Magnetic Field in a Time-Dependent Capacitor

Transcription

1 Magnetic Field in a Time-Dependent Capacito 1 Poblem Kik T. McDonald Joseph Heny Laboatoies, Pinceton Univesity, Pinceton, NJ 8544 (Octobe 3, 23) Reconside the classic example of the use of Maxwell s displacement cuent to calculate the magnetic field in the midplane of a capacito with cicula plates of adius R while the capacito is being chaged by a time-dependent cuent I(t). In paticula, conside the displacement cuent density, ɛ E/ t in MKSA units fo vacuum between the plates, to consist of a collection of small, close-packed wies that extend fom one plate of the capacito to the othe. Sum the magnetic field fom this set of vitual wies to find the total field. 1 2 Solution Aspects of this poblem ae consideed in [1, 2, 3, 4, 5]. Fom the cicula symmety of the capacito, and Ampèe/Maxwell s law that ( ) E B = μ J conduction + ɛ t (1) in vacuum, we infe that the magnetic field is puely azimuthal, B = B(, z)ˆφ, and independent of azimuth in a cylindical coodinate system (, θ, z) whose axis is that of the capacito (and feed wies). We suppose that the thickness of the capacito is small compaed to its adius R, so that we may appoximate the electic field as being unifom and in the ẑ diection inside the capacito, and zeo outside. Then we have E = Eẑ, and E = σ ɛ = Q π ɛ (<R), (2) whee the chage Q on the capacito is elated to the chaging cuent accoding to I = dq/dt. Hence, on the midplane of the capacito eq. (1) becomes B = μ J displacement, (3) whee J displacement = I π ẑ ( <), (>R). (4) 1 The essence of this poblem is to show that the magnetic field due to a cicula bundle of n fine wies, each caying cuent I/n, is the same as that of a single, fat wie that caies cuent I. 1

2 This is a quasi-static appoximation that ignoes the adiation of the acceleating chages in the cicuit. By consideing a loop of adius, centeed on the oigin and in the midplane of the capacito, the integal fom of eq. (3) immediately tells us that B() = μ I 2π 1 (<), ( >R). Outside the capacito, the magnetic field has the same fom as that of a wie which caies cuent I. Maxwell invented the concept of displacement cuent to insue that eq. (1) would lead to such esults. The paticula challenge of the pesent poblem is to imagine that the displacement cuent consists of small filaments, the sum of whose fields should give eq. (5). The displacement cuent in a vitual wie of adius a R that uns between the plates of the capacito is I displacement = πa 2 J displacement = I a2 R. (6) 2 The magnetic field due to this vitual wie ciculates aound the wie, and has magnitude (5) B vitual = μ I displacement 2πs = μ Ia 2 2π s, (7) whee s is the pependicula distance fom the obseve to the vitual wie. Stictly speaking, this fom holds only fo distance s small compaed to the length of the vitual wie. Howeve, having accepted that the electic field is unifom within the capacito, we can use fom (7) without futhe eo. We now must add up the contibutions fom all the vitual wies to find the total magnetic field at an abitay point in the midplane. Because of the azimuthal symmety, we can take this point to be (,, ), which lies on the x axis, without loss of geneality. We expect the total magnetic at this point to be along the y axis, so we sum up only the y components of the fields (7) of the vitual wies. The cente of an abitay vitual wie is witten as (,θ,). The distance s fom the obsevation point to the vitual wie is s = cos θ, (8) as shown in the figue. The unit vecto along the diection of s has components ŝ = ( cos θ, sin θ), (9) s so that a unit vecto pependicula to ŝ, which is in the diection of the magnetic field of the vitual wie, has components ˆB vitual = ( sin θ, cos θ,) s, (1) 2

