Introduction to electromagnetism: Difference between revisions

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{{short description|A nonNon-technical introduction to topics in electromagnetism}}

{{about|a conceptual understanding of the topic|a more detailed mathematical treatment|electromagnetic fieldelectromagnetism}}

'''Electromagnetism''' is one of the [[Fundamental interaction|fundamental forces]] of nature. Early on, [[electricity]] and [[magnetism]] were studied separately and regarded as separate phenomena. [[Hans Christian Ørsted]] discovered that the two were related – [[electric current]]s give rise to magnetism. [[Michael Faraday]] discovered the converse, that magnetism could [[electromagnetic induction|induce]] electric currents, and [[James Clerk Maxwell]] put the whole thing together in a unified theory of [[electromagnetism]]. [[Maxwell's equations]] further indicated that [[electromagnetic wave]]s existed, and the experiments of [[Heinrich Hertz]] confirmed this, making [[radio]] possible. Maxwell also postulated, correctly, that [[light]] was a form of electromagnetic wave, thus making all of [[optics]] a branch of electromagnetism. [[Radio wave]]s differ from light only in that the [[wavelength]] of the former is much longer than the latter. [[Albert Einstein]] showed that the [[magnetic field]] arises through the [[Classical electromagnetism and special relativity|relativistic motion]] of the [[electric field]] and thus magnetism is merely a side effect of electricity. The modern theoretical treatment of electromagnetism is as a [[quantum field]] in [[quantum electrodynamics]].

In many situations of interest to [[electrical engineering]], it is not necessary to apply quantum theory to get correct results. [[Classical physics]] is still an accurate approximation in most situations involving [[macroscopic]] objects. With few exceptions, quantum theory is only necessary at the [[atomic scale]] and a simpler classical treatment can be applied. Further simplifications of treatment are possible in limited situations. [[Electrostatics]] deals only with stationary [[electric charge]]s so magnetic fields do not arise and are not considered. [[Permanent magnet]]s can be described without reference to electricity or electromagnetism. [[Circuit theory]] deals with [[electrical network]]s where the fields are largely confined around current carrying [[Electrical conductor|conductors]]. In such circuits, even Maxwell's equations can be dispensed with and simpler formulations used. On the other hand, a quantum treatment of electromagnetism is important in [[chemistry]]. [[Chemical reaction]]s and [[chemical bond]]ing are the result of [[quantum mechanical]] interactions of [[electron]]s around [[atom]]s. Quantum considerations are also necessary to explain the behaviour of many electronic devices, for instance the [[tunnel diode]].

== Electric charge ==

[[File:CoulombsLaw scal-2.svgpng|thumb|[[282x282px|Coulomb's law|Coulomb's force]]tells forus that like charges (top)repel and opposite charges (bottom)attract.]]

Electromagnetism is one of the [[Fundamental interaction|fundamental forces of nature]] alongside [[gravity]], the [[Strong interaction|strong force]] and the [[Weak interaction|weak force]]. Whereas gravity acts on all things that have [[mass]], electromagnetism acts on all things that have [[electric charge]]. Furthermore, as there is the [[conservation of mass]] according to which mass cannot be created or destroyed, there is also the [[conservation of charge]] which means that the charge in a closed system (where no charges are leaving or entering) must remain constant.<ref name=":0">{{Cite book|last=Purcell, Edward M.|title=Electricity and magnetism|date=21 January 2013|isbn=978-1-107-01402-2|edition=Third|___location=Cambridge|pages=3–4|oclc=805015622}}</ref> The fundamental law that describes the gravitational force on a massive object in [[classical physics]] is [[Newton's law of gravity]]. Analogously, [[Coulomb's law]] is the fundamental law that describes the force that charged objects exert on one another. It is given by the formula

[[File:VFPt charges plus minus thumb.svg|thumb|[[Field line|Electric field lines]] point from positive charges to negative charges.]]

: <math>F=k_\text{e}{q_1q_2\over r^2}</math>

where ''F'' is the force, ''k''_e is the [[Coulomb constant]], ''q''₁ and ''q''₂ are the [[Magnitude (mathematics)|magnitudes]] of the two charges, and ''r''² is the square of the distance between them. It describes the fact that like charges repel one another whereas opposite charges attract one another and that the stronger the charges of the particles, the stronger the force they exert on one another. The law is also an [[Inverse-square law|inverse square law]] which means that as the distance between two particles is doubled, the force on them is reduced by a factor of four.<ref>{{Cite book|last=Walker, Jearl, 1945-|title=Fundamentals of physics|date=2011|publisher=Wiley|others=Halliday, David, 1916-2010., Resnick, Robert, 1923-2014.|isbn=978-0-470-46911-8|edition=9th|___location=Hoboken, NJ|pages=578|oclc=435710913}}</ref>

== Electric and magnetic fields ==

[[File:VFPt_charges_plus_minus_thumb.svg|thumb|[[Field line|Electric field lines]] point from positive charges to negative charges.]]

{{Multiple image

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| image2 = Openstax college-physics 22.17 Lorentz-force-right-hand.jpg

| caption2footer = The force exerted on a positive charge by an electric field (topleft) and a magnetic field (bottomright) combine to give the [[Lorentz force]].

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In physics, [[Field (physics)|fields]] are entities that interact with matter and can be described mathematically by assigning a value to each point in space and time. [[Vector field]]s are fields which are assigned both a numerical value and a direction at each point in space and time. Electric charges produce a vector field called the [[electric field]]. The numerical value of the electric field, also called the electric field strength, determines the strength of the electric force that a charged particle will feel in the field and the direction of the field determines which direction the force will be in. By convention, the direction of the electric field is the same as the direction of the force on positive charges and opposite to the direction of the force on negative charges.<ref name=":2">{{Cite web|last=Pumplin|first=Jon|date=2000|title=Electric field lines|url=https://web.pa.msu.edu/courses/2000fall/phy232/lectures/efields/efieldlines.html|access-date=18 October 2018|website=Michigan State University Physics}}</ref><ref name=":3">{{Cite web|last=Nave|first=R|title=Electric Field|url=http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elefie.html|access-date=16 October 2018|website=Georgia State University Hyperphysics}}</ref> Because positive charges are repelled by other positive charges and are attracted to negative charges, this means the electric fields point away from positive charges and towards negative charges. These properties of the electric field are encapsulated in the equation for the electric force on a charge written in terms of the electric field:

