|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{electromagnetism|cTopic=Electrodynamics}}
| | Alyson is the name people use to call me and I think it seems quite good when you say it. Invoicing is my profession. Doing ballet is some thing she would by no means give up. For years he's been residing in Mississippi and he doesn't plan on changing it.<br><br>Feel free to visit my web page; [http://bluelinkx.com/profile/rofrank best psychics] |
| In [[electromagnetism]], '''displacement current''' is a quantity appearing in [[Maxwell's equations]] that is defined in terms of the rate of change of [[electric displacement field]]. Displacement current has the units of electric [[current density]], and it has an associated [[magnetic field]] just as actual currents do. However it is not an electric current of moving [[Electric charge|charges]], but a time-varying [[electric field]]. In materials, there is also a contribution from the slight motion of charges bound in atoms, [[dielectric polarization]].
| |
| | |
| The idea was conceived by [[James Clerk Maxwell]] in his 1861 paper [http://books.google.com/books?id=v1YEAAAAYAAJ&pg=PA14#v=onepage&q&f=false On Physical Lines of Force, Part III] in connection with the displacement of electric particles in a [[dielectric]] medium. Maxwell added displacement current to the [[electric current]] term in [[Ampère's circuital law|Ampère's Circuital Law]]. In his 1865 paper [[A Dynamical Theory of the Electromagnetic Field]] Maxwell used this amended version of [[Ampère's circuital law|Ampère's Circuital Law]] to derive the [[electromagnetic wave equation]]. This derivation is now generally accepted as a historical landmark in physics by virtue of uniting electricity, magnetism and optics into one single unified theory. The displacement current term is now seen as a crucial addition that completed Maxwell's equations and is necessary to explain many phenomena, most particularly the existence of [[electromagnetic wave]]s.
| |
| | |
| == Explanation ==
| |
| | |
| The [[electric displacement field]] is defined as:
| |
| | |
| :<math> \boldsymbol{D} = \varepsilon_0 \boldsymbol{E} + \boldsymbol{P}\ .</math>
| |
| | |
| where:
| |
| :''ε<sub>0</sub>'' is the [[permittivity]] of free space
| |
| :'''E''' is the [[electric field intensity]]
| |
| :'''P''' is the [[polarization (electrostatics)|polarization]] of the medium
| |
| | |
| Differentiating this equation with respect to time defines the ''displacement current density'', which therefore has two components in a [[dielectric]]:<ref name=Jackson>{{cite book |title=Classical Electrodynamics |author=John D Jackson |edition=3rd Edition |publisher=Wiley |year=1999 |page=238 |isbn=0-471-30932-X}}</ref>
| |
| | |
| :<math> \boldsymbol{J}_ \boldsymbol{D} = \varepsilon_0 \frac{\partial \boldsymbol{E}}{\partial t} + \frac{\partial \boldsymbol{P}}{\partial t}\ .</math>
| |
| | |
| The first term on the right hand side is present in material media and in free space. It doesn't necessarily involve any actual movement of charge, but it does have an associated magnetic field, just as does a current due to charge motion. Some authors apply the name ''displacement current'' to only this contribution.<ref name=Griffiths>For example, see {{cite book |author=David J Griffiths |page=323 |title=Introduction to Electrodynamics |edition=3rd Edition |isbn=0-13-805326-X |publisher=Pearson/Addison Wesley |year=1999}} and {{cite book |author=Tai L Chow |title=Introduction to Electromagnetic Theory |page=204 |publisher=Jones & Bartlett |year=2006 |isbn=0-7637-3827-1 |url=http://books.google.com/books?id=dpnpMhw1zo8C&pg=PA153&dq=isbn=0-7637-3827-1#PPA204,M1}}</ref>
| |
| | |
| The second term on the right hand side is associated with the [[electric polarization|polarization]] of the individual molecules of the dielectric material. Polarization results when the charges in molecules move a little under the influence of an applied [[electric field]]. The positive and negative charges in molecules separate, causing an increase in the state of polarization '''''P'''''. A changing state of polarization corresponds to charge movement and so is equivalent to a current.
