37 Maxwell s Equations

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1 37 Maxwell s quatins In this chapter, the plan is t summarize much f what we knw abut electricity and magnetism in a manner similar t the way in which James Clerk Maxwell summarized what was knwn abut electricity and magnetism near the end f the nineteenth century Maxwell nt nly rganized and summarized what was knwn, but he added t the knwledge Frm his wrk, we have a set f equatins knwn as Maxwell s quatins His wrk culminated in the discvery that light is electrmagnetic waves In building up t a presentatin f Maxwell s quatins, I first want t revisit ideas we encuntered in chapter 2 and I want t start that revisit by intrducing an easy way f relating the directin in which light is traveling t the directins f the electric and magnetic fields that are the light Recall the idea that a charged particle mving in a statinary magnetic field q v P experiences a frce given by F q v P This frce, by the way, is called the Lrentz Frce Fr the case depicted abve, by the righthand rule fr the crss prduct f tw vectrs, this frce wuld be directed ut f the page F q v P 324

2 Viewing the exact same situatin frm the reference frame in which the charged particle is at rest we see a magnetic field mving sideways (with velcity v v P ) thrugh the particle Since we have changed nthing but ur viewpint, the particle is experiencing the same frce v F q We intrduce a middleman by adpting the attitude that the mving magnetic field desn t really exert a frce n the charged particle, rather it causes an electric field which des that Fr the frce t be accunted fr by this middleman electric field, the latter must be in the directin f the frce The existence f light indicates that the electric field is caused t exist whether r nt there is a charged particle fr it t exert a frce n v The bttm line is that wherever yu have a magnetic field vectr mving sideways thrugh space yu have an electric field vectr, and, the directin f the velcity f the magnetic field vectr is cnsistent with directin f v directin f Yu arrive at the same result fr the case f an electric field mving sideways thrugh space (Recall that in chapter 2, we discussed the fact that an electric field mving sideways thrugh space causes a magnetic field) The purpse f this brief review f material frm chapter 2 was t arrive at the result directin f v directin f This directin relatin will cme in handy in ur discussin f tw f the fur equatins knwn as Maxwell s quatins One f Maxwell s quatins is called Faraday s Law It brings tgether a cuple f things we have already talked abut, namely, the idea that a changing number f magnetic field lines thrugh a lp r a cil induces a current in that lp r cil, and, the idea that a magnetic field vectr that is mving sideways thrugh a pint in space causes an electric field t exist at that pint in space The frmer is a manifestatin f the latter Fr instance, suppse yu have an increasing number f dwnward directed magnetic field lines thrugh a hrizntal lp The idea is that fr the number f magnetic field lines thrugh the lp t be increasing, there must be magnetic field lines mving sideways thrugh the cnducting material f the lp (t get inside the perimeter f the lp) This causes an electric field in the cnducting material f the 325

3 lp which in turn pushes n the charged particles f the cnducting material f the lp and thus results in a current in the lp We can discuss the prductin f the electric field at pints in space ccupied by the cnducting lp even if the cnducting lp is nt there If we cnsider an imaginary lp in its place, the magnetic field lines mving thrugh it t the interir f the lp still prduce an electric field in the lp; there are simply n charges fr that field t push arund the lp Suppse we have an increasing number f dwnward-directed magnetic field lines thrugh an imaginary lp Viewed frm abve the situatin appears as: increasing The big idea here is that yu can t have an increasing number f dwnward-directed magnetic field lines thrugh the regin encircled by the imaginary lp withut having, either, dwnwarddirected magnetic field lines mving transversely and inward thrugh the lp int the regin encircled by the lp, r, upward-directed magnetic field lines mving transversely and utward thrugh the lp ut f the regin encircled by the lp ither way yu have magnetic field lines cutting thrugh the lp and with each magnetic field cutting thrugh the lp there has t be an assciated electric field with a cmpnent tangent t the lp Our technical expressin fr the number f magnetic field lines thrugh the lp is the magnetic flux, given, in the case f a unifrm (but time-varying) magnetic field by Φ A (37-3) where A is the area f the regin encircled by the lp 326