Electromagnetism is one of the [[Fundamental interaction|fundamental forces of nature]] alongside [[gravity]], the [[Strong interaction|strong force]] and the [[Weak interaction|weak force]] . Whereas gravity acts on all things that have mass, electromagnetism acts on all things that have [[electric charge]]. But unlike gravity, whilst mass can only be positive, charge can be both positive and negative. Furthermore, whilst positive masses exert an attractive [[Newton's law of universal gravitation|gravitational force]] on one another, positive charges exert an attractive [[Coulomb's law|electric force]] only on oppositely charged negative charges (and vice versa) and a repulsive electric force on other positive charges (negative charges also repel other negative charges).<ref name=":0">{{Cite book|last=Purcell, Edward M.|url=https://www.worldcat.org/oclc/805015622|title=Electricity and magnetism|date=21 January 2013|publisher=|isbn=978-1-107-01402-2|edition=Third|___location=Cambridge|pages=3–4|oclc=805015622}}</ref> The electric force between charged particles is called the Coulomb force and is described by [[Coulomb's law]] which states that the electric force between two charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them:<ref>{{Cite book|last=Walker, Jearl, 1945-|url=https://www.worldcat.org/oclc/435710913|title=Fundamentals of physics|date=2011|publisher=Wiley|others=Halliday, David, 1916-2010., Resnick, Robert, 1923-2014.|isbn=978-0-470-46911-8|edition=9th|___location=Hoboken, NJ|pages=578|oclc=435710913}}</ref>

<math display="block">F = qE</math>

where ''F'' is the force on a charge ''q'' in an electric field ''E''.<ref name=":3" /><ref>{{Cite book|last=Purcell, Edward M. |title=Electricity and magnetism |date=21 January 2013| isbn=978-1-107-01402-2 |edition=Third |___location=Cambridge |pages=7 |oclc=805015622}}</ref>

As well as producing an electric field, charged particles will produce a [[magnetic field]] when they are in a state of motion that will be felt by other charges that are in motion (as well as [[permanent magnet]]s).<ref>{{Cite web|title=The Feynman Lectures on Physics Vol. II Ch. 1: Electromagnetism |url=https://feynmanlectures.caltech.edu/II_01.html#Ch1-S2 | access-date=2018-10-30 |website=feynmanlectures.caltech.edu}}</ref> The direction of the force on a moving charge from a magnetic field is perpendicular to both the direction of motion and the direction of the magnetic field lines and can be found using the [[right-hand rule]]. The strength of the force is given by the equation

<math>F=k_e{q_1q_2\over r^2}</math>

<math display="block">F = qvB \sin\theta</math>

where ''F'' is the force on a charge ''q'' with speed ''v'' in a magnetic field ''B'' which is pointing in a direction of angle ''θ'' from the direction of motion of the charge.<ref name=":6">{{Cite web| title=Magnetic forces| url=http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magfor.html#c2 | access-date=2020-11-26 | website=hyperphysics.phy-astr.gsu.edu}}</ref>

where ''F'' is the Coulomb force, ''k_e'' is the [[Coulomb constant]], ''q₁'' and ''q₂'' are the charges of the two particles, and ''r²'' is the square of the distance between them.

Electric charge has several important properties:

* it is ''quantised'': this means that it can only take integer multiple values of the [[elementary charge]] ''e'' of an electron or proton (i.e. it can only take values of ''q'' = 0, ±''e'', ±2''e'', ±3''e'' , ...).<ref name=":1">{{Cite book|last=Serway|first=Raymond A.|title=Physics for Scientists and Engineers, Technology Update|publisher=Cengage Learning|year=2015|isbn=9781305465398|edition=9th|___location=|pages=692}}</ref> Although it is only a matter of definition, by convention the electron is said to have a negative charge −''e'' and the proton is said to have a positive charge +''e'' .<ref name=":0" /><ref name=":1" /> The first measurement of and experimental confirmation of the quantisation of charge was [[Robert Andrews Millikan|Robert Millikan's]] [[oil drop experiment]] in which the electric force on the particle is set to exactly counter the gravitational force that pulls it down, and the [[terminal velocity]] of this particle can be used to calculate its charge.<ref>{{Cite web|last=|first=|last2=|first2=|date=|title=UChicago Breakthroughs: 1910s|url=https://www.uchicago.edu/breakthroughs/1910s/|url-status=live|archive-url=|archive-date=|access-date=2020-11-26|website=The University of Chicago|language=en}}</ref><ref>{{Cite web|last=|first=|date=|title=Robert Millikan|url=http://www.aps.org/programs/outreach/history/historicsites/millikan.cfm|url-status=live|archive-url=|archive-date=|access-date=2020-11-26|website=APS physics|language=en}}</ref> This experiment is still one of the best confirmations of the quantisation of charge; one large experiment concluding in 2015 used over 100 million oil drops finding no evidence for charges that were not integer multiple values of ''e.<ref>{{cite web|last=|first=|date=January 2007|title=SLAC – Fractional Charge Search – Results|url=http://www.slac.stanford.edu/exp/mps/FCS/FCS_rslt.htm|url-status=live|archive-url=|archive-date=|accessdate=26 November 2020|website=|publisher=Stanford Linear Accelerator Center}}</ref>''

* it is ''conserved'': according to the [[Charge conservation|law of charge conservation]], the overall charge of a [[closed system]] (where no charge can leave or enter) cannot change. Quantum theory tells us that charges can be created but only in the [[pair production]] of oppositely charged [[Particle|particles]] and [[Antiparticle|antiparticles]] whose charges exactly cancel out so that charge is always conserved overall.<ref name=":0" /> Research suggests that the overall charge in the universe is neutral so we know that all the positive charges and negative charges in the universe will always cancel out in total.<ref>S. Orito; M. Yoshimura (1985). "Can the Universe be Charged?". ''Physical Review Letters''. '''54''' (22): 2457–60. {{Bibcode|1985PhRvL..54.2457O}}. {{doi|10.1103/PhysRevLett.54.2457}}. {{PMID|10031347}}.</ref><ref>E. Masso; F. Rota (2002). "Primordial helium production in a charged universe". ''Physics Letters B''. '''545''' (3–4): 221–25. {{arXiv|astro-ph/0201248}}. {{Bibcode|2002PhLB..545..221M}}. {{doi|10.1016/S0370-2693(02)02636-9}}.</ref>

* it produces [[Electric field|electric fields]]: by convention, electric [[Field line|field lines]] start at positive charges and end at negative charges, pointing in the direction of the electric force on a positive charge in the field (and in the opposite direction to the direction of the force on negative charges).<ref name=":2">{{Cite web|last=Pumplin|first=Jon|date=2000|title=Electric field lines|url=https://web.pa.msu.edu/courses/2000fall/phy232/lectures/efields/efieldlines.html|access-date=18 October 2018|website=Michigan State University Physics}}</ref><ref name=":3">{{Cite web|last=Nave|first=R|title=Electric Field|url=http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elefie.html|access-date=16 October 2018|website=Georgia State University Hyperphysics}}</ref> Electric field lines are drawn more densely the stronger the electric field to visualise the strength of the electric force on charged particles in the field.<ref name=":2" /> The electric field is defined as the force on a charge per unit charge so that Coulomb's law can be rewritten in terms of the electric field as shown:<ref name=":3" /><ref>{{Cite book|last=Purcell, Edward M.|url=https://www.worldcat.org/oclc/805015622|title=Electricity and magnetism|date=21 January 2013|publisher=|isbn=978-1-107-01402-2|edition=Third|___location=Cambridge|pages=7|oclc=805015622}}</ref>

