Sott Hughes 10 Marh 2005 Massahusetts Institute of Tehnology Department of Physis 8.022 Spring 2004 Leture 10: Magneti fore; Magneti fields; Ampere s law 10.1 The Lorentz fore law Until now, we have been onerned with eletrostatis the fores generated by and ating upon harges at rest. We now begin to onsider how things hange when harges are in motion 1. A simple apparatus demonstrates that something wierd happens when harges are in motion: If we run urrents next to one another in parallel, we find that they are attrated when the urrents run in the same diretion; they are repulsed when the urrents run in opposite diretions. This is despite the fat the wires are ompletely neutral: if we put a stationary test harge near the wires, it feels no fore. Figure 1: Left: parallel urrents attrat. Right: Anti-parallel urrents repel. Furthermore, experiments show that the fore is proportional to the urrents double the urrent in one of the wires, and you double the fore. Double the urrent in both wires, and you quadruple the fore. 1 We will deviate a bit from Purell s approah at this point. In partiular, we will defer our disussion of speial relativity til next leture. 89
This all indiates a fore that is proportional to the veloity of a moving harge; and, that points in a diretion perpendiular to the veloity. These onditions are sreaming for a fore that depends on a ross produt. What we say is that some kind of field B the magneti field arises from the urrent. (We ll talk about this in detail very soon; for the time being, just aept this.) The diretion of this field is kind of odd: it wraps around the urrent in a irular fashion, with a diretion that is defined by the right-hand rule: We point our right thumb in the diretion of the urrent, and our fingers url in the same sense as the magneti field. With this sense of the magneti field defined, the fore that arises when a harge moves through this field is given by F = q v B, where is the speed of light. The appearane of in this fore law is a hint that speial relativity plays an important role in these disussions. If we have both eletri and magneti fields, the total fore that ats on a harge is of ourse given by F = q ( E + v ) B. This ombined fore law is known as the Lorentz fore. 10.1.1 Units The magneti fore law we ve given is of ourse in gs units, in keeping with Purell s system. The magneti fore equation itself takes a slightly different form in SI units: we do not inlude the fator of 1/, instead writing the fore F = q v B. 90
This is a very important differene! It makes omparing magneti effets between SI and gs units slightly nasty. Notie that, in gs units, the magneti field has the same overall dimension as the eletri field: v and are in the same units, so B must be fore/harge. For historial reasons, this ombination is given a speial name: 1 dyne/esu equals 1 Gauss (1 G) when the fore in question is magneti. (There is no speial name for this ombination when the fore is eletri.) In SI units, the magneti field does not have the same dimension as the eletri field: B must be fore/(veloity harge). The SI unit of magneti field is alled the Tesla (T): the Tesla equals a Newton/(oulomb meter/se). To onvert: 1 T = 10 4 G. 10.2 Consequenes of magneti fore Suppose I shoot a harge into a region filled with a uniform magneti field: B v The magneti field B points out of the page; the veloity v initially points to the right. What motion results from the magneti fore? At every instant, the magneti fore points perpendiular to the harge s veloity exatly the fore needed to ause irular motion. It is easy to find the radius of this motion: if the partile has harge q and mass m, then F mag = F entripetal qvb = mv2 R R = mv qb. 91
If the harge q is positive, the partile s trajetory veers to the right, vie versa if its negative. This kind of qualitative behavior bending the motion of harges along a urve is typial of magneti fores. Notie that magneti fores do no work on moving harges: if we imagine the harge moves for a time dt, the work that is done is dw = F ( d s = F ) v dt v = q B v dt = 0. The zero follows from the fat that v B is perpendiular to v. 10.3 Fore on a urrent Sine a urrent onsists of a stream of freely moving harges, a magneti field will exert a fore upon any flowing urrent. We an work out this fore from the general magneti fore law. Consider a urrent I that flows down a wire. This urrent onsists of some linear density of freely flowing harges, λ, moving with veloity v. (The diretion of the harges motion is defined by the wire: they are onstrained by the wire s geometry to flow in the diretion it points.) Look at a little differential length dl of this wire (a vetor, sine the wire defines the diretion of urrent flow). The amount of harge ontained in this differential length is dq = λ dl. The differential of fore exerted on this piee of the wire is then d F = (λ dl) v B. There are two equivalent ways to rewrite this in terms of the urrent. First, beause the urrent is effetively a vetor by virtue of the veloity of its onstituent harges, we put I = λ v and find d F = dl I B. Seond, we an take the urrent to be a salar, and use the geometry of the wire to define the vetor: d F = I d l B. These two formulas are ompletely equivalent to one another. Let s fous on the seond version. The total fore is given by integrating: F = I d l B. If we have a long, straight wire whose length is L and is oriented in the ˆn diretion, we find F = IL ˆn B. This formula is often written in terms of the fore per unit length: F/L = (I/)ˆn B. 92
10.4 Ampere s law We ve talked about the fore that a magneti field exerts on harges and urrent; but, we have not yet said anything about where this field omes from. I will now give, without any proof or motivation, a few key results that allow us to determine the magneti field in many situations. The main result we need is Ampere s law: B d s = 4π I enl. C In words, if we take the line integral of the magneti field around a losed path, it equals 4π/ times the urrent enlosed by the path. Ampere s law plays a role for magneti fields that is similar to that played by Gauss s law for eletri fields. In partiular, we an use it to alulate the magneti field in situations that are suffiiently symmetri. An important example is the magneti field of a long, straight wire: In this situation, the magneti field must be onstant on any irular path around the wire. The amount of urrent enlosed by this path is just I, the urrent flowing in the wire: B d s = B(r)2πr = 4π I B(r) = 2I r. The magneti field from a urrent thus falls off as 1/r. Reall that we saw a similar 1/r law not so long ago the eletri field of a long line harge also falls off as 1/r. As we ll see fairly soon, this is not a oinidene. The diretion of this field is in a irulational sense the B field winds around the wire aording to the right-hand rule 2. This diretion is often written ˆφ, the diretion of 2 In priniple, we ould have defined it using a left-hand rule. This would give a fully onsistent desription of physis provided we swithed the order of all ross produts (whih is idential to swithing the sign of all ross produts). 93