4 Faraday s Law, as it appears in Maxwell s quatins, is a relatin between the rate f change f the magnetic flux thrugh the lp and the electric field (prduced by this changing flux) in the lp T arrive at it, we cnsider an infinitesimal segment dl f the lp and the infinitesimal cntributin t the rate f change f the magnetic flux thrugh the lp resulting frm magnetic field lines mving thrugh that segment dl int the regin encircled by the lp We g t a magnified view f this infinitesimal segment dl f the lp It is such a tiny piece f the lp that it lks straight v dl If the magnetic field depicted abve is mving sideways tward the interir f the lp with a speed v then all the magnetic field lines in the regin f area A dl, will, in time, mve leftward a distance That is, they will all mve frm utside the lp t inside the lp creating a change f flux, in time, f d φ da dφ dl Nw, if I divide bth sides f this equatin by the time in which the change ccurs, we have d φ dl which I can write as r φ dlv φ v dl 37-1 Fr the case at hand, lking at the diagram, we see that and v are at right angles t each ther s the magnitude f v is just v In that case, since v (frm equatin 2-1 with v in place f v ), we have v Replacing the prduct v appearing n the right side P f equatin 37-1 ( φ v dl ) yields φ dl which I cpy at the tp f the fllwing page: 327

5 φ dl We can generalize this t the case where the velcity vectr v is nt perpendicular t the d In that case the cmpnent f that is alng d l, times the length dl itself, is just dl and ur equatin becmes infinitesimal lp segment in which case is nt alng l φ d In this expressin, the directin f l d is determined nce ne decides n which f the tw directins in which a magnetic field line can extend thrugh the regin enclsed by the lp is defined t make a psitive cntributin t the flux thrugh the lp The directin f l d is then the ne which relates the sense in which l d pints arund the lp, t the psitive directin fr magnetic field lines thrugh the lp, by the right hand rule fr smething curly smething straight With this cnventin the minus sign is needed t make the dt prduct have the same sign as the sign f the nging change in flux Cnsider fr instance the case depicted in the diagram: l v dl We are lking at a hrizntal lp frm abve Dwnward is depicted as int the page Calling dwnward the psitive directin fr flux makes clckwise, as viewed frm abve, the psitive sense fr the d l s in the lp, meaning the d l n the right side f the lp is pinting tward the bttm f the page (as depicted) Fr a dwnward-directed magnetic field mving leftward int the lp, must be directed tward the tp f the page (frm directin f v directin f ) Since is in the ppsite directin t that f d l, d l must be negative ut mvement f dwnward-directed magnetic field lines int the regin encircled by the lp, what with dwnward being cnsidered the psitive directin fr flux, means a psitive rate f change f flux The left side f φ dl is thus psitive With d l being negative, we need the minus sign in frnt f it t make the right side psitive t Nw φ is the rate f change f magnetic flux thrugh the regin encircled by the lp due t the magnetic field lines that are entering that regin thrugh the ne infinitesimal d l that we 328

6 have been cnsidering There is a φ fr each infinitesimal d l making up the lp Thus there are an infinite number f them Call the infinite sum f all the φ s Φ and ur equatin becmes: Φ d l which is typically written dl Φ The integral is called a line integral because we integrate ver a curve, namely the curve that is the lp, and the circle n the integral sign indicates that the curve in questin is clsed The relatin is called Faraday s Law and is ne f Maxwell s equatins The same kinds f cnsideratins fr the case f an electric field mving sideways thrugh the perimeter f an imaginary lp leads t d l This is ur lp frm f the idea that an electric field mving sideways thrugh an empty pint in space causes a magnetic field t exist at that pint in space It is an incmplete versin f ne f Maxwell s quatins and, as it stands, is knwn as Maxwell s xtensin t Ampere s Law The thing is, unlike the case in which magnetic fields mve sideways thrugh the perimeter f a lp, there is a way in which electric fields can mve sideways thrugh the perimeter f a lp, withut there being a change in the number f electric field lines thrugh the regin enclsed by the lp In fact this happens whenever there is an electric current thrugh the regin enclsed by the lp A dwnward current thrugh a hrizntal lp can be mdeled as a vertical infinite line f psitive charge (perhaps in a statinary sheath f negative charge, als infinite in length, s the verall charge f any length f the cmbinatin is zer) mving dwnward µ e Φ 329