| |
| | |
| This polarization is the displacement current as it was originally conceived by Maxwell. Maxwell made no special treatment of the vacuum, treating it as a material medium. For Maxwell, the effect of '''''P''''' was simply to change the [[relative permittivity]] ''ε<sub>r</sub>'' in the relation '''''D''''' = ''ε<sub>r</sub>ε<sub>0</sub>'' '''''E'''''.
| |
| | |
| The modern justification of displacement current is explained below.
| |
| | |
| ===Isotropic dielectric case===
| |
| In the case of a very simple dielectric material the [[constitutive relation]] holds:
| |
| | |
| :<math> \boldsymbol{D} = \varepsilon \boldsymbol{E} \ , </math>
| |
| | |
| where the [[permittivity]] ''ε = ε<sub>0</sub> ε<sub>r</sub>'',
| |
| * ''ε<sub>r</sub>'' is the relative permittivity of the dielectric and
| |
| * ''ε<sub>0</sub>'' is the [[electric constant]].
| |
| | |
| In this equation the use of ''ε'', accounts for
| |
| the polarization of the dielectric.
| |
| | |
| The [[scalar (physics)|scalar]] value of displacement current may also be expressed in terms of [[electric flux]]:
| |
| | |
| :<math> I_\mathrm{D} =\varepsilon \frac{\partial \Phi_E}{\partial t}. </math>
| |
| | |
| The forms in terms of ''ε'' are correct only for linear [[isotropic]] materials. More generally ''ε'' may be replaced by a [[tensor]], may depend upon the electric field itself, and may exhibit time dependence (dispersion).
| |
| | |
| For a linear isotropic dielectric, the polarization '''''P''''' is given by:
| |
| | |
| :<math>\boldsymbol{P} = \varepsilon_0 \chi_e \boldsymbol{E} = \varepsilon_0 (\varepsilon_r - 1) \boldsymbol{E}</math>
| |
| | |
| where ''χ<sub>e</sub>'' is known as the [[electric susceptibility]] of the dielectric. Note that:
| |
| | |
| :<math>\varepsilon = \varepsilon_r \varepsilon_0 = (1+\chi_e)\varepsilon_0. </math>
| |
| | |
| ==Necessity==
| |
| Some implications of the displacement current follow, which agree with experimental observation, and with the requirements of logical consistency for the theory of electromagnetism.
| |
| | |
| ===Generalizing Ampère's circuital law===
| |
| ====Current in capacitors====
| |
| An example illustrating the need for the displacement current arises in connection with capacitors with no medium between the plates. Consider the charging capacitor in the figure. The capacitor is in a circuit that transfers charge (on a wire external to the capacitor) from the left plate to the right plate, charging the capacitor and increasing the electric field between its plates. The same current enters the right plate (say ''I'' ) as leaves the left plate. Although current is flowing through the capacitor, no actual charge is transported through the vacuum between its plates. Nonetheless, a magnetic field exists between the plates as though a current were present there as well. The explanation is that a ''displacement current'' ''I<sub>D</sub>'' flows in the vacuum, and this current produces the magnetic field in the region between the plates according to [[Ampère's law]]:<ref name=Palmer>{{cite book |title=Advanced University Physics |page=214 |author= Stuart B. Palmer, Mircea S. Rogalski |url=http://books.google.com/books?id=TF6Igz5lJLgC&pg=PP1&dq=Physics+inauthor:%22Stuart+B+Palmer%22&as_pt=ALLTYPES#PPA214,M1 |isbn=2-88449-065-5 |publisher=Taylor & Francis |year=1996}}</ref><ref name=Serway>{{cite book |title=Principles of Physics |page=807 |author=Raymond A. Serway, John W. Jewett |publisher=Thomson Brooks/Cole |year=2006 |isbn=0-534-49143-X |url=http://books.google.com/books?id=1DZz341Pp50C&pg=PA807 }}</ref>
| |
| | |
| [[File:Current continuity in capacitor.JPG|thumb |200px | An electrically charging capacitor with an imaginary cylindrical surface surrounding the left-hand plate. Right-hand surface ''R'' lies in the space between the plates and left-hand surface ''L'' lies to the left of the left plate. No conduction current enters cylinder surface ''R'', while current ''I'' leaves through surface ''L''. Consistency of Ampère's law requires a displacement current ''I<sub>D</sub> = I'' to flow across surface ''R''.]]