: <math>\mathbf E_i=k_e{q_i\over r^2}\qquad \Longrightarrow \qquad \mathbf F_{12} = q_2 \mathbf E_1

\quad \And \quad \mathbf F_{21} = q_1\mathbf E_2 \qquad \Longrightarrow \qquad

\mathbf F = q\mathbf E </math>

:where <math display="inline">\mathbf E_i</math> is the electric field generated by charge <math display="inline">q_i</math> and <math display="inline">\mathbf F_{12}</math> is the force of charge ''q₁'' on ''q₂'' (and vice versa for <math display="inline">\mathbf F_{21}</math>). The final equation gives the general equation for the force exerted on a charged particle by an electric field.

* moving charges also produce [[Magnetic field|magnetic fields]]: moving charges (such as charged [[Free particle|free particles]] and [[Electric current|electric currents]]) and [[Magnet|permanent magnets]] produce magnetic fields that attract other moving charges and magnets.<ref>{{Cite web|title=The Feynman Lectures on Physics Vol. II Ch. 1: Electromagnetism|url=http://www.feynmanlectures.caltech.edu/II_01.html#Ch1-S2|access-date=2018-10-30|website=www.feynmanlectures.caltech.edu}}</ref> The direction of the force on a moving charge from a magnetic field is perpendicular to both the direction of motion and the direction of the magnetic field lines and can be found using the [[right-hand rule]] .<ref name=":6">{{Cite web|title=Magnetic forces|url=http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magfor.html#c2|access-date=2020-11-26|website=hyperphysics.phy-astr.gsu.edu}}</ref> The magnitude of the force <math display="inline">|\mathbf F|</math> is given by the equation<ref name=":6" />

: <math>|\mathbf F| = q|\mathbf v \times \mathbf B| = q|\mathbf v| |\mathbf B|\sin\theta</math>

:where ''q'' is the charge of the particle and <math display="inline">|\mathbf v \times \mathbf B|</math> is the magnitude of the [[cross product]] between the velocity of the charge '''v''' and the magnetic field <math display="inline">\mathbf B</math> which is equal to the product of their magnitudes times the sine of the angle between them <math display="inline">\theta</math>.

The overall electromagnetic force on a charged particle is a combination of the electric and magnetic forces on it and is called the [[Lorentz force]]:<ref name=":6" /><ref>{{Cite book|last=Purcell, Edward M.|url=https://www.worldcat.org/oclc/805015622|title=Electricity and magnetism|date=21 January 2013|publisher=|isbn=978-1-107-01402-2|edition=Third|___location=Cambridge|pages=277|oclc=805015622}}</ref>

<math>\mathbf F=q(\mathbf E + \mathbf v \times \mathbf B)</math>

In all equations shown, symbols in bold are [[Vector (mathematics and physics)|vector quantities]] and the electric and magnetic fields are [[Vector field|vector fields]]. For more information on the mathematics used here, see [[cross product]] and [[vector calculus]].

== Electricity ==

=== Electric flux and Gauss' law ===

{{Multiple image

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| caption1 = If there is no charge enclosed by a closed surface, then the amount of electric field flowing into it must exactly cancel with the electric field flowing out of it.

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| image2 = GaussLaw1VFPt Earths Magnetic Field Confusion.svg

| caption2 = TheBecause amountthe flow of electricmagnetic fluxfield throughout theof a closed surface (above)must dependscancel onwith the amountflow ofinto chargeit, enclosedmagnets bymust it.have both North and TheSouth fluxpoles alsowhich dependscannot onbe otherseparated factorsinto (left)monopoles.

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| caption3 = The amount of flux flowing into the enclosed volume is exactly cancelled by the flux flowing out of it because there are no charges enclosed.

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[[Flux]] can be thought of as the flow of the electric or magnetic field through a surface. Flux flowing through a surface is analogous to the flow of a fluid through a surface; the greater the density of flow and the greater the size of the surface, the more that can flow through it and the greater the angle between the surface and the direction of flow, the less that can flow through.<ref name=":4">{{Cite book|last=Grant, I. S. (Ian S.)|url=https://www.worldcat.org/oclc/21447877|title=Electromagnetism|date=1990|publisher=Wiley|others=Phillips, W. R. (William Robert)|isbn=0-471-92711-2|edition=2nd|series=The Manchester Physics Series|___location=Chichester [England]|pages=17–22|oclc=21447877}}</ref> [[Gauss's law|Gauss' law]] is the first of [[Maxwell's equations]] and states that the [[electric flux]] <math display="inline">\Phi_E</math> through a closed surface is proportional to the amount of charge enclosed within it:<ref name=":4" /><ref>{{Cite web|title=Gauss's Law|url=http://hyperphysics.phy-astr.gsu.edu/hbase/electric/gaulaw.html|access-date=2018-10-30|website=hyperphysics.phy-astr.gsu.edu}}</ref>

<math>\Phi_E = \frac{Q}{\varepsilon_0}</math>

where ''Q'' is the total charge enclosed by the surface, and <math display="inline">\varepsilon_0</math> is the [[permittivity of free space]].

This means that the more electric charge there is, the more electric flux is produced. From the equation, we can see that when there is a net positive charge inside the surface (with flux flowing out of the enclosed volume because electric field lines start at positive charges), the electric flux is defined as positive and when there is a net negative charge inside the surface (with flux flowing into the enclosed volume), the electric flux is defined as negative.

