20 Faraday s Law and Maxwell s Extension to Ampere s Law

Transcription

1 Chapter 20 Faraday s Law and Maxwell s Extensin t Ampere s Law 20 Faraday s Law and Maxwell s Extensin t Ampere s Law Cnsider the case f a charged particle that is ming in the icinity f a ming bar magnet as depicted in the fllwing diagram: p N S When we iew the situatin frm the reference frame f the magnet, what we see (as depicted just abe) is a charged particle ming in a statinary magnetic field. We hae already studied the fact that a magnetic field exerts a frce F = q P n a charged particle ming in that magnetic field. Nw let s lk at the same phenmenn frm the pint f iew f the charged particle: N S (where = ). P Surely we aren t ging t change the frce exerted n the charged particle by the magnetic field f the magnet just by lking at the situatin frm a different reference frame. In fact we e already addressed this issue. What I said was that it is the relatie mtin between the magnet and the charged particle that matters. Whether the charged particle is ming thrugh magnetic field lines, r the magnetic field lines, due t their mtin, are ming sideways thrugh the particle, the particle experiences a frce. Nw here s the new iewpint n this situatin: What we say is, that the ming magnetic field desn t really exert a frce n the statinary charged 174

2 Chapter 20 Faraday s Law and Maxwell s Extensin t Ampere s Law particle, but rather, that by ming sideways thrugh the pint at which the particle is lcated, the magnetic field creates an electric field at that lcatin, and it is the electric field that exerts the frce n the charged particle. In this iewpint, we hae, at the lcatin f the statinary charged particle, an electric field that is exerting a frce n the particle, and a magnetic field that is exerting n frce n the particle. At this stage it might seem that it wuld be necessary t designate the magnetic field as sme special kind f magnetic field that desn t exert a frce n a charged particle despite the relatie elcity between the charged particle and the magnetic field. Instead, what we actually d is t characterize the magnetic field as being at rest relatie t the charged particle. S, as iewed frm the reference frame in which the magnet is at rest: p N S the particle experiences a frce F directed ut f the page in the diagram abe due t its mtin thrugh the magnetic field. And, as iewed frm the reference frame in which the charged particle is at rest: N S the particle finds itself in a statinary magnetic field but experiences the same frce F because it als finds itself in an electric field directed ut f the page. 175

3 Chapter 20 Faraday s Law and Maxwell s Extensin t Ampere s Law S we hae tw mdels fr explaining the frce n the statinary charged particle in the case depicted by: N S In mdel 1 we simply say that in terms f the Lrentz Frce F = q, what matters is the relatie elcity between the particle and the magnetic field and t calculate the frce we identify the elcity P f the particle relatie t the magnetic field as being rightward at magnitude p = in the diagram abe s F = q P (where q is the charge f the particle). In mdel 2 we say that the apparent mtin f the magnetic field causes there t be an electric field and a statinary magnet field s the particle experiences a frce F = q E. Of curse we are using tw different mdels t characterize the same frce. In rder fr bth mdels t gie the same result we must hae: where: E = P (20-1) E is the electric field at an empty pint in space due t the mtin f that pint relatie t a magnetic field ectr that exists at that pint in space, is the elcity f the empty pint in space relatie t the magnetic field ectr, and P is the magnetic field ectr. Physicists hae fund mdel 2 t be mre fruitful, especially when attempting t explain magnetic waes. The idea that a magnetic field in apparent sideways mtin thrugh a pint in space causes there t be an electric field at that pint in space, is referred t as Faraday s Law f Inductin. Our mnemnic fr Faraday s Law f Inductin is: A changing magnetic field causes an electric field. 176

