Hoofstuk 29 Magnetiese Velde a.g.v Elektriese Strome Nadat hierdie hoofstuk afghandel is, moet die student: Magnetiese veld as gevolg van n stroom kan bereken; Die regterhandreëls kan neerskryf en toepas; Die krag tussen twee parallelle strome kan aflei. (29-1)
Magnetiese veld gevorm om n lang reguit stroomdraende draad Die grootte v/d magneetveld gevorm om die draad by punt P n afstand R vanaf die draad word gegee deur: B = µ i 0 2π R Die magnetiese veldlyne vorm konsentriese sirkels met die draad by hul middelpunte. Die magneetveldvektor B is n raaklyn aan die magneetveldlyne. B se rigting kan m.b.v die regterhandreël vir n reguit geleier bepaal word. Hou die geleier in jou regterhand sodat die regterduim in die rigting wat die konvensionele stroom vloei, wys. Dan krul jou vingers in die rigting van die magneetveld, B (29-3)
φ Bewys van Vgl. 29-4: Consider the wire element of length ds shown in the figure. The element generates at point P a magnetic field of µ 0i ds sinθ magnitude db =. Vector db is pointing 2 4π r into the page. The magnetic field generated by the whole wire is found by integration: 0 B = db = 2 db = 0 0 µ i ds sinθ 2π 2 r r = s + R sinθ = sin φ = R/ r = R/ s + R µ 0i Rds µ 0i s µ 0i B = 2 2 3/2 2π = = 2 R 2 2 s + R π s + R 2πR 2 2 2 2 dx ( 2 2 x + a ) 3/2 ( ) 0 0 = x a x + a 2 2 2 B = µ 0 i 2π R 1 db = µ 0i ds r 3 4π r (29-4)
Die magnetiese veld by P a.g.v slegs die onderste of die boonste helfde van die oneindige lang draad in fig.29-5 is die helfde van hierdie waarde, dus B = µ iφ 0 4π R
Krag tussen Twee Parallelle Strome Fig. 29-10 Twee parallelle drade wat stroom in dieselfde rigting dra, trek mekaar aan. B a is die magneetveld by draad b a.g.v. die stroom in draad a. F ba is die krag uitgeoefen op draad b omdat dit n stroom dra in magneetveld B a. M.b.v die regterhandreël vir reguit geleiers is die rigting van B a by draad b afwaarts.
Die rigting van F ab kan met die regterhandreël v n kruisproduk bepaal word as direk na draad a. Net so is F direk na draad b. As die magneetveld B a bekend is, kan die krag F ab wat dit op draad b veroorsaak bepaal word m.b.v vgl. 28-26 met L, die lengtevektor v/d draad, selfde rigting as konv. stroom in die draad. In fig. 29-10, is vektore L & B loodreg op mekaar, dus
Om die krag op een stroomdraende geleier te vind as gevolg van n tweede stroomdraende geleier, word die veld as gevolg van die tweede geleier by die eerste gleier gevind. Daarna word die krag op die eerste geleier deur daardie veld bepaal. Die metode kan dan ook verder gebruik word om die krag op draad a as gevolg van die stroom in draad b te bepaal. Daar word gevind dat die krag reguit na draad b is; Twee drade met parallelle strome in dieselfde rigting, trek mekaar aan. Netso, indien die strome in teenoorgestelde rigtings is (sogenaand antiparallel), stoot die drade mekaar af. Die krag tussen strome in parallelle drade was die basis vir die definiese van die ampere (NB Sien bladsy 693 in die handboek).
Definisie van n ampere:
B ds =µ i 0 enc Ampere's Law The law of Biot-Savart combined with the principle of superposition can be used to determine B if we know the distribution of currents. In situations that have high symmetry we can use Ampere's law instead, because it is simpler to apply. Ampere's law can be derived from the law of Biot-Savart, with which it is mathematically equivalent. Ampere's law is more suitable for advanced formulations of electromagnetism. It can be expressed as follows: The line integral B ds of the magnetic field B along any closed path is equal to the total current enclosed inside the path multiplied by µ. The closed path used is known as an " Amperian loop. " In its present form Ampere's law is not complete. A missing term was added by Clark Maxwell. The complete form of Ampere's law will be discussed in Chapter 32. (29-7) 0
Magnetic Field Outside a Long Straight Wire We already have seen that the magnetic field lines of the magnetic field generated by a long straight wire that carries a current circles, i have the form of which are concentric with the wire. We choose an Amperian loop that reflects the cylindrical symmetry of the problem. The loop is also a circle of radius r that has its center on the wire. The magnetic field is tangent to the loop and has a constant magnitude B : B ds = Bds cos 0 = B ds = 2π rb = µ i = µ i B = Not e : 0 enc µ 0i 2π r Ampere's law holds true for any closed path. We choose to use the path that makes thecalculation of B as easy as possible. (29-9) 0
The Solenoid The solenoid is a long, tightly wound helical wire coil in which the coil length is much larger than the coil diameter. Viewing the solenoid as a collection of single circular loops, one can see that the magnetic field inside is approximately uniform. The magnetic field inside the solenoid is parallel to the solenoid axis. The sense of B can be determined using the right-hand rule. We curl the fingers of the right hand along the direction of the current in the coil windings. The thumb of the right hand points along B. The magnetic field outside the solenoid is much weaker and can be taken to be approximately zero. (29-11)
B = 4π x 10-7 x 0.29 x 200 = 7.29 x 10-5 T