| |
| | |
| :<math>\oint_C \mathbf{B}\ \boldsymbol{ \cdot}\ \mathrm{d}\boldsymbol{\ell} = \mu_0 I_D \ .</math>
| |
| | |
| where
| |
| :*<math>\oint_C </math> is the closed [[line integral]] around some closed curve ''C''.
| |
| :*<math>\mathbf{B} </math> is the [[magnetic field]] in [[tesla (unit)|tesla]].
| |
| :* <math>\boldsymbol{ \cdot}\ </math> is the vector [[dot product]].
| |
| :*<math>\mathrm{d}\boldsymbol{\ell} </math> is an [[infinitesimal]] element ([[differential (mathematics)|differential]]{{disambiguation needed|date=September 2013}}) of the curve ''C'' (that is, a vector with magnitude equal to the length of the infinitesimal line element, and direction given by the tangent to the curve ''C'').
| |
| :*<math>\mu_0 \!\ </math> is the [[magnetic constant]] also called the permeability of free space.
| |
| :*<math>I_D \!\ </math> is the net displacement current that links the curve ''C''.
| |
| | |
| The magnetic field between the plates is the same as that outside the plates, so the displacement current must be the same as the conduction current in the wires, that is,
| |
| | |
| :<math>I_D = I \ ,</math>
| |
| | |
| which extends the notion of current beyond a mere transport of charge.
| |
| | |
| Next, this displacement current is related to the charging of the capacitor. Consider the current in the imaginary cylindrical surface shown surrounding the left plate. A current, say ''I'', passes outward through the left surface ''L'' of the cylinder, but no conduction current (no transport of real charges) enters the right surface ''R''. Notice that the electric field between the plates ''E'' increases as the capacitor charges. That is, in a manner described by [[Gauss's law]], assuming no dielectric between the plates:
| |
| | |
| :<math> Q(t) =\varepsilon_0 \oint_{\mathcal S} d \mathbf{\mathcal S} \ \boldsymbol{ \cdot} \ \boldsymbol{ E} (t) \ , </math>
| |
| | |
| where ''S'' refers to the imaginary cylindrical surface. Assuming a parallel plate capacitor with uniform electric field, and neglecting fringing effects around the edges of the plates, differentiation provides:<ref name=Palmer>{{cite book |title=Advanced University Physics |author= Stuart B. Palmer, Mircea S. Rogalski |page= 214 |publisher=Taylor & Francis |year=1996 |isbn=2-88449-065-5 |url=http://books.google.com/books?id=TF6Igz5lJLgC&pg=PP1&dq=Physics+inauthor:%22Stuart+B+Palmer%22&as_pt=ALLTYPES#PPA214,M1}}</ref>
| |
| | |
| :<math> \frac {dQ}{dt} = \mathit I =\varepsilon_0 \oint_{\mathcal S} d \mathbf{\mathcal S} \ \boldsymbol{ \cdot} \ \frac {\partial \boldsymbol {E} }{\partial t } \approx -{ S}\ \varepsilon_0 \frac {\partial E}{\partial t} \ , </math>
| |
| | |
| where the sign is negative because charge leaves this plate (the charge is decreasing), and where ''S'' is the area of the face ''R''. The electric field at face ''L'' is zero because the field due to charge on the right-hand plate is terminated by the equal but opposite charge on the left-hand plate. Under the assumption of a uniform electric field distribution inside the capacitor, the displacement current density ''J<sub>D</sub>'' is found by dividing by the area of the surface:
| |
| | |
| :<math> J_D = \frac{I_D}{ S}= -\frac{I}{ S}= \varepsilon_0 \frac {\partial E}{\partial t} = \frac {\partial D}{\partial t} \ , </math>
| |
| | |
| where ''I'' is the current leaving the cylindrical surface (which must equal −''I<sub>D</sub>'' as the two currents sum to zero) and ''J<sub>D</sub>'' is the flow of charge per unit area into the cylindrical surface through the face ''R''.
| |
| | |
| [[File:Displacement current in capacitor.svg|thumb|200px|Example showing two surfaces ''S<sub>1</sub>'' and ''S<sub>2</sub>'' that share the same bounding contour ''∂S''. However, ''S<sub>1</sub>'' is pierced by conduction current, while ''S<sub>2</sub>'' is pierced by displacement current.]]