3 Combining eqs. (7) and (1), the y component of the magnetic field of the vitual wie is B vitual,y = μ Ia 2 cos θ 2π s s = μ Ia 2 ( cos θ) 2π ( cos θ). (11) We integate ove the aea of the midplane, noting that an aea element d dθ contains d dθ/πa 2 vitual wies of adius a. B(,, ) = R d 2π dθ πa 2 B vitual,y(s) = μ I R 2π d 2π 2 cos θ dθ cos θ. (12) The θ integal must be evaluated sepaately fo the cases > and <. Fo > we have 2π 2π cos θ dθ cos θ = 1 1 dθ ( > ), (13) using Dwight 859.1, , and Thus, fo an obseve outside the capacito plates, we find B(,, ) = μ I R d = μ I π 2π ( >R), (14) in ageement with eq. (5). Fo <, the angula integal is 2π cos θ dθ cos θ = 1 2π dθ cos θ cos θ = (< ). (15) 2 3

4 That is, thee is no contibution to the total magnetic field at adius due to the vitual wies at adius >, as could be anticipated fom Ampèe s law. Hence, fo an obseve at <Rwe integate ove the vitual wies with <to find B(,, ) = μ I d = μ I π 2π (<R), (16) which is also in ageement with eq. (5). A Appendix. The Effects of Cuents in the Capacito Plates Since the chage on the capacito plates is time dependent, thee must be cuents flowing on those plates. What contibution, if any, do these adial cuents make to the magnetic field? A.1 Magnetic Field in the Plane of the Capacito, but Outside It One way to addess this question is via Ampèe s law, as illustated in the figue below. Ampèe s law in integal fom states that the integal of the tangential component of the magnetic field aound a loop is equal to (μ times) the cuent though the loop. To detemine the cuent though the loop, we conside a suface that is bounded by the loop and use the cuent that cosses that suface. The left figue above shows a section though a capacito with cicula plates of adius R that is fed by a time-dependent cuent I(t). We desie the magnetic field at adius a>ron a cicula loop located in the midplane of the capacito. We fist take the Ampèian suface to be a kind of cylindical cup such that the base of the cup is pieced by the ighthand lead wie of the capacito (left figue). Then the only cuent though the suface is the cuent I in the wie. The tangential component B(a) of the magnetic field at adius a is independent of the position aound the loop, so B(a, t) dl =2πaB(a, t)=μ I(t), (17) 4

5 o B(a, t)= μ I(t) 2πa, (18) as noted ealie in eq. (5). If we choose the Ampèian suface bounded by the loop of adius a to be the disc in the plane of the loop (second figue), then no eal (i.e., conduction) cuent passes though the loop. Howeve, since thee is a time-dependent electic field at the disc, thee is a non-zeo displacement cuent acoss it, of magnitude E I displacement (t) = J displacement da = ɛ t da = ɛ d Q π = dq = I(t), (19) dt π ɛ dt ecalling eq. (2) fo the electic field in a capacito. Hence, use of the Ampèian suface in the middle figue also leads to eq. (18) fo the magnetic field at adius a. We could also choose the Ampèian suface to be that shown in the thid figue on p. 4, which consists of the oute potion of the disc in the plane of the loop, plus a cup of adius whose bottom is pieced by the ighthand lead of the capacito. In this case, thee ae 3 contibutions to the cuent though the suface: 1. The displacement cuent fo adii >, E I displacement (>,t)= ɛ > t da = ɛ d Q π( 2 )= 1 2 I(t). (2) dt π ɛ 2. The adial cuent I (, t) atadius on the capacito plate. 3. The cuent I(t) in the lead of the capacito. Fo Ampèe s law to calculate the same magnetic field no matte which Ampèian suface is used, Maxwell noted that the total cuent acoss any suface that is bounded by the same loop must be the same. Applying this esult to the fist and thid sufaces, we have I(t) =I displacement (>,t) I (, t)+i(t), (21) wheeweuseaminussignfothetemi (, t) since the adial cuent cosses the Ampèian suface in the opposite sense to that of the cuent I(t) in the lead wie. Thus, we conclude that I (, t) =I displacement (>,t)= 1 2 I(t). (22) When we conside the Ampèian suface in the fouth pat of the figue on p. 4, we see that I(t) =I displacement (<,t)+i (, t), (23) and we conclude that as found in eq. (22). I (, t) =I(t) I displacement (<,t)=(i(t) π2 π I(t) = 5 ( 1 2 ) I(t), (24)

