Chapter 28: Sources of Magnetic Field
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1 Chapte 28: Souces of Magnetic Field In the last chapte, we talked about what happens to a chage moving in a magnetic field B, but we didn t conside how the B got thee in the fist place. We just assumed a magnetic field got ceated somehow, and then talked about the magnetic foce F mag that this field would exet on point chages moving in the field and wies caying cuents in the field. In this chapte, we finally addess the question of how you make a B. Key idea: Moving electic chages make a B. 1
2 The Biot-Savat Law: Magnetic Field of a Cuent Element Conside a wie of abitay shape caying a steady cuent I. The cuent I is the esult of individual chages moving within the wie. This cuent poduces a magnetic field in the space suounding the wie. How do we wite down the magnetic field that this cuent poduces at a point P a distance away fom the wie? Conside an infinitesimal segment ( element ) of the wie. Let the length of this cuent element be d and define a vecto d that has magnitude equal to d and that points in the diection of the cuent I. Let be the vecto fom this cuent element to the point P. The vecto ˆ is then a unit vecto in the diection of. The segment of the wie of infinitesimal length d poduces some infinitesimal contibution to the total field at P. We call this infinitesimal contibution to the field db. Expeimentally, we find that db has the following popeties: 2
3 db is popotional to I db is popotional to d db 1 is popotional to 2 db is popotional to sinφ (φ = angle between d and ˆ ) db is pependicula to d and pependicula to ˆ We summaize all of the above findings as follows: μ ˆ 0 Id db = 2 4π This is called the Biot-Savat Law, afte the Fench physicists Jean-Baptiste Biot and Felix Savat. In magnitude, the infinitesimal contibution to the field fom just one cuent element is: μ Id ˆ 0 μ0 Idsinφ db = = 2 2 4π 4π (1) (2) 3
4 The constant μ 0 is called the pemeability of fee space. In SI units, it has the value: 7 T m μ0 = 4π 10 (3) A Magnetic Field of a Cuent-Caying Wie To find the total magnetic field at P due to the whole wie, we must add up all the infinitesimal contibutions db fom all segments of the wie. This means we must integate ove the whole wie: μ ˆ 0 Id B = 2 4π Equation (4) gives the magnetic field due to a cuent distibution. It is the magnetic analogue of the equation we used to calculate electic fields due to continuous chage distibutions in Chapte 21: dq 1 dq E = k ˆ= ˆ 4πε (4) 4
5 Note: The equation fo B follows a 2 1 law, just like the equation fo E. The diections of B and E ae diffeent: E is adial, but B is not. You cannot have an isolated cuent element, but you can have an isolated chage element (i.e., a point chage). Theefoe, you always have to integate, as in Eq. (4), to get the B due to any eal cuent distibution. 5
6 Ampee s Law Ampee s law gives an easy way (easie than the Biot-Savat law) to detemine the B poduced by cuent distibutions in cases of a high degee of symmety. Ampee s law elates the line integal of the tangential component of B aound some closed loop to the net cuent enclosed by the loop. Conside a long (essentially infinitely long), staight wie caying a steady cuent I. Now conside an imaginay cicula loop (the Ampeian loop ) centeed on the wie. At each point this cicle, the B poduced by the wie is tangent to the cicle and has the same magnitude as at any othe point on the cicle. Let d be an infinitesimal bit of aclength on the cicula loop. Then eveywhee on the loop: B d = Bd cos0 = Bd Integating aound the entie closed cicula loop once gives: B d = Bd = B d = B( 2π ) But we know B fom the Biot-Savat law: ( ) 6
7 μ0i B = ( ) 2π Plugging ( ) into ( ) gives: B d =μ0i The equation immediately above is also coect fo any abitay distibution of steady cuents passing though the loop, and fo any abitay shape of loop enclosing the cuent distibution, as long as the cuent I is undestood to mean the net cuent enclosed by the loop. Thus, fo any distibution of cuents passing though any abitay closed loop: B d =μ0iencl, (5) in which I encl means the net cuent enclosed by the loop. Eq. (5) is called Ampee s law. We use it to find the B poduced by a distibution of cuents, just as we used Gauss s law to find E due to a distibution of chages. But, like Gauss s law, Ampee s law is useful fo finding the field only in cetain special cases of symmety (specifically, when the B can be extacted fom the integal). 7
8 Foce Between Paallel Conductos When cuents flow in two paallel wies, each wie ceates a B at the location of the othe wie. This B exets a foce on the othe wie. Fo example, let I 1 be the cuent flowing (fom left to ight, say) in Wie #1, which is one of two long, staight, paallel wies. Similaly, let I 2 flow (also fom left to ight) in the othe wie, Wie #2, which is a distance d below Wie #1. The magnitude of the magnetic field poduced by Wie #1 is: μ0i1 B1 =, 2π 1 in which 1 means the distance fom Wie #1. At the location of Wie #2, 1 = d, and we have: μ0i1 B1 = ( ) 2π d The diection of this field is into the boad, pependicula to Wie #2. 8
9 This field B 1 exets a foce on Wie #2 given by Fmag = IL B. Applied to Wie #2, this equation gives: F = IL B, in which F 12 is the foce that Wie #1 exets on Wie #2. In magnitude, this foce is: F12 = ILB 2 1sinφ = ILB 2 1sin90 = ILB 2 1 ( ) Plugging ( ) into ( ) gives: μ0i1 F12 = I2L 2π d μ0iil 1 2 F12 = (6) 2π d By the RHR fo Fmag = IL B, the vecto F 12 points up, towad Wie #1. Thus, Wie #2 is attacted to Wie #1. By Newton s Thid Law, the foce F that Wie #2 exets on Wie #1 is of the same magnitude, downwad. 21 9
10 Because the wies wee assumed to be vey long (i.e., essentially infinitely long), it s often moe convenient to think about the foce pe unit length athe than the total foce. Fom Eq. (6), the foce pe unit length that each wie exets on the othe has magnitude: F μ0ii 1 2 = (7) L 2π d 10
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