| |
| | |
| Combining these results, the magnetic field is found using the integral form of [[Ampère's law]] with an arbitrary choice of contour provided the displacement current density term is added to the conduction current density (the Ampère-Maxwell equation):<ref name="Feynman" >from {{cite book
| |
| | last = Feynman
| |
| | first = Richard P.
| |
| | authorlink =
| |
| | coauthors = Robert Leighton, Matthew Sands
| |
| | title = The Feynman Lectures on Physics, Vol. 2
| |
| | publisher = Addison-Wesley
| |
| | year = 1963
| |
| | location = Massachusetts, USA
| |
| | pages = 18–4
| |
| | url =
| |
| | doi =
| |
| | isbn =0-201-02116-1 }}</ref>
| |
| | |
| :<math>\oint_{\partial S} \boldsymbol{B} \cdot d\boldsymbol{\ell} = \mu_0 \int_S (\boldsymbol{J} + \epsilon_0 \frac {\partial \boldsymbol{E}}{\partial t}) \cdot d\boldsymbol{S} \, </math>
| |
| | |
| This equation says that the integral of the magnetic field '''''B''''' around a loop ''∂S'' is equal to the integrated current '''''J''''' through any surface spanning the loop, plus the displacement current term ''ε<sub>0 </sub>∂'''E''' / ∂t'' through the surface. Applying the Ampère-Maxwell equation to surface ''S<sub>1</sub>'' we find:
| |
| | |
| :<math>B = \frac {\mu_0 I}{2 \pi r}\,</math>
| |
| | |
| However, applying this law to surface ''S<sub>2</sub>'', which is bounded by exactly the same curve <math>\partial S </math>, but lies between the plates, provides:
| |
| | |
| :<math>B = \frac {\mu_0 I_D}{2 \pi r}\,</math>
| |
| | |
| Any surface that intersects the wire has current ''I'' passing through it so [[Ampère's law]] gives the correct magnetic field. Also, any surface bounded by the same loop but passing between the capacitor's plates has no charge transport flowing through it, but the ''ε<sub>0 </sub>∂'''E''' / ∂t'' term provides a second source for the magnetic field besides charge conduction current. Because the current is increasing the charge on the capacitor's plates, the electric field between the plates is increasing, and the rate of change of electric field gives the correct value for the field '''''B''''' found above.
| |
| | |
| ====Mathematical formulation====
| |
| In a more mathematical vein, the same results can be obtained from the underlying differential equations. Consider for simplicity a non-magnetic medium where the [[Magnetic permeability#Relative permeability|relative magnetic permeability]] is unity, and the complication of [[Magnetization current#Magnetization current|magnetization current]] is absent.
| |
| The current leaving a volume must equal the rate of decrease of charge in a volume. In differential form this [[Current density#Continuity equation|continuity equation]] becomes:
| |
| | |
| :<math> \nabla \boldsymbol{\cdot J_f} = -\frac {\partial \rho_f}{\partial t} \ , </math>
| |
| | |
| where the left side is the divergence of the free current density and the right side is the rate of decrease of the free charge density. However, [[Ampère's law]] in its original form states:
| |
| | |
| :<math> \boldsymbol{ \nabla \times B} = \mu_0 \boldsymbol J_f \ , </math>
| |
| | |
| which implies that the divergence of the current term vanishes, contradicting the continuity equation. (Vanishing of the ''divergence'' is a result of the [[Vector calculus identities#Divergence of the curl|mathematical identity]] that states the divergence of a ''curl'' is always zero.) This conflict is removed by addition of the displacement current, as then:<ref name=Cloude>{{cite book |title=An Introduction to Electromagnetic Wave Propagation and Antennas |author=Raymond Bonnett, Shane Cloude |isbn=1-85728-241-8 |publisher=Taylor & Francis |year=1995 |url=http://books.google.com/books?id=gME9zlyG304C&pg=PA16&dq=wave+%22displacement+current%22#PPA16,M1 |page=16}}</ref><ref name=Slater>{{cite book |title=Electromagnetism |author=JC Slater and NH Frank |page=84 |url=http://books.google.com/books?id=GYsphnFwUuUC&pg=PA83&dq=displacement+%22ampere%27s+law%22#PPA84,M1 |isbn=0-486-62263-0 |publisher=Courier Dover Publications |year=1969 |edition=Reprint of 1947 edition}}</ref>
| |
| | |