6 A.2 Diect Calculation of the Radial Cuent We digess slightly to veify eq. (24) fo the adial cuent by an appoach that does not involve the displacement cuent. Conside a ing of inne adius and oute adius + d < R on the ighthand capacito plate of the figue on p. 4. The aea of this ing is da =2π d. Since the chage Q(t) on the plate is unifomly distibuted (in ou appoximation), the chage δq on the ing is δq(, t) =Q da A = 2Q d. (25) R2 Consevation of chage tells us that the time ate of change of chage on the ing is equal to the diffeence of the adial cuents at and + d, I ( + d, t) I (, t) = dδq, (26) dt noting that the adial cuent is inwads on the ighthand capacito plate. Hence, di (, t) d = 2d dq d dt = 2I(t)d, (27) using eq. (25), and noting that when I(t) > the chage on the ighthand plate is deceasing with time Thus, di (, t) d = 2I(t), (28) which integates to I (, t) =A 2 I(t). (29) Since the adial cuent vanishes when = R we see that A = I, andthat I (, t) = 1 2 I(t), (3) as found peviously. 2 2 If the fequency of the sinusoidal cuent is high enough, the cuent flows only on the exteio of the conductos. In this case we can speak of the adial cuent I,out () that flows outwads fom the lead wie on the outside of a capacito plate, and the cuent I,in () that flows inwads fom the oute edge onto the inside of the plate. If we continue to ignoe finge-field effects then thee is no chage accumulation on the outside of the plate, while the chage distibution on the inside of the plate is unifom. Theefoe, I,out () =I (no chage accumulation on the outside of the plate). The inside cuent has the geneal fom of eq. (29) (so as to povide a unifom chage accumulation on the inside of the plate), while at the oute im of the plate we have I,in () =A I = I,out (R) = I. Hence, we have A =,and I,out () =I, ) I,in () = I 2 R, I,total() =I (1 2. (31) 2 The magnetic field due to the cuents on the (thin) capacito plates is that due to I,total (), and hence is the same as that deduced by ou othe appoaches. 6

7 A.3 Magnetic Field Inside the Capacito We seek the value of the magnetic field at a adius a<rinside the capacito, as shown in the figue below. We calculate the field thee ways, using the thee Ampèian sufaces illustated above. Fist, we take the Ampèian suface to be the disc bounded by the loop. Then, no conduction cuent passes though the suface, and we conside only the displacement cuent. Ampèe s law tells us that B(a, t) dl =2πaB(a, t)=μ I though = μ I displacement (<a,t)=μ I(t) a2 R. 2 (32) Thus, a B(a, t) =μ I(t) 2πR, 2 (33) in ageement with eq. (5). If we wish to avoid use of the displacement cuent, we can use the Ampèian suface shown in the middle of the figue on p. 6, fo which Ampèe s law becomes [ ] 2πaB(a, t)=μ I though = μ [I(t) I (a, t)] = μ I(t) 1 1 a2 = μ I(t) a2 R, 2 (34) so the magnetic field is again given by eq. (33). We could avoid use of the adial cuent by consideing the Ampèian suface in the ight figue on p. 6. In this case Ampèe s law tells us that [ ] 2πaB(a, t)=μ I though = μ [I(t) I displacement (>a,t)] = μ I(t) 1 1 a2 = μ I(t) a2 R, 2 (35) and again the magnetic field is given by eq. (33). The magnetic field eq.(33) inside the capacito is not equal to the magnetic field that exists at the same adius a about a wie when no capacito is pesent. Howeve, we cannot give a unique answe to the question as to why the field is diffeent. We can say that the field is due to the cuent in the wie plus the adial cuent in the capacito plates. O, we can say that the field is due to the displacement cuent at adius <a. O, we can say that the field is due to the cuent in the wie plus (actually minus) the displacement cuent at adius >a. 3 3 The fact that multiple appoaches to calculating the magnetic field give the same answe is a stength athe than a defect of the appoach of Ampèe and Maxwell, and should not be taken as a sign that the displacement cuent is fictitious. 7