4 Chapter 20 Faraday s Law and Maxwell s Extensin t Ampere s Law The acceleratin experienced by a charged particle in the icinity f a magnet, when the charged particle is ming relatie t the magnet represents an experimental result that we hae characterized in terms f the mdel described in the preceding part f this chapter. The mdel is useful in that it can be used t predict the utcme f, and pride explanatins regarding, related physical prcesses. Anther experimental result is that a particle that has a magnetic diple mment and is ming in an electric field with a elcity that is neither parallel nr antiparallel t the electric field, des (except fr tw special magnetic diple mment directins) experience angular acceleratin. We interpret this t mean that the particle experiences a trque. Recalling that a particle with a magnetic diple mment that is at rest in an electric field experiences n trque, but ne that is at rest in a magnetic field des indeed experience a trque (as lng as the magnetic diple mment and the magnetic field it is in are nt parallel r antiparallel t each ther), yu might think that we can mdel the fact that a particle with a magnetic diple mment experiences a trque when it is ming relatie t an electric field, by defining a magnetic field caused by the apparent mtin f the electric field relatie t the particle. Yu wuld be right. T build such a mdel, we cnsider a charged particle that is ming in an electric field prduced by a lng line f charge that is unifrmly distributed alng the line. We start by depicting the situatin in the reference frame in which the particle is at rest and the line f charge is ming: E Nte that we hae tw different ways f accunting fr the magnetic field due t the ming line f charge, at the lcatin f the particle with a magnetic diple mment. The ming line f charge is a current s we can think f the magnetic field as being caused by the current. = I The ther ptin is t iew the magnetic field as being caused by the electric field lines ming sideways thrugh the particle. There is, hweer, nly ne magnetic field, s, the tw different ways f accunting fr it must yield the same result. We are ging t arrie at an expressin fr 177

5 Chapter 20 Faraday s Law and Maxwell s Extensin t Ampere s Law the magnetic field due t the mtin f an electric field by frcing the tw different ways f accunting fr the magnetic field t be cnsistent with each ther. First, we ll simply use Ampere s law t determine the magnetic field at the lcatin f the particle. Let s define the linear charge density (the charge per length) f the line f charge t be λ and the distance that the particle is frm the line f charge t be r. Suppse that in an amunt f time dt the line f charge mes a distance dx. Then the amunt f charge passing a fixed pint n the line alng which the charge is ming, in time dt, wuld be λ dx. Diiding the latter by dt yields λ dx/dt which can be expressed as λ and is just the rate at which charge is flwing past the fixed pint, that is, it is the current I. In ther wrds, the ming line f charge is a current I = λ. ack in chapter 17 we gae the experimental result fr the magnetic field due t a lng straight wire carrying current I in the frm f an equatin that we called Ampere s Law. It was equatin 17-2; it read: I = 2π r and it applies here. substituting I = λ int this expressin fr yields λ = (20-2) 2π r y the right hand rule fr smething curly smething straight we knw that the magnetic field is directed int the page at the lcatin f the particle that has a magnetic diple mment, as depicted in the fllwing diagram: r Nw let s wrk n btaining an expressin fr the same magnetic field frm the iewpint that it is the electric field ming sideways thrugh the lcatin f the particle that causes the magnetic field. First we need an expressin fr the electric field due t the line f charge, at the lcatin f the particle, that is, at a distance r frm the line f charge. The way t get that is t cnsider the line f charge as cnsisting f an infinite number f bits f charged material, each f which is a segment f infinitesimal length dx f the line f charge. Since the line f charge has a linear charge density λ, this means that each f the infinitesimal segments dx has charge λ dx. T get the electric field at the lcatin f the particle that has a magnetic diple mment, all we hae t d is t add up all the cntributins t the electric field at the lcatin f the particle, due t all the infinitesimal segments f charged material making up the line f charge. Each cntributin is gien by Culmb s Law fr the Electric Field. The difficulty is that there are an infinite 178