| :<math> \boldsymbol{ \nabla \times B} = \mu_0 \left(\boldsymbol J +\varepsilon_0 \frac {\partial \boldsymbol E}{\partial t}\right) = \mu_0 \left( \boldsymbol J_f +\frac {\partial \boldsymbol D}{\partial t}\right) \ , </math>
| |
| | |
| and
| |
| | |
| :<math>\boldsymbol{ \nabla \cdot } \left( \boldsymbol{\nabla \times B}\right ) = 0 = \mu_0 \left( \nabla \cdot \boldsymbol J_f +\frac {\partial }{\partial t} \boldsymbol {\nabla \cdot D } \right ) \ , </math>
| |
| | |
| which is in agreement with the continuity equation because of [[Gauss's law]]:
| |
| | |
| :<math> \boldsymbol {\nabla \cdot D} = \rho_f \ . </math>
| |
| | |
| ===Wave propagation===
| |
| The added displacement current also leads to wave propagation by taking the curl of the equation for magnetic field.<ref name=Slater2>{{cite book |title=Electromagnetism |page=91 |author=JC Slater and NH Frank |edition=''op. cit.'' |isbn=0-486-62263-0 |url=http://books.google.com/books?id=GYsphnFwUuUC&pg=PA83&dq=displacement+%22ampere%27s+law%22#PPA91,M1}}</ref>
| |
| | |
| :<math> \boldsymbol{J_D} = \epsilon_0\frac { \partial \boldsymbol{E} } { \partial t }</math>
| |
| | |
| Substituting this form for '''''J''''' into [[Ampère's law]], and assuming there is no bound or free current density contributing to '''''J''''' :
| |
| | |
| :<math> \boldsymbol{ \nabla \times B} = \mu_0 \boldsymbol {J_D} \ , </math>
| |
| | |
| with the result:
| |
| | |
| :<math>\boldsymbol{ \nabla \times}\left( \boldsymbol {\nabla \times B} \right ) = \mu_0 \epsilon_0 \frac {\partial}{\partial t} \boldsymbol {\nabla \times E} \ . </math>
| |
| | |
| However,
| |
| :<math>\boldsymbol {\nabla \times E} = -\frac{\partial }{\partial t} \boldsymbol B \ , </math>
| |
| | |
| leading to the [[wave equation]]:<ref name=King>{{cite book |page=182 |title=Wave Motion |author=J Billingham, A C King |isbn=0-521-63450-4 |publisher=Cambridge University Press |url=http://books.google.com/books?id=bNePaHM20LQC&pg=PA179&dq=wave+%22displacement+current%22#PPA181,M1 |year=2006}}</ref>
| |
| :<math>-\boldsymbol{ \nabla \times}\left( \boldsymbol {\nabla \times B} \right ) = \nabla^2 \boldsymbol B =\mu_0 \epsilon_0 \frac {\partial^2}{\partial t^2} \boldsymbol {B } = \frac{1}{c^2} \frac {\partial^2}{\partial t^2} \boldsymbol {B } \ , </math>
| |
| | |
| where use is made of the vector identity that holds for any vector field '''''V'''('''r''', t)'':
| |
| | |
| :<math>\boldsymbol{\nabla \times}\left( \boldsymbol { \nabla \times V}\right ) = \boldsymbol {\nabla}\left(\boldsymbol{\nabla \cdot V}\right ) - \nabla^2 \boldsymbol V \ , </math>
| |
| | |
| and the fact that the divergence of the magnetic field is zero. An identical wave equation can be found for the electric field by taking the ''curl'':
| |
| | |
| :<math>\boldsymbol {\nabla \times } \left( \boldsymbol {\nabla \times E} \right) = -\frac {\partial}{\partial t}\boldsymbol {\nabla \times } \boldsymbol{B}=-\mu_0 \frac {\partial}{\partial t} \left( \boldsymbol J + \epsilon_0\frac {\partial}{\partial t} \boldsymbol E \right) \ .</math>
| |
| | |
| If '''''J, P''''' and ''ρ'' are zero, the result is:
| |
| | |
| :<math>\nabla^2 \boldsymbol E =\mu_0 \epsilon_0 \frac {\partial^2}{\partial t^2} \boldsymbol {E } = \frac{1}{c^2} \frac {\partial^2}{\partial t^2} \boldsymbol {E } \ . </math>
| |
| | |
| The electric field can be expressed in the general form:
| |
| | |
| :<math> \boldsymbol{E} = - \boldsymbol{\nabla} \varphi - \frac { \partial \boldsymbol{A} } { \partial t } \ ,</math>
| |
| | |
| where ''φ'' is the [[electric potential]] (which can be chosen to satisfy [[Poisson's equation]]) and '''''A''''' is a [[Magnetic potential|vector potential]]. The '''∇'''''φ'' component on the right hand side is the Gauss's law component, and this is the component that is relevant to the conservation of charge argument above. The second term on the right-hand side is the one relevant to the electromagnetic wave equation, because it is the term that contributes to the ''curl'' of '''''E'''''. Because of the vector identity that says the ''curl'' of a ''gradient'' is zero, '''∇'''''φ'' does not contribute to '''∇×''E'''''.