8 A.4 Magnetic Field Outside the Capacito at a<r We seek the value of the magnetic field at a adius a<r, but outside the capacito, as shown in the figue below. We calculate the field thee ways, using the thee Ampèian sufaces illustated above. Fist, we take the Ampèian suface to be the disc bounded by the loop. Then, the only cuent though the suface is the cuent in the wie lead. Hence, Ampèe s law tells us that B(a, t) dl =2πaB(a, t)=μ I though = μ I(t). (36) Thus, I(t) B(a, t)=μ 2πa, (37) in ageement with eq. (5). But, we can avoid diect use of the cuent in the capacito lead by eithe of the second o thid Ampèian sufaces in the figue on p. 7. Fo the second suface, Ampèe s law becomes [ ] a 2 2πaB(a, t) = μ I though = μ [I displacement (<a,t)+i (a, t)] = μ I(t) R + 1 a2 2 = μ I(t), so the magnetic field is again given by eq. (37). We could avoid use of the adial cuent by consideing the Ampèian suface in the ight figue on p. 7. In this case Ampèe s law tells us that (38) 2πaB(a, t)=μ I though = μ I displacement (<R,t)=μ I(t), (39) and again the magnetic field is given by eq. (37). A.5 Comments Use of the thid figue on p. 7 pemits us to calculate the magnetic field anywhee aound the capacito without diect use of any conduction cuent. Howeve, this should not be constued as evidence that time-dependent magnetic fields ae only due to displacement cuents, since we can equally well pefom calculations that do not use the displacement cuents. Rathe, we follow Maxwell in noting that both conduction and displacement cuents exist, and eithe o both types of cuents may be useful in analyzing paticula physical situations. 8

9 The calculations in secs. A.1 and A.3-4 have all been based on Ampèe s law, which gives quick esults in situation with high symmety. This appoach is not, howeve, pactical in geneal. An altenative appoach is to use what is sometimes called the etaded Biot-Savat law (see, fo example, [6]) B(x,t)= μ 4π Jcond (x,t = t s/c) ŝ d 3 x + μ s 2 4πc Jcond (x,t = t s/c) ŝ s whee s = x x, s = s, ŝ = s/s, andj cond is the conduction cuent. 4 d 3 x, (4) A.6 Magnetic Field Inside a Naow-Gap Capacito Via the Biot- Savat Law In the pesent example of a capacito with a low-fequency cuent, etadation and time vaiation of the cuent can be ignoed. In this case eq. (4) educes to the familia fom of the Biot-Savat law. A detailed calculation via the Biot Savat law of the magnetic field in the gap between the capacito plates equies use of the adial cuents as well as the cuents in the lead wies. The appoximation of the pesent poblem is that the electic field inside the capacito is unifom, which coesponds to the gap between the plates being much smalle than thei adius. In this appoximation, the contibutions to the magnetic fields fom the adial cuents in the two capacito plates cancel fo points outside the capacito and the magnetic field is the same as that due to the wie leads, which value is unaffected by the small gap in the leads at the location of the capacito. The calculation of the field at a point inside a naow-gap capacito via the Bio-Savat law can be made as follows. The lead wies un along the z axis, and the capacito plates lie in the planes z = ±ɛ whee ɛ R. We desie the field at the point (x, y, z) =(a,, ) whee a<r. Then the Biot-Savat calculation is B(a,, ) = μ 4π leads Idzẑ s s 3 + μ 4π plates I 2π daea ˆ t t 3, (41) whee s =(a,,z) is the vecto distance between the obsevation point and a point on the lead wie, t is the vecto distance between the obsevation point and a point on the capacito plates, and I ˆ/2π is the adial cuent density on the plates, whee I is given by eq. (3). We fist evaluate the integal ove the leads. Hee, ẑ s = aŷ, ands =(a 2 + z 2 ) 1/2. Thus, B leads (a,, ) = 2 μ 4π aiŷ dz ɛ (a 2 + z 2 ) = μ ( Iŷ 1 ɛ ) μ I (42) 3/2 2πa a 2πaŷ, just as if the capacito wasn t thee. In the integal ove the plates, the main contibution comes fom points (, θ, z) (a + δx, δy ± ɛ) on the plate that ae vey close to the obsevation point, whee δx ɛ, δy ɛ. Fo this egion the vecto distance is appoximately paallel to the z axis, and we use the appoximation t = ±ɛz. Also,ˆ ˆx in this egion, so ˆ t ±ɛŷ. Since a in this egion, we have I () I (a) = I(1 a 2 / ). 4 This pesciption is not evidence that the displacement cuent is nonexistent, but only that thee ae methods of calculation that don t use it. 9

