbounded) vector.

Vector addition and subtraction

Let a=a₁i + a₂j + a₃k and b=b₁i + b₂j + b₃k. The sum of a and b is: :

\mathbf+\mathbf
=(a_1+b_1)\mathbf
+(a_2+b_2)\mathbf
+(a_3+b_3)\mathbf

The addition may be represented graphically by placing the start of the arrow b at the tip of the arrow a, and then drawing an arrow from the start of a to the tip of b. The new arrow drawn represents the vector a + b, as illustrated below: Pythagorean theorem This addition method is sometimes called the parallelogram rule because a and b form the sides of a parallelogram and a + b is one of the diagonals. If a and b are bound vectors, then the addition is only defined if a and b have the same base point, which will then also be the base point of a + b. One can check geometrically that a + b = b + a and (a + b) + c = a + (b + c). The difference of a and b is: :

\mathbf-\mathbf
=(a_1-b_1)\mathbf
+(a_2-b_2)\mathbf
+(a_3-b_3)\mathbf

Subtraction of two vectors can be geometrically defined as follows: to subtract b from a, place the ends of a and b at the same point, and then draw an arrow from the tip of b to the tip of a. That arrow represents the vector a − b, as illustrated below: parallelogram If a and b are bound vectors, then the subtraction is only defined if they share the same base point which will then also become the base point of their difference. This operation deserves the name "subtraction" because (a − b) + b = a. In physics, vectors of different physical dimension may occur in the same diagram. However, adding or subtracting them (graphically or otherwise) is meaningless.

Scalar multiplication

A vector may also be multiplied by a real number r. In mathematics numbers are often called scalars to distinguish them from vectors, and this operation is therefore called scalar multiplication. The resulting vector is: :

r\mathbf=(ra_1)\mathbf
+(ra_2)\mathbf
+(ra_3)\mathbf

The length of ra is |r||a|. If the scalar is negative, it also changes the direction of the vector by 180^o. Two examples (r = -1 and r = 2) are given below: real number Here it is important to check that the scalar multiplication is compatible with vector addition in the following sense: r(a + b) = ra + rb for all vectors a and b and all scalars r. One can also show that a - b = a + (-1)b. The set of all geometrical vectors, together with the operations of vector addition and scalar multiplication, satisfies all the axioms of a vector space. Similarly, the set of all bound vectors with a common base point forms a vector space. This is where the term "vector space" originated. In physics, scalars also have a unit. The scale of acceleration in the diagram is e.g. 2 m/s² : cm, and that of force 5 N : cm. Thus a scale ratio of 2.5 kg : 1 is used for mass. Similarly, if displacement has a scale of 1:1000 and velocity of 0.2 cm : 1 m/s, or equivalently, 2 ms : 1, a scale ratio of 0.5 : s is used for time.

Unit vector

Main article: Unit vector A unit vector is any vector with a length of one. If you have a vector of arbitrary length, you can use it to create a unit vector. This is known as normalizing a vector. Unit vector To normalize a vector a = [a₁, a₂, a₃], scale the vector by the inverse of its length ||a||. That is: :

\mathbf=\frac=\frac\mathbf+\frac\mathbf+\frac\mathbf

Dot product

Main article: Dot product The dot product of two vectors a and b (sometimes called inner product, or, since its result is a scalar, the scalar product) is denoted by a·b and is defined as: :

\mathbf\cdot\mathbf
=\left\|\mathbf\right\|\left\|\mathbf\right\|\cos(\theta)

where ||a|| and ||b|| denote the norm (or length) of a and b, and θ is the measure of the angle between a and b (see trigonometric function for an explanation of cosine). Geometrically, this means that a and b are drawn with a common start point and then the length of a is multiplied with the length of that component of b that points in the same direction as a. This operation is often useful in physics; for instance, work is the dot product of force and displacement.

Cross product

The cross product (also vector product or outer product) differs from the dot product primarily in that the result of a cross product of two vectors is a vector. While everything that was said above can be generalized in a straightforward manner to more than three dimensions, the cross product is only meaningful in three dimensions (although a related product exists in seven dimensions - see below). The cross product, denoted a×b, is a vector perpendicular to both a and b and is defined as: :

\mathbf\times\mathbf
=\left\|\mathbf\right\|\left\|\mathbf\right\|\sin(\theta)\,\mathbf

where θ is the measure of the angle between a and b, and n is a unit vector perpendicular to both a and b. The problem with this definition is that there are two unit vectors perpendicular to both b and a. Which vector is the correct one depends upon the orientation of the vector space, i.e. on the handedness of the coordinate system. The coordinate system i, j, k is called right handed, if the three vectors are situated like the thumb, index finger and middle finger (pointing straight up from your palm) of your right hand. Graphically the cross product can be represented by this figure Image:crossproduct.png In such a system, a×b is defined so that a, b and a×b also becomes a right handed system. If i, j, k is left-handed, then a, b and a×b is defined to be left-handed. Because the cross product depends on the choice of coordinate systems, its result is referred to as a pseudovector. Fortunately, in nature cross products tend to come in pairs, so that the "handedness" of the coordinate system is undone by a second cross product. The length of a×b can be interpreted as the area of the parallelogram having a and b as sides.

Scalar triple product

The scalar triple product (also called the box product or mixed triple product) isn't really a new operator, but a way of applying the other two multiplication operators to three vectors. The scalar triple product is denoted by (a b c) and defined as: :

(\mathbf\ \mathbf\ \mathbf)
=\mathbf\cdot(\mathbf\times\mathbf)

It has three primary uses. First, the absolute value of the box product is the volume of the parallelepiped which has edges that are defined by the three vectors. Second, the scalar triple product is zero if and only if the three vectors are linearly dependent, which can be easily proved by considering that in order for the three vectors to not make a volume, they must all lie in the same plane. Third, the box product is positive if and only if the three vectors a, b and c are oriented like the coordinate system i, j and k. In coordinates, if the three vectors are thought of as rows, the scalar triple product is simply the determinant of the 3-by-3 matrix having the three vectors as rows. The scalar triple product is linear in all three entries and anti-symmetric in the following sense: : Technically, the scalar triple product is not a scalar, it is a pseudoscalar: under a coordinate inversion (x goes to −x), it flips sign.

Vectors as directional derivatives

A vector may also be defined as a directional derivative: consider a function

f(x^\alpha)

and a curve

x^\alpha (\sigma)

. Then the directional derivative of

f

is a scalar defined as

\frac = \frac\frac.

where the index

\alpha

is summed over the appropriate number of dimensions (e.g. from 1 to 3 in 3-dimensional Euclidian space, from 0 to 3 in 4-dimensional spacetime, etc.). Then consider a vector tangent to

x^\alpha (\sigma)

t^\alpha = \frac.

We can rewrite the directional derivative in differential form (without a given function

f

) as

\frac = t^\alpha\frac.

Therefore any directional derivative can be identified with a corresponding vector, and any vector can be identified with a corresponding directional derivative. We can therefore define a vector precisely: :

\mathbf \equiv a^\alpha \frac.