If there is no charge enclosed by the surface, then the electric flux must be zero. This means that when there is no charge enclosed by the surface either there are no field lines going through the surface at all or the flux flowing in through the surface must cancel out with the flux flowing out of the surface.<ref>{{Cite web|last=|first=|date=|title=The Feynman Lectures on Physics Vol. II Ch. 4: Electrostatics, S5: The flux of E|url=https://www.feynmanlectures.caltech.edu/II_04.html#Ch4-S5|url-status=live|archive-url=|archive-date=|access-date=2020-11-27|website=www.feynmanlectures.caltech.edu}}</ref>

=== Electric potential and potential energy ===

The [[electric potential energy]] of a system is defined as the amount of [[Work (physics)|physical work]] it would take to move all the charges in the sytem from very far away to the configuration that they are currently in and can be thought of as the energy stored in the electric field for a given configuration of charges.<ref>{{Cite book|last=Grant, I. S. (Ian S.)|url=https://www.worldcat.org/oclc/21447877|title=Electromagnetism|date=1990|publisher=Wiley|others=Phillips, W. R. (William Robert)|isbn=0-471-92711-2|edition=2nd|___location=Chichester [England]|pages=33|oclc=21447877}}</ref> Another way of thinking about the electric potential energy is as analogously to [[Gravitational energy|gravitational potential energy]]; like a mass released from high up will convert its gravitational potential energy to kinetic energy as it falls to the ground, separated charges will convert their electric potential energy to kinetic energy as they are accelerated either attractively towards one another or repulsively away from one another.<ref name=":7">{{Cite book|last=Young, Hugh D., Freedman, Roger A.|url=https://www.worldcat.org/oclc/897436903|title=Sears and Zemansky's University Physics with Modern Physics|publisher=[[Pearson]]|year=2016|isbn=978-0-321-97361-0|edition=14th|___location=Boston|pages=776–778, 783|oclc=897436903}}</ref>

The [[electric potential]] of a system is defined as the electric potential energy per unit charge:<ref name=":7" />

<math>\phi = {U_E \over Q} </math>

where <math display="inline">\phi</math> is the electric potential, ''U_E'' is the electric potential energy, and ''Q'' is the total charge of the system. The [[Voltage|potential difference]] (also known as voltage) between two points is defined as the work required to move a charge between those two points.<ref name=":7" /> Another equivalent definition of the electric potential is in terms of the electric field. For a static electric field, the electric field is defined to be minus the [[gradient]] of the electric potential and so the electric field can be thought of as a field that points away from high potentials towards low potentials.<ref>{{Cite book|last=Grant, I. S. (Ian S.)|url=https://www.worldcat.org/oclc/21447877|title=Electromagnetism|date=1990|publisher=Wiley|others=Phillips, W. R. (William Robert)|isbn=0-471-92711-2|edition=2nd|___location=Chichester [England]|pages=65|oclc=21447877}}</ref> Electric fields point from positive charges to negative charges (and opposite charges attract) so this definition tells us that positive charges are attracted to low potentials and negative charges are attracted to high potentials.

== Magnetism ==

=== Gauss' law for magnetism ===

[[File:VFPt Earths Magnetic Field Confusion.svg|thumb|Magnets must have North and South poles so cannot be monopoles like electric charges. Therefore, the [[magnetic flux]] going out of a closed surface always cancels with the flux going in through the closed surface.]]

The second of [[Maxwell's equations|Mawell's equations]] is [[Gauss's law for magnetism|Gauss' law for magnetism]] which states that the [[magnetic flux]] <math display="inline">\Phi_B</math> through a closed surface is always equal to zero:<ref name=":5">{{Cite book|last=Purcell, Edward M.|url=https://www.worldcat.org/oclc/805015622|title=Electricity and magnetism|date=21 January 2013|publisher=|isbn=978-1-107-01402-2|edition=Third|___location=Cambridge|pages=322, 437|oclc=805015622}}</ref>

<math>\Phi_B = 0</math>

This law has colloquially been called "no magnetic monopoles" because it means that magnetic fields do not begin or end at single monopolar [[Magnetic monopole|magnetic charges]] (unlike electric fields which begin at positive charges and end at negative charges) but that magnets must always have more than one pole.<ref name=":5" /> For example, [[Magnet|permanent magnets]] have a North and a South pole and so are [[Magnetic dipole|magnetic dipoles]] and there can also be [[Quadrupole magnet|quadrupole magnets]] with four poles.<ref>{{Cite web|title=Quadrupole Magnetic Field|url=http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magquad.html|access-date=2020-11-27|website=hyperphysics.phy-astr.gsu.edu}}</ref>

=== Magnets ===

[[Magnet|Magnets]] are materials that produce their own magnetic fields. All magnets have North and South poles and the magnetic field produced by them points from the North to the South pole. Like electric charges, opposite magnetic poles attract one another and like magnetic poles repel one another but, unlike electric charges, magnetic poles cannot exist on their own (as shown by Gauss' law for magnetism) and so North and South poles must come together.<ref name=":8">{{Cite web|title=Magnets and Electromagnets|url=http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html#c1|access-date=2020-11-27|website=hyperphysics.phy-astr.gsu.edu}}</ref>

The combination of the electric and magnetic forces on a charged particle is called the [[Lorentz force]].<ref name=":6" /><ref>{{Cite book|last=Purcell, Edward M. |title=Electricity and magnetism|date=21 January 2013|isbn=978-1-107-01402-2|edition=Third|___location=Cambridge|pages=277|oclc=805015622}}</ref> [[Classical electromagnetism]] is fully described by the Lorentz force alongside a set of equations called [[Maxwell's equations]]. The first of these equations is known as [[Gauss's law]]. It describes the electric field produced by charged particles and by [[charge distribution]]s. According to Gauss's law, the [[flux]] (or flow) of electric field through any [[closed surface]] is proportional to the amount of charge that is enclosed by that surface.<ref name=":4">{{Cite book|last=Grant, I. S. (Ian S.) |title=Electromagnetism | date=1990 | publisher=Wiley | others=Phillips, W. R. (William Robert) | isbn=0-471-92711-2 |edition=2nd|series=The Manchester Physics Series| ___location=Chichester [England] |pages=17–22 |oclc=21447877}}</ref><ref>{{Cite web|title=Gauss's Law |url=http://hyperphysics.phy-astr.gsu.edu/hbase/electric/gaulaw.html|access-date=2018-10-30|website=hyperphysics.phy-astr.gsu.edu}}</ref> This means that the greater the charge, the greater the electric field that is produced. It also has other important implications. For example, this law means that if there is no charge enclosed by the surface, then either there is no electric field at all or, if there is a charge near to but outside of the closed surface, the flow of electric field into the surface must exactly cancel with the flow out of the surface.<ref>{{Cite web|title=The Feynman Lectures on Physics Vol. II Ch. 4: Electrostatics, S5: The flux of E|url=https://feynmanlectures.caltech.edu/II_04.html#Ch4-S5|access-date=2020-11-27 |website=feynmanlectures.caltech.edu}}</ref> The second of Maxwell's equations is known as [[Gauss's law for magnetism]] and, similarly to the first Gauss's law, it describes flux, but instead of [[electric flux]], it describes [[magnetic flux]]. According to Gauss's law for magnetism, the flow of magnetic field through a closed surface is always zero. This means that if there is a magnetic field, the flow into the closed surface will always cancel out with the flow out of the closed surface. This law has also been called "no magnetic monopoles" because it means that any magnetic flux flowing out of a closed surface must flow back into it, meaning that positive and negative magnetic poles must come together as a [[magnetic dipole]] and can never be separated into [[magnetic monopole]]s.<ref name=":5">{{Cite book| last=Purcell | first = Edward M.| title=Electricity and magnetism |date=21 January 2013| isbn=978-1-107-01402-2| edition=Third|___location=Cambridge |pages=322 |oclc=805015622}}</ref> This is in contrast to electric charges which can exist as separate positive and negative charges.