10 We set up a new, pola coodinate system (ρ, φ) inthex-y plane, centeed on the point (x, y) =(a, ). Then, t =(ɛ 2 + ρ 2 ) 1/2. The integand is now independent of angle φ, sowe wite the aea element as daea = 2πρ dρ. The magnetic field due to the cuents in the plates is theefoe, B plates (a,, ) 2 μ 4π = μ ɛi 4πa I 1 a2 2πa 1 a2 πɛŷ ŷ 2 ɛ = μ I 2πa 2ρ dρ (ɛ 2 + ρ 2 ) 3/2 1 a2 ŷ. (43) The total magnetic field at the obsevation point inside the capacito is the sum of eqs. (42) and (43), B(a,, ) = μ [ ] I 1 1 a2 ŷ = μ Ia ŷ, (44) 2πa 2πR2 as found peviously via Ampèe s law. A.7 A Final Comment Ampèe s law and the Biot-Savat law epesent somewhat diffeent ways of thinking about the oigin of the magnetic fields. In both views we say that the magnetic field is due to cuents. Ampèe s law implements the idea that the magnetic field at a point can be detemined only fom knowledge of the cuents on some suface that encloses that point. In contast, the Bio-Savat law equies knowledge of the cuents in all space. Thus, Ampèe s law is moe economical in its equiements fo infomation about the physical system. But, as Maxwell was the fist to note, the moe elegant appoach of Ampèe equies a moe sophisticated conception of the meaning of the tem cuent. The ewad fo mastey of these enlagements of the notion of cuent was ou undestanding of electomagnetic wave phenomena. Refeences [1] A.P. Fench and J.P. Tessman, Displacement Cuent and Magnetic Fields, Am.J.Phys. 31, 21 (1963), [2] R.M. Whitme, Calculation of Magnetics and Electic Fields fom Displacement Cuents, Am. J. Phys. 33, 481 (1965), [3] A. Shadowitz, The Electomagnetic Field (McGaw-Hill, 1977; Dove epint, 1978), sec [4] A.J. Dahm, Calculation of the displacement cuent using the integal fom of Ampee s law, Am. J. Phys. 46, 1227 (1978), 1

11 [5] D.F. Batlett, Conduction cuent and the magnetic field in a paallel plate capacito, Am. J. Phys. 58, 1168 (199), [6] K.T. McDonald, The Relation Between Expessions fo Time-Dependent Electomagnetic Fields Given by Jefimenko and by Panofsky and Phillips (Dec. 5, 1996), 11