| |
| | |
| == History and interpretation ==
| |
| | |
| Maxwell's displacement current was postulated in part III of his 1861 paper '[[Media:On Physical Lines of Force.pdf|On Physical Lines of Force]]'. Few topics in modern physics have caused as much confusion and misunderstanding as that of displacement current.<ref name=Siegel2>{{cite book |title=Innovation in Maxwell's Electromagnetic Theory |author= Daniel M. Siegel |isbn=0-521-53329-5 |page=85 |url=http://books.google.com/books?id=AbQq85U8K0gC&pg=PA123&dq=Kirchhoff+displacement+current#PPA85,M1 |publisher=Cambridge University Press |year=2003}}</ref> This is in part due to the fact that Maxwell used a sea of molecular vortices in his derivation, while modern textbooks operate on the basis that displacement current can exist in free space. Maxwell's derivation is unrelated to the modern day derivation for displacement current in the vacuum, which is based on consistency between [[#To obtain consistency between Ampère's law and current continuity|Ampère's law for the magnetic field and the continuity equation]] for electric charge.
| |
| | |
| Maxwell's purpose is stated by him at (Part I, p. 161):
| |
| | |
| {{quote|I propose now to examine magnetic phenomena from a mechanical point of view, and to determine what tensions in, or motions of, a medium are capable of producing the mechanical phenomena observed.}}
| |
| | |
| He is careful to point out the treatment is one of analogy:
| |
| | |
| {{quote|The author of this method of representation does not attempt to explain the origin of the observed forces by the effects due to these strains in the elastic solid, but makes use of the mathematical analogies of the two problems to assist the imagination in the study of both.}}
| |
| | |
| In part III, in relation to displacement current, he says
| |
| | |
| {{quote|I conceived the rotating matter to be the substance of certain cells, divided from each other by cell-walls composed of particles which are very small compared with the cells, and that it is by the motions of these particles, and their tangential action on the substance in the cells, that the rotation is communicated from one cell to another.}}
| |
| | |
| Clearly Maxwell was driving at magnetization even though the same introduction clearly talks about dielectric polarization.
| |
| | |
| Maxwell concluded, using Newton's equation for the speed of sound (''Lines of Force'', Part III, equation (132)), that “light consists of transverse undulations in the same medium that is the cause of electric and magnetic phenomena.”