6 Chapter 20 Faraday s Law and Maxwell s Extensin t Ampere s Law number f cntributins. Yu will be ding such calculatins when yu study chapter 30 f this textbk. At this stage, we simply pride the result fr the electric field due t an infinitely lng line f charge haing a cnstant alue f linear charge density λ: Multiplying bth sides by e yields E = e E = λ 2πre λ 2πr The expressin n the right side f this equatin appears in equatin 20-2, Substituting e E fr λ where the latter appears in equatin 20-2 yields: 2πr = e E λ =. 2π r This represents the magnitude f the magnetic field that is experienced by a particle when it is ming with speed p = relatie t an electric field E when the elcity is perpendicular t E. Experimentally we find that a particle with a magnetic diple mment experiences n trque (and hence n magnetic field) if its elcity is parallel r antiparallel t the electric field E. As such, we can make ur result mre general (nt nly gd fr the case when the elcity is perpendicular t the electric field) if we write, E in place f E. = e E T E E Starting with the preceding equatin, we can bundle bth the magnitude and the directin (as determined frm Ampere s Law and the right hand rule when we treat the ming line f charge as a current, and as depicted in the diagram abe) f the magnetic field int ne equatin by writing: = e E 179

7 Chapter 20 Faraday s Law and Maxwell s Extensin t Ampere s Law We can express in terms f the elcity P f the particle relatie t the line f charge E p (instead f the elcity f the line f charge relatie t the particle) just by recgnizing that Substituting this expressin ( ( = e E ) yields: P P = = ) int ur expressin fr the magnetic field = e P E In this mdel, where we accunt fr the trque experienced by a particle that has a magnetic diple mment when that particle is ming in an electric field, by defining a magnetic field = e P E which depends bth n the elcity f the particle relatie t the electric field and the electric field itself, the electric field itself is cnsidered t exert n trque n the charged particle. At this stage it might seem that it wuld be necessary t designate the electric field as sme special kind f electric field that desn t exert a trque n a charged particle despite the relatie elcity between the charged particle and the electric field. Instead, what we actually d is t characterize the electric field as being at rest relatie t the charged particle. S, as iewed frm the reference frame in which the line f charge is at rest: E p 180

8 Chapter 20 Faraday s Law and Maxwell s Extensin t Ampere s Law the particle that has a magnetic diple mment experiences a trque due t its mtin thrugh the electric field. And, as iewed frm the reference frame in which the particle is at rest: E p the particle that has a magnetic diple mment finds itself in a statinary electric field but experiences the same trque because it als finds itself in a magnetic field directed, in the diagram abe, int the page. One way f saying what is ging n here is t say that, lsely speaking: A changing electric field causes a magnetic field. The phenmenn f a changing electric field causing a magnetic field is referred t as Maxwell s Extensin t Ampere s Law. S far, in this chapter we hae addressed tw majr pints: A magnetic field ming sideways thrugh a pint in space causes there t be an electric field at that pint in space, and, an electric field ming sideways thrugh a pint in space causes there t be a magnetic field at that pint in space. In the remainder f this chapter we find that putting these tw facts tgether yields smething interesting. Expressing what we hae fund in terms f the pint f iew in which pint P is fixed and the field is ming thrugh pint P with speed =, we hae: a magnetic field ectr ming with elcity transersely thrugh a pint in space will cause an electric field E = at that pint in space; and; an electric field ectr ming with elcity transersely thrugh a pint in space will cause a magnetic field = e E at that pint in space. The wrd cause is in qutes because there is neer any time delay. A mre precise way f putting it wuld be t say that wheneer we hae a magnetic field ectr ming transersely thrugh a pint in space, there exists, simultaneusly, an electric field E = at that pint in space, and wheneer we hae an electric field ectr ming transersely thrugh a pint in space there exists, simultaneusly, a magnetic field = E at that pint in space. e P 181