Materials that are attracted to magnets and which can be themselves magnetised are called [[ferromagnetic materials]]. Ferromagnetic materials can be magnetised because when their electron's [[Spin magnetic moment|spin magnetic moments]] are aligned with an external magnetic field, they sustain their own internal magnetic field even when the external magnetic field is removed. Examples of ferromagnetic materials which can be magnetised with external magnetic fields to create magnets are [[iron]], [[nickel]] and [[cobalt]].<ref name=":9">{{Cite web|title=Ferromagnetism|url=http://hyperphysics.phy-astr.gsu.edu/hbase/Solids/ferro.html#c4|access-date=2020-11-27|website=hyperphysics.phy-astr.gsu.edu}}</ref>

=== The Biot–Savart law ===

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| caption1 = The right-hand grip rule for a straight wire (left) and for a solenoidal wire (middle). Electrical current passed through a solenoidal wire around an iron core can produce an [[electromagnet]].

| image2 = Coil right-hand rule.svg

| caption2 = The magnetic field at a point produced by a moving charge as it travels (right). At first the point is at 90° to the charge so the sine component of the B-field equals one. When the charge moves away from the point, the angle changes and so the B-field decreases.

| image3 = Biot-Savart Superposition.svg

| caption3 =

| perrow =

| width1 = 235

| width2 = 200

| width3 = 245

| footer = The [[right-hand grip rule]] for a straight wire (left) and for a coiled wire (right). Electrical current passed through a wire coiled around an iron core can produce an [[electromagnet]].

}}

The third of Maxwell's equations is called the [[Ampere-maxwell law|Ampère–Maxwell law]]. It states that a magnetic field can be generated by an [[electric current]].<ref>{{Cite web|title=Ampere's Law|url=http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/amplaw.html|access-date=2020-11-27|website=hyperphysics.phy-astr.gsu.edu}}</ref> The direction of the magnetic field is given by Ampère's [[right-hand grip rule]]. If the wire is straight, then the magnetic field is curled around it like the gripped fingers in the right-hand rule. If the wire is wrapped into coils, then the magnetic field inside the coils points in a straight line like the outstretched thumb in the right-hand grip rule.<ref>{{Cite book|last=Grant, I. S. (Ian S.)|title=Electromagnetism|date=1990|publisher=Wiley|others=Phillips, W. R. (William Robert)| isbn=0-471-92711-2|edition=2nd| series=The Manchester Physics Series| ___location=Chichester [England]| pages=125|oclc=21447877}}</ref> When electric currents are used to produce a [[magnet]] in this way, it is called an [[electromagnet]]. Electromagnets often use a wire curled up into [[solenoid]] around an iron core which strengthens the magnetic field produced because the iron core becomes magnetised.<ref name=":8">{{Cite web |title=Magnets and Electromagnets |url=http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html#c1 |access-date=2020-11-27| website=hyperphysics.phy-astr.gsu.edu}}</ref><ref name=":9">{{Cite web|title=Ferromagnetism|url=http://hyperphysics.phy-astr.gsu.edu/hbase/Solids/ferro.html#c4 |access-date=2020-11-27 |website=hyperphysics.phy-astr.gsu.edu}}</ref> Maxwell's extension to the law states that a time-varying electric field can also generate a magnetic field.<ref name=":5" /> Similarly, [[Faraday's law of induction]] states that a magnetic field can produce an electric current. For example, a magnet pushed in and out of a coil of wires can produce an electric current in the coils which is proportional to the strength of the magnet as well as the number of coils and the speed at which the magnet is inserted and extracted from the coils. This principle is essential for [[transformer]]s which are used to transform currents from high [[voltage]] to low voltage, and vice versa. They are needed to convert high voltage [[mains electricity]] into low voltage electricity which can be safely used in homes. Maxwell's formulation of the law is given in the [[Maxwell–Faraday equation]]—the fourth and final of Maxwell's equations—which states that a time-varying magnetic field produces an electric field.

[[Ampère's circuital law]] states that an electric current will induce a magnetic field.<ref>{{Cite web|title=Ampere's Law|url=http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/amplaw.html|access-date=2020-11-27|website=hyperphysics.phy-astr.gsu.edu}}</ref>

[[File:EM Spectrum Properties (Amplitude Corrected).svg|thumb|440x440px|The [[electromagnetic spectrum]]]]

Together, Maxwell's equations provide a single uniform theory of the electric and magnetic fields and Maxwell's work in creating this theory has been called "the second great unification in physics" after the first great unification of [[Newton's law of universal gravitation]].<ref>{{Cite journal|last=Editors|first=AccessScience|date=2014|title=Unification theories and a theory of everything| url=https://www.accessscience.com/content/unification-theories-and-a-theory-of-everything/BR0814141|journal=Access Science| language=en|doi=10.1036/1097-8542.BR0814141|url-access=subscription}}</ref> The solution to Maxwell's equations in [[free space]] (where there are no charges or currents) produces [[wave equation]]s corresponding to [[electromagnetic waves]] (with both electric and magnetic components) travelling at the [[speed of light]].<ref>{{Cite book| last=Grant | first = Ian S. | title=Electromagnetism|date=1990 |publisher=Wiley | others=Phillips, W. R. (William Robert) | isbn=0-471-92711-2 | edition=2nd|series=The Manchester Physics Series| ___location=Chichester [England]| pages=365 |oclc=21447877}}</ref> The observation that these wave solutions had a wave speed exactly equal to the speed of light led Maxwell to hypothesise that light is a form of electromagnetic radiation and to posit that other electromagnetic radiation could exist with different wavelengths.<ref name="ADTEF">{{cite journal | last=Maxwell|first=James Clerk| year=1865 |title=A dynamical theory of the electromagnetic field | url=http://upload.wikimedia.org/wikipedia/commons/1/19/A_Dynamical_Theory_of_the_Electromagnetic_Field.pdf| url-status=live |journal=Philosophical Transactions of the Royal Society of London |volume=155 |pages=459–512 |bibcode=1865RSPT..155..459M |doi=10.1098/rstl.1865.0008 |archive-url=https://web.archive.org/web/20110728140123/http://upload.wikimedia.org/wikipedia/commons/1/19/A_Dynamical_Theory_of_the_Electromagnetic_Field.pdf|archive-date=28 July 2011|quote=Light and magnetism are affections of the same substance (p.499)|s2cid=186207827}}</ref> The existence of electromagnetic radiation was proved by [[Heinrich Hertz]] in a series of experiments ranging from 1886 to 1889 in which he discovered the existence of [[radio wave]]s. The full [[electromagnetic spectrum]] (in order of increasing frequency) consists of radio waves, [[microwave]]s, [[Infrared|infrared radiation]], [[visible light]], [[Ultraviolet|ultraviolet light]], [[X-ray]]s and [[gamma ray]]s.<ref>{{Cite web|date=2011-08-25|title=Introduction to the Electromagnetic Spectrum and Spectroscopy {{!}} Analytical Chemistry {{!}} PharmaXChange.info | url=https://pharmaxchange.info/2011/08/introduction-to-the-electromagnetic-spectrum-and-spectroscopy/| access-date=2020-11-26 | website=pharmaxchange.info |language=en-US}}</ref>