| |
| | |
| But although the above quotations point towards a magnetic explanation for displacement current, for example, based upon the divergence of the above ''curl'' equation, Maxwell's explanation ultimately stressed linear polarization of dielectrics:
| |
| | |
| {{quote|This displacement...is the commencement of a current...The amount of displacement depends on the nature of the body, and on the electromotive force so that if ''h'' is the displacement, ''R'' the electromotive force, and ''E'' a coefficient depending on the nature of the dielectric:
| |
| :::<math>R = -4\pi \mathrm E^2 h \ ; </math>
| |
| and if ''r'' is the value of the electric current due to displacement
| |
| :::<math>r = \frac{dh}{dt}\ , </math>
| |
| These relations are independent of any theory about the mechanism of dielectrics; but when we find electromotive force producing electric displacement in a dielectric, and when we find the dielectric recovering from its state of electric displacement...we cannot help regarding the phenomena as those of an elastic body, yielding to a pressure and recovering its form when the pressure is removed.—Part III – ''The theory of molecular vortices applied to statical electricity'' , pp. 14–15}}
| |
| | |
| With some change of symbols (and units): ''r → J'', ''R → −E'' and the material constant E<sup>−2</sup> → ''4π ε<sub>r</sub>ε<sub>0</sub>'' these equations take the familiar form:
| |
| | |
| ::<math>J = \frac{d}{dt} \frac {1}{4 \pi \mathrm E^2} \mathit E = \frac{d}{dt} \varepsilon_r\varepsilon_0 \mathit E = \frac{d}{dt} \mathit D \ . </math>
| |
| | |
| When it came to deriving the electromagnetic wave equation from displacement current in his 1865 paper [[A Dynamical Theory of the Electromagnetic Field]], he got around the problem of the non-zero divergence associated with Gauss's law and dielectric displacement by eliminating the Gauss term and deriving the wave equation exclusively for the solenoidal magnetic field vector.
| |
| | |
| Maxwell's emphasis on polarization diverted attention towards the electric capacitor circuit, and led to the common belief that Maxwell conceived of displacement current so as to maintain conservation of charge in an electric capacitor circuit. There are a variety of debatable notions about Maxwell's thinking, ranging from his supposed desire to perfect the symmetry of the field equations to the desire to achieve compatibility with the continuity equation.<ref name=Nahin>{{cite book |title=Oliver Heaviside: The Life, Work, and Times of an Electrical Genius of the Victorian Age |url=http://books.google.com/books?id=e9wEntQmA0IC&pg=PA109&dq=history+Maxwell+symmetry+of+field+equations&as_pt=ALLTYPES |page=109 |author=Paul J. Nahin |isbn=0-8018-6909-9 |year=2002 |publisher=Johns Hopkins University Press }}</ref><ref name=Stepin>{{cite book |title=Theoretical Knowledge |author=Vyacheslav Stepin |url=http://books.google.com/books?id=4LEns8rzBOEC&pg=PA202&dq=history+Maxwell+symmetry+of+field+equations&as_pt=ALLTYPES |page= 202|isbn=1-4020-3045-2 |year=2002 |publisher=Springer}}</ref>
| |
| | |
| ==See also==
| |
| *[[Electromagnetic wave equation]]
| |
| *[[Ampère's law]]
| |
| *[[Capacitance#Capacitance and 'displacement current'|Capacitance]]
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| ==Maxwell's papers==
| |
| *[http://blazelabs.com/On%20Faraday%27s%20Lines%20of%20Force.pdf ''On Faraday's Lines of Force''] Maxwell's paper of 1855
| |
| * ''On Physical Lines of Force'', [http://books.google.com/books?id=R10EAAAAYAAJ&pg=PA161#v=onepage&q&f=false Part I], [http://books.google.com/books?id=R10EAAAAYAAJ&pg=PA281#v=onepage&q&f=false Part II], [http://books.google.com/books?id=R10EAAAAYAAJ&pg=PA338#v=onepage&q&f=false Part II cont.], [http://books.google.com/books?id=v1YEAAAAYAAJ&pg=PA12#v=onepage&q&f=false Part III], [http://books.google.com/books?id=v1YEAAAAYAAJ&pg=PA85#v=onepage&q&f=false Part IV] - Philosophical Magazine
| |
| *[[Media:On Physical Lines of Force.pdf|''On Physical Lines of Force'']] Maxwell's paper of 1861
| |
| *[[Media:A Dynamical Theory of the Electromagnetic Field.pdf|''A Dynamical Theory of the Electromagnetic Field'']] Maxwell's paper of 1864
| |
| | |
| ==Further reading==
| |
| *[http://dx.doi.org/10.1119/1.1969140 AM Bork] ''Maxwell, Displacement Current, and Symmetry'' (1963)
| |
| *[http://dx.doi.org/10.1119/1.1974263 AM Bork] ''Maxwell and the Electromagnetic Wave Equation'' (1967)
| |
| | |
| {{DEFAULTSORT:Displacement Current}}
| |
| [[Category:Electric current]]
| |
| [[Category:Electrodynamics]]
| |
| [[Category:Electromagnetism]]
| |
| | |
| [[hu:Eltolási áram]]
| |