9 Chapter 20 Faraday s Law and Maxwell s Extensin t Ampere s Law Cnsider the fllwing circuit. Assume that we are lking dwn n the circuit frm abe, meaning that int the page is dwnward, and ut f the page is upward. I want yu t fcus yur attentin n the rightmst wire f that circuit. As sn as smene clses that switch we are ging t get a current thrugh that wire and that current is ging t prduce a magnetic field. y means f the right-hand rule fr smething curly smething straight, with the current being the smething straight, and ur knwledge that straight currents cause magnetic fields that make lps arund the current, we can deduce that there will be an upward-directed (pinting ut f the page) magnetic field at pints t the right f the wire. In steady state, we understand that the upward-directed magnetic field ectrs will be eerywhere t the right f the wire with the magnitude f the magnetic field ectr being smaller the greater the distance the pint in questin is frm the wire. Nw the questin is, hw lng des it take fr the magnetic field t becme established at sme pint a specified distance t the right f the wire? Des the magnetic field appear instantly at eery pint t the right f the wire r des it take time? James Clerk Maxwell decided t explre the pssibility that it takes time, in ther wrds, that the magnetic field deelps in the icinity f the wire and mes utward with a finite elcity. Here I want t talk abut the leading edge f the magnetic field, the expanding bundary within which the magnetic field already exists, and utside f which, the magnetic field des nt yet exist. With each passing infinitesimal time interal anther infinitesimal layer is added t the regin within which the magnetic field exists. While this is mre a case f magnetic field ectrs grwing sideways thrugh space, the effect f the mtin f the leading edge thrugh space is the same, at the grwing bundary, as magnetic field ectrs ming thrugh space. As such, I am ging t refer t this magnetic field grwth as mtin f the magnetic field thrugh space. 182

10 Chapter 20 Faraday s Law and Maxwell s Extensin t Ampere s Law T keep the drawing uncluttered I m ging t shw just ne f the infinite number f magnetic field ectrs ming rightward at sme unknwn elcity (and it is this elcity that I am curius abut) as the magnetic field due t the wire becmes established in the unierse. Again, what I m saying is that, as the magnetic field builds up, what we hae, are rightwardming upward (pinting ut f the page, tward yu) magnetic field lines due t the current that just began. Well, as a magnetic field ectr mes thrugh whateer lcatin it is ming thrugh, it causes an electric field E =. 183

11 Chapter 20 Faraday s Law and Maxwell s Extensin t Ampere s Law At any pint P thrugh which the magnetic field ectr passes, an electric field exists cnsistent with E =. What this amunts t is that we hae bth a magnetic field and an electric field ming rightward thrugh space. ut we said that an electric field ming transersely thrugh space causes a magnetic field. Mre specifically we said that it is always accmpanied by a magnetic field gien by = e E. Nw we e argued arund in a circle. The current causes the magnetic field and its mement thrugh space causes an electric field whse mement thrugh space causes the magnetic field. Again, the wrd causes here shuld really be interpreted as exists simultaneusly with. Still, we hae tw explanatins fr the existence f ne and the same magnetic field and the tw explanatins must be cnsistent with each ther. Fr that t be the case, if we take ur expressin fr the magnetic field caused by the mtin f the electric field, = E e and substitute int it, ur expressin E = fr the electric field caused by the mtin f the magnetic field, we must btain the same that, in this circular argument, is causing itself. Let s try it. Substituting E = int = E, we btain: e = e ( ) All right. Nting that is perpendicular t bth and, meaning that the magnitude f the crss prduct, in each case, is just the prduct f the magnitudes f the multiplicand ectrs, we btain: 2 = e 184

12 Chapter 20 Faraday s Law and Maxwell s Extensin t Ampere s Law which I cpy here fr yur cnenience: 2 = e Again, it is ne and the same n bth sides, s, the nly way this equatin can be true is if 2 e is exactly equal t 1. Let s see where that leads us: e 2 = 1 = e 2 1 = 1 e = 4π T m A 12 2 C N m 2 = m s Ww! That s the speed f light! When James Clerk Maxwell fund ut that electric and magnetic fields prpagate thrugh space at the (already knwn) speed f light he realized that light is electrmagnetic waes. 185