A specific case is given by the [[Biot–Savart law]] which states that when there are no time-varying electric or magnetic fields, the strength of a magnetic field produced by a steady [[Electric current|current]] in a long, straight wire is proportional to the strength of the current and inversely proportional to the distance from the wire.<ref>{{Cite book|last=Grant, I. S. (Ian S.)|url=https://www.worldcat.org/oclc/21447877|title=Electromagnetism|date=1990|publisher=Wiley|others=Phillips, W. R. (William Robert)|isbn=0-471-92711-2|edition=2nd|series=The Manchester Physics Series|___location=Chichester [England]|pages=138|oclc=21447877}}</ref> The direction of the magnetic field can be found using Ampère's [[right-hand grip rule]] which shows that the magnetic field will be curled around the current-carrying wire clockwise or anticlockwise depending on the direction of current flow.<ref>{{Cite book|last=Grant, I. S. (Ian S.)|url=https://www.worldcat.org/oclc/21447877|title=Electromagnetism|date=1990|publisher=Wiley|others=Phillips, W. R. (William Robert)|isbn=0-471-92711-2|edition=2nd|series=The Manchester Physics Series|___location=Chichester [England]|pages=125|oclc=21447877}}</ref> The right-hand grip rule can also be used for current passing through a solenoidal wire producing a magnetic field inside the coil. This principle is utilised by [[Electromagnet|electromagnets]] which consist of a wire coiled around an iron core. Current is passed through the wire creating a magnetic field in the iron core. This magnetic field aligns the spins of the electrons in the iron which contribute to magnetic field making it stronger.<ref name=":8" /><ref name=":9" />

The Biot–Savart law for a charged particle states that the magnetic field ''B(r)'' produced by a moving charged particle is proportional to the charge ''q'' and velocity ''v'' of the particle and inversely proportional to the square of the distance away from it ''r²'':<ref>{{Cite book|last=Griffiths, David J. (David Jeffery), 1942-|url=https://www.worldcat.org/oclc/1021068059|title=Introduction to electrodynamics|date=29 June 2017|publisher=|isbn=978-1-108-42041-9|edition=Fourth|___location=Cambridge, United Kingdom|pages=462|oclc=1021068059}}</ref>

<math>B(r) = {\mu_0 \over 4\pi} {q |\mathbf v \times \mathbf \hat{r}| \over r^2} =

{\mu_0 \over 4\pi} {qv \sin\theta \over r^2}</math>

where <math display="inline">\mu_0</math> is the [[Permeability Of Free Space|permeability of free space]] and <math display="inline">|\mathbf v \times \mathbf \hat{r}|</math> is the magnitude of the cross product between the velocity and a unit vector <math display="inline">\mathbf \hat{r}</math> pointing from the the charge to the point where the magnetic field is being calculated which is equal to the magnitude of the velocity times the sine of the angle <math display="inline">\theta</math> between the direction of motion of the charge and the direction of <math display="inline">\mathbf \hat{r}</math>.

== Electromagnetic unification ==

=== Maxwell's equations and electromagnetic radiation ===

[[File:EM_Spectrum_Properties_edit.svg|link=https://en.wikipedia.org/wiki/File:EM_Spectrum_Properties_edit.svg|thumb|473x473px|The [[electromagnetic spectrum]]]]

[[Maxwell's equations]] consist of Gauss' laws for electricity and magnetism (as described above) as well as the [[Maxwell-Faraday equation]] and the [[Ampère–Maxwell equation]].<ref name=":5" /> The Maxwell-Faraday equation states that a time-varying magnetic field produces an electric field whilst the Ampère–Maxwell equation extends Ampère's circuital law to include the statement that a time-varying electric field (as well as an electric current) will produce a magnetic field.<ref name=":5" /> Together Maxwell's equations provide a single uniform theory of electromagnetism and Maxwell's work in creating this theory has been called "the second great unification in physics" after the first great unification of [[Newton's law of universal gravitation]].<ref>{{Cite journal|last=Editors|first=AccessScience|date=2014|title=Unification theories and a theory of everything|url=https://www.accessscience.com/content/unification-theories-and-a-theory-of-everything/BR0814141|journal=Access Science|language=en|doi=10.1036/1097-8542.BR0814141}}</ref> The solution to Maxwell's equations in [[free space]] (where there are no charges or currents) produces [[Wave equation|wave equations]] corresponding to [[electromagnetic waves]] (with both electric and magnetic components) travelling at the [[speed of light]].<ref>{{Cite book|last=Grant, I. S. (Ian S.)|url=https://www.worldcat.org/oclc/21447877|title=Electromagnetism|date=1990|publisher=Wiley|others=Phillips, W. R. (William Robert)|isbn=0-471-92711-2|edition=2nd|series=The Manchester Physics Series|___location=Chichester [England]|pages=365|oclc=21447877}}</ref> The observation that these wave solutions had a wave speed equal to the speed of light led Maxwell to conclude that light is a form of electromagnetic radiation and to posit that other electromagnetic radiation could exist with different wavelengths.<ref name="ADTEF">{{cite journal|last=Maxwell|first=James Clerk|year=1865|title=A dynamical theory of the electromagnetic field|url=http://upload.wikimedia.org/wikipedia/commons/1/19/A_Dynamical_Theory_of_the_Electromagnetic_Field.pdf|url-status=live|journal=Philosophical Transactions of the Royal Society of London|volume=155|pages=459–512|bibcode=1865RSPT..155..459C|doi=10.1098/rstl.1865.0008|archiveurl=https://web.archive.org/web/20110728140123/http://upload.wikimedia.org/wikipedia/commons/1/19/A_Dynamical_Theory_of_the_Electromagnetic_Field.pdf|archivedate=28 July 2011|quote=Light and magnetism are affections of the same substance (p.499)|via=|s2cid=186207827}}</ref> The existence of electromagnetic radiation was proved by [[Heinrich Hertz]] in a series of experiments ranging from 1886 to 1889 in which he discovered the existence of [[Radio wave|radio waves]].<ref>{{Cite book|last=Huurdeman, Anton A.|url=https://www.worldcat.org/oclc/50251955|title=The worldwide history of telecommunications|date=2003|publisher=J. Wiley|isbn=0-471-20505-2|___location=New York|pages=202–204|oclc=50251955}}</ref> The full [[electromagnetic spectrum]] (in order of increasing frequency) consists of radio waves, [[Microwave|microwaves]], [[Infrared|infrared radiation]], [[visible light]], [[Ultraviolet|ultraviolet light]], [[X-ray|X-rays]] and [[Gamma ray|gamma rays]].<ref>{{Cite web|date=2011-08-25|title=Introduction to the Electromagnetic Spectrum and Spectroscopy {{!}} Analytical Chemistry {{!}} PharmaXChange.info|url=https://pharmaxchange.info/2011/08/introduction-to-the-electromagnetic-spectrum-and-spectroscopy/|access-date=2020-11-26|website=pharmaxchange.info|language=en-US}}</ref>

