The Law of Biot-Savart & RHR P θ
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1 The Law of iot-savat & RHR P R dx x Jean-aptiste iot élix Savat Phys 122 Lectue 19 G. Rybka
2 Recall: Potential Enegy of Dipole Wok equied to otate a cuentcaying loop in a magnetic field Potential enegy U = - Wok done by field leads to U τd Þ U = µ x µ. With U = 0 at = 90 (whee toque is maximum)
3 Plotting U = -µ U = +µcos U = 0 U = µcos WHCH ORENTATON HAS THE MOST POTNTAL ENERGY? U()
4 Now some new stuff but fist
5 Right-Hand-Rule Pe-Summay = qv = L τ = = µ d = τ ˆ 0 d s 2 µ 4π t is going to get easie
6 Calculation of Electic ield Two ways to calculate the Electic ield: Coulomb's Law: E = 1 πε q " "ute foce" Gauss' Law ε 0 E ds = q "High symmety" What ae the analogous equations fo the Magnetic ield?
7 Calculation of Magnetic ield Two ways to calculate the Magnetic ield: iot-savat Law: d = µ 0 dl 4π 3 "ute foce" Ampee's Law dl = µ 0 "High symmety" These ae the analogous equations fo the Magnetic ield
8 iot-savat Law bits and pieces dl Diffeential magnetic field fom a diffeential cuent element d = X d µ 4π 0 dl ˆ 2 µ = 4π Tm/A pemeability of fee space, ield hee? Apply RHR fom above ield points into the page ntegate along the whole wie So, the magnetic field ciculates aound the wie
9 Magnetic ield of Staight Wie Calculate field at point P using iot-savat Law: µ dx d = 0 4π 3 Which way is? à RHR + µ dx = d = 0 ( ) sin 4π 3 Rewite in tems of R,: dx P is out of page R x R = sin Þ tan = R x Þ x = Rcot dx = R 1 d 2 sin Þ dx(sin ) 2 = sind R
10 Magnetic ield of Staight Wie π 0 d = µ 0 4 sin π R dx P R is out of page x π µ = 0 4πR 0 sind Þ = µ 0 [ ] 4πR cos π 0 = µ 0 2 πr
11 How do we check this esult? i.e. that we expect fom wie to be popotional to /R? Measue ORCE on cuent-caying wie due to the magnetic field PRODUCED by ANOTHER cuent caying wie a d b How does foce depend on cuents and sepaation?
12 oce on 2 Paallel Cuent-Caying Wies Calculate foce on length L of wie b due to field of wie a: The field at b due to a is given by: Calculate foce on length L of wie a due to field of wie b: The field at a due to b is given by: d a = µ π a 0 2 L d b a L d b a Þ oce on b = d L L b a a b b π µ 2 0 = = d b = µ π b 0 2 Þ oce on a = d L L b a b a a π µ 2 0 = =
13 oce etween Cuent-Caying Wies towads us d = 2 L Anothe towads us = µ o L 2πd Conclusion: Cuents in same diection attact towads us d Anothe away fom us Conclusion: Cuents in opposite diection epel
14 A long staight wie is caying cuent fom left to ight. Two identical chages ae moving with equal speed. Compae the magnitude of the foce on chage a moving diectly to