inreasing polar angle φ. The full vetor magneti field is thus written 10.4.1 Field of a plane of urrent B = 2I r ˆφ. The magneti field of the long wire an be used to derive one more important result. Suppose we take a whole bunh of wires and lay them next to eah other: θ y x L The urrent in eah wire is taken to go into the page. Suppose that the total amount of urrent flowing in all of the wires is I, so that the urrent per unit length is K = I/L. What is the magneti field at a distane y above the enter of the plane? This is fairly simple to work out using superposition. First, from the symmetry, you should be able to see that only the horizontal omponent of the magneti field (pointing to the right) will survive. For the vertial omponents, there will be equal and opposite ontributions from wires left and right of the enter. To sum what s left, we set up an integral: B = 2 ˆx L/2 L/2 (I/L)dx x2 + y 2 os θ. In the numerator under the integral, we are using (I/L)dx as the urrent arried by a wire of width dx. Doing the trigonometry, we replae osθ with something a little more useful: B = 2 ˆx L/2 = 2Ky ˆx L/2 L/2 (I/L)dx x2 + y 2 L/2 dx x 2 + y 2. y x2 + y 2 This integral is doable, but not partiularly pretty (you end up with a mess involving artangents). A more tratable form is obtaining by taking the limit of L : using we find B = ˆx 2πKy y dx x 2 + y 2 = π y, 94
= +ˆx 2πK = ˆx 2πK y > 0 y < 0 (above the plane) (below the plane) The most important thing to note here is the hange as we ross the sheet of urrent: B = 4πK Does this remind you of anything? It should! When we ross a sheet of harge we have E = 4πσ. The sheet of urrent plays a role in magneti fields very similar to that played by the sheet of harge for eletri fields. 10.4.2 Fore between two wires Combining the result for the magneti field from a wire with urrent I 1 with the fore per unit length upon a long wire with urrent I 2 tells us the fore per unit length that arises between two wires: F L = 2I 1I 2 2 r. Using right-hand rule, you should be able to onvine yourself quite easily that this fore is attrative when the urrents flow in the same diretion, and is repulsive when they flow in opposite diretions. 10.4.3 SI units In SI units, Ampere s law takes the form C B d S = µ 0 I enl where the onstant µ 0 = 4π 10 7 Newtons/amp 2 is alled the magneti permeability of free spae. To onvert any gs formula for magneti field to SI, multiply by µ 0 (/4π). For example, the magneti field of a wire beomes The fore between two wires beomes B = µ 0I 2πr ˆφ. F L = µ 0I 1 I 2 2πr If you try to reprodue this fore formula, remember that the magneti fore in SI units does not have the fator 1/... 95
10.5 Divergene of the B field Let s take the divergene of straight wire s magneti field using Cartesian oordinates. We put r = x 2 + y 2. With a little trigonometry, you should be able to onvine yourself that ˆφ = ŷ os φ ˆx sin φ xŷ = x2 + y yˆx 2 x2 + y, 2 so B = 2I [ xŷ x 2 + y yˆx ] 2 x 2 + y 2. The divergene of this field is B = 2I = 0. [ ] 2yx (x 2 + y 2 ) 2xy 2 (x 2 + y 2 ) 2 We ould have guessed this without doing any alulation: if we make any small box, there will be just as many field lines entering it as leaving. Although we have only done this in detail for this very speial ase, it turns out this result holds for magneti fields in general: B = 0 Reall that we found E = 4πρ the divergene of the eletri field told us about the density of eletri harge. The result B = 0 thus tells us that there is no suh thing as magneti harge. This is atually not a foregone onlusion: there are reasons to believe that very small amounts of magneti harge may exist in the universe, reated by proesses in the big bang. If any suh harge exists, it would reate a Coulomb-like magneti field, with a form just like the eletri field of a point harge. No onlusive evidene for these monopoles has ever been found; but, absene of evidene is not evidene of absene. 10.6 What is the magneti field??? This magneti field is, so far, just a onstrut that may seem like I ve pulled out of the air. I haven t pulled it out of the air just for kiks observations and measurements demonstrate that there is an additional field that only ats on moving harges. But what exatly is this field? Why should there exist some field that only ats on moving harges? The answer is to be found in speial relativity. The defining postulate of speial relativity essentially tells us that physis must be onsistent in every frame of referene. Frames of referene are defined by observers moving, with respet to eah other, at different veloities. Consider, for example, a long wire in some laboratory that arries a urrent I. In this lab frame, the wire generates a magneti field. Suppose that a harge moves with veloity v parallel to this wire. The magneti field of the wire leads to an attrative fore between the harge and the wire. 96
Suppose we now examine this situation from the point of view of the harge (the harge frame ). From the harge s point of view, it is sitting perfetly still. If it is sitting still, there an be no magneti fore! We appear to have a problem: in the lab frame, there is an attrative magneti fore. In the harge frame, there an t possibly be an attrative magneti fore. But for physis to be onsistent in both frames of referene, there must be some attrative fore in the harge frame. What is it??? There s only thing it an be: in the harge s frame of referene, there must be an attrative ELECTRIC field. In other words, what looks like a pure magneti field in one frame of referene looks (at least in part) like an eletri field in another frame of referene. To understand how this happens, we must begin to understand speial relativity. This is our next topi. 97