=== Special relativity ===

{{Multiple image

| align =

| directiontotal_width = vertical450

| total_width =

| image1 = Relativistic electromagnetism fig5.svg

| alt1 =

| caption1 = The lab frame

| image2 = Relativistic electromagnetism fig6.svg

| caption2 = The lab frame (top) and the electron's rest frame (bottom).

}}

AccordingA tofurther unification of electromagnetism came with Einstein's [[Special relativity|special theory of relativity]]. According to special relativity, observers moving at different speeds relative to one another occupy different [[Frame of reference|observational frames of reference]]. If one observer is in motion relative to another observer then they experience [[length contraction]] where unmoving objects appear closer together to the observer in motion than to the observer at rest. Therefore, if an electron is moving at the same speed as the current in a neutral wire, then they experience the flowing electrons in the wire as standing still relative to it and the positive charges as contracted together. In the [[lab frame]], the electron is moving and so feels a magnetic force from the current in the wire but because the wire is neutral it feels no electric force. But in the electron's [[rest frame]], the positive charges seem closer together compared to the flowing electrons and so the wire seems positively charged. Therefore, in the electron's rest frame it feels no magnetic force (because it is not moving relativein toits itselfown frame) but it does feel an electric force due to the positively charged wire. This result from relativity proves that magnetic fields are just electric fields in a different reference frame (and vice versa) and so the two are different manifestations of the same underlying [[electromagnetic field]].<ref>{{Cite book| last=Purcell|first=Edward M. | title=Electricity and magnetism |date=2013 |publisher=| isbn=978-1107014022| edition=Third| ___location=Cambridge |pages=235–68 |oclc=805015622}}</ref><ref>{{Cite web| title=The Feynman Lectures on Physics Vol. II Ch. 13: Magnetostatics | url=httphttps://www.feynmanlectures.caltech.edu/II_13.html#Ch13-S6 |access-date=2018-10-30 |website=www.feynmanlectures.caltech.edu}}</ref><ref>A. French (1968) ''Special Relativity'', chapter 8 – Relativity and electricity, pp. 229–65, W.W. Norton.</ref>

== Conductors, insulators and circuits ==

Line 148 ⟶ 79:

=== Conductors ===

[[File:Electrostatic induction.svg|thumb|301x301px|The charges in a perfect conductor rearrange so that the electric field is always zero inside.]]

A [[Electrical conductor|conductor]] is a material that allows electrons to flow easily. The most effective conductors are usually [[Metal|metals]] because they can be described fairly accurately by the [[free electron model]] in which electrons delocalize from the [[Atomic nucleus|atomic nuclei]], leaving positive [[Ion|ions]] surrounded by a cloud of free electrons.<ref>{{Cite book|last=Hook, J. R., Hall, H. E.|url=https://www.worldcat.org/oclc/868939953|title=Solid State Physics|date=2010|publisher=John Wiley & Sons|isbn=978-1-118-72347-0|edition=2nd|___location=Chichester, West Sussex, U.K.|pages=76–77|oclc=868939953}}</ref> Examples of good conductors include [[copper]], [[Aluminium|aluminum]], and [[silver]]. Wires in electronics are often made of copper.<ref>{{Cite web|title=What Metals Make Good Conductors of Electricity?|url=https://sciencing.com/metals-make-good-conductors-electricity-8115694.html|access-date=2020-11-27|website=Sciencing|date=10 April 2018 |language=en}}</ref>

The main properties of conductors are:<ref>{{Cite book|last=Purcell|first=Edward M.|title=Electricity and magnetism|date=2013|publisher=|isbn=978-1107014022|edition=Third|___location=Cambridge|page=129|oclc=805015622}}</ref>

# ''The electric field is zero inside a perfect conductor.'' Because charges are free to move in a conductor, when they are disturbed by an external electric field they rearrange themselves such that the field that their configuration produces exactly cancels the external electric field inside the conductor.

Line 157 ⟶ 88:

# ''The net [[electric flux]] through a surface is proportional to the charge enclosed by the surface.'' This is a restatement of [[Gauss's law|Gauss' law]].

In some materials, the electrons are bound to the atomic nuclei and so are not free to move around but the energy required to set them free is low. In these materials, called [[Semiconductor|semiconductors]], the conductivity is low at low temperatures but as the temperature is increased the electrons gain more [[thermal energy]] and the conductivity increases.<ref>{{Cite web|title=The Feynman Lectures on Physics Vol. III Ch. 14: Semiconductors|url=https://www.feynmanlectures.caltech.edu/III_14.html|access-date=2020-11-26|website=www.feynmanlectures.caltech.edu}}</ref> Silicon is an example of a semiconductors that can be used to create [[Solar panel|solar panelscells]] which become more conductive the more energy they receive from [[Photon|photons]] from the sun.<ref>{{Cite web|title=How a Solar Cell Works|url=https://www.acs.org/content/acs/en/education/resources/highschool/chemmatters/past-issues/archive-2013-2014/how-a-solar-cell-works.html|access-date=2020-11-26|website=American Chemical Society|language=en}}</ref>

[[Superconductivity|Superconductors]] are materials that exhibit little to no [[Electrical resistance and conductance|resistance]] to the flow of electrons when cooled below a certain critical temperature. Superconductivity can only be explained by the quantum mechanical [[Pauli exclusion principle]] which states that no two [[Fermion|fermions]] (an electron is a type of fermion) can occupy exactly the same [[quantum state]]. In superconductors, below a certain temperature the electrons form [[boson]] bound pairs which do not follow this principle and this means that all the electrons can fall to the same [[energy level]] and move together uniformly in a current.<ref>{{Cite web|title=The Feynman Lectures on Physics Vol. III Ch. 21: The Schrödinger Equation in a Classical Context: A Seminar on Superconductivity|url=https://www.feynmanlectures.caltech.edu/III_21.html#Ch21-S5|access-date=2020-11-26|website=www.feynmanlectures.caltech.edu}}</ref>

=== Insulators ===

[[File:Conductorenequilibrio.gif|thumb|In a dielectric material, an electric field can polarise the material.]]