7 An imaginary lp that is fixed in space Current I THROUGH flwing directly away frm yu Think f it as an infinite line f psitive charge (viewed end n) that is mving away frm yu ecause the electric field lines are thse f the charged particles making up the current and thse particles are mving away frm yu, the electric field lines are mving away frm yu The mving charge (the current) causes electric field lines t be mving transversely thrugh the perimeter f the lp thus causing a magnetic field in that perimeter, which, fr the case depicted is clckwise as viewed frm abve We typically leave ut the middleman electric field when discussing this effect and say that a current thrugh the area enclsed by a lp causes a magnetic field in that lp and call the phenmenn Ampere s Law Careful analysis (that is nt particularly difficult but I want t shrten this discussin) f the phenmenn allws us t write Ampere s Law in the frm dl µ I THROUGH Nte that the left side is the same as the left side f what we called Maxwell s extensin t Ampere s Law integral l µ e Φ Maxwell s extensin cvers the case in which there is a changing number f electric field lines thrugh the regin enclsed by the lp but n current thrugh that regin Ampere s law cvers the case in which there is a current thrugh the regin enclsed by the lp but n changing number f electric field lines thrugh that regin If we have bth a current and a changing number f electric field lines thrugh a lp, then we have t add the tw cntributins t the magnetic field in the perimeter f the lp This results in the equatin d d l µ I THROUGH + µ e Φ which is Ampere s Law with Maxwell s xtensin It is ne f the fur equatins knwn as Maxwell s quatins S far in this chapter we have discussed tw f the fur equatins knwn as Maxwell s quatins The ther tw (bth f which we have already encuntered) are Gauss s Law fr the electric field: da Q NCLOSD e 33

8 and Gauss s Law fr the magnetic field: da That s all fur f Maxwell s quatins Here we list them in tabular frm with the crrespnding name and cnceptual statement beside each ne: Maxwell s quatins dl da da dl Q NCLOSD e Φɺ µ I + µ e Φ THROUGH ɺ Name and Crrespnding Cnceptual Statement Gauss s Law fr the lectric Field essentially a revised frm f Culmb s Law It states that a charged particle r a distributin f charge causes an electric field Gauss s Law fr the Magnetic Field It states that there is n such thing as a magnetic mnple Faraday s Law It states that a changing magnetic field causes an electric field Ampere s Law with Maxwell s xtensin It states that a current causes a magnetic field, and, that a changing electric field causes a magnetic field The frm f each f the equatins given abve is referred t as the integral frm f the crrespnding Maxwell s quatin If fr each equatin, ne takes the limit as the clsed surface r lp becmes infinitesimal in size, ne arrives at the differential frm f the crrespnding Maxwell s quatin In differential frm the equatins are expressed in terms f the vectr differential peratrs, the divergence ( delta dt ) and the curl ( delta crss ) Yu aren t required t knw what these peratrs mean r hw t use them, but, yu are required t be able t recgnize Maxwell s quatins when yu see them in differential frm, t be able t assciate each ne with the crrespnding integral frm f the equatin, and t be able t assciate each ne with the crrespnding name and cnceptual statement The fllwing tw tables will help yu meet these requirements: 331

9 Integral Frm f Maxwell s quatins QNCLOSD da e dl da dl Φɺ µ I + µ e Φ THROUGH ɺ Differential Frm f Maxwell s quatins r /e d µ J + µ e d The symbl r represents the charge density and J represents the current per area Differential Frm f Name and Crrespnding Maxwell s quatins Cnceptual Statement Gauss s Law fr the lectric Field essentially a revised frm f Culmb s Law It states that a charged r/e particle r a distributin f charge causes an electric field Gauss s Law fr the Magnetic Field It states that there is n such thing as a magnetic mnple d Faraday s Law It states that a changing magnetic field causes an electric field Ampere s Law with Maxwell s xtensin It states d µ J + µ e that a current causes a magnetic field, and, that a changing electric field causes a magnetic field 332

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