[[Insulator (electricity)|Insulators]] are material which are highly [[Electrical resistivity and conductivity|resistive]] to the flow of electrons and so are often used to cover conducting wires for safety. In insulators, electrons are tightly bound to atomic nuclei and the energy to free them is very high so they are not free to move and are resistive to induced movement by an external electric field.<ref>{{Cite web|title=Conductors and Insulators|url=http://hyperphysics.phy-astr.gsu.edu/hbase/electric/conins.html|access-date=2020-11-27|website=hyperphysics.phy-astr.gsu.edu}}</ref> However, some insulators, called [[Dielectric|dielectrics]], can be [[Polarizability|polarised]] under the influence of an external electric field so that the charges are minutely displaced forming [[Dipole|dipoles]] that create a positive and negative side.<ref>{{Cite web|title=Dielectric {{!}} physics|url=https://www.britannica.com/science/dielectric|access-date=2020-11-27|website=Encyclopedia Britannica|language=en}}</ref> Dielectrics are used in [[Capacitor|capacitors]] to allow them to store more electric potential energy in the electric field between the capacitor plates.<ref name=":10">{{Cite web|title=Dielectrics|url=http://hyperphysics.phy-astr.gsu.edu/hbase/electric/dielec.html|access-date=2020-11-27|website=hyperphysics.phy-astr.gsu.edu}}</ref>

=== Capacitors ===

[[File:Parallel plate capacitor.svg|thumb|A parallel plate capacitor.]]

A [[capacitor]] is an [[electronic component]] that stores electrical potential energy in an electric field between two oppositely charged conducting plates. If one of the conducting plates has a [[charge density]] of +''Q/A'' and the other has a charge of -''Q/A'' where ''A'' is the area of the plates, then there will be an electric field between them. The potential difference between two parallel plates ''V'' can be derived mathematically as<ref name=":11">{{Cite book|last=Grant, I. S. (Ian S.)|url=https://www.worldcat.org/oclc/21447877|title=Electromagnetism|date=1990|publisher=Wiley|others=Phillips, W. R. (William Robert)|isbn=0-471-92711-2|edition=2nd|series=The Manchester Physics Series|___location=Chichester [England]|pages=41–42|oclc=21447877}}</ref>

<math>V = {Qd \over \varepsilon_0 A}</math>

Line 177 ⟶ 108:

If a dielectric is placed between the plates then the permittivity of free space is multiplied by the [[relative permittivity]] of the dielectric and the capacitance increases.<ref name=":10" /> The maximum energy that can be stored by a capacitor is proportional to the capacitance and the square of the potential difference between the plates<ref name=":11" />

<math>E = \frac 1 2 CV^2</math>

=== Inductors ===

An [[inductor]] is an electronic component that stores energy in a magnetic field inside a coil of wire. A current-carrying coil of wire induces a magnetic field according to [[Ampère's circuital law]]. The greater the current ''I'', the greater the energy stored in the magnetic field and the lower the [[inductance]] which is defined <math display="inline">L= \Phi_B/I</math> where <math display="inline">\Phi_B</math> is the magnetic flux produced by the coil of wire. The inductance is a measure of the circuitscircuit's resistance to a change in current and so inductors with high inductances can also be used to oppose [[alternating current]].<ref>{{Cite book|last=Purcell, Edward M.|url=https://www.worldcat.org/oclc/805015622|title=Electricity and magnetism|date=21 January 2013|publisher=|isbn=978-1-107-01402-2|edition=Third|___location=Cambridge|pages=374|oclc=805015622}}</ref>

=== Other circuit components ===

Line 190 ⟶ 121:

|[[Resistor]]

|Impedes the flow of current

|[[File:Resistor_symbol_America.svg|link=https://en.wikipedia.org/wiki/File:Resistor_symbol_America.svg|center|120x120px]]

|-

|[[Electric battery|Battery]]

Line 214 ⟶ 145:

|[[Inductor]]

|Stores energy in magnetic fields, resists change in current

|[[File:Inductor_symbol.svg|link=https://en.wikipedia.org/wiki/File:Inductor_symbol.svg|center|108x108px]]

|}

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| caption1 =

| image2 = KVL.png

| caption2 = KirchoffKirchhoff's junction rule (above):

I₁ + I₂ + I₃ = I₄ + I₅

KirchoffKirchhoff's loop rule (below):

V₁ + V₂ + V₃ + V₄ = 0

Line 242 ⟶ 173:

<math>P = IV = V^2/R = I^2R</math>

[[Kirchhoff's circuit laws|Kirchhoff's junction rule]] states that the current going into a junction (or node) must equal the current that leaves the node. This comes from [[charge conservation]], as current is defined as the flow of charge over time. If a current splits as it exits a junction, the sum of the resultant split currents is equal to the incoming circuit.<ref name=":12">{{Cite book|last=Young, H. D., Freedman, R. A.|url=https://www.worldcat.org/oclc/897436903|title=Sears and Zemansky's University Physics with Modern Physics|publisher=[[Pearson PLC|Pearson]]|year=2016|isbn=978-0-321-97361-0|edition=14th|___location=Boston|pages=872–878|oclc=897436903}}</ref>

[[Kirchhoff's circuit laws|Kirchhoff's loop rule]] states that the sum of the voltage in a closed loop around a circuit equals zero. This comes from the the fact that the electric field is [[Conservative vector field|conservative]] which means that no matter the path taken, the potential at a point doesn'tdoes not change when you get back there.<ref name=":12" />

Rules can also tell us how to add up quantities such as the current and voltage in [[series and parallel circuits]].<ref name=":12" />

Line 254 ⟶ 185:

For parallel circuits, the voltage remains the same for each component and the currents and resistances are related as shown:

<math>V_{tot} = V_1 = V_2 = V_3 = \ldots \qquad {1 \over R_{tot}} = {1 \over R_1} + {1 \over R_2} + {1 \over R_3} + \ldots \qquad I_{tot} = {1 \over I_1} + {1 \over I_2} + {1 \over I_3} + \ldots</math>

== See also ==

Line 265 ⟶ 196:

[[Category:Electromagnetism]]

[[Category:Introductory articles|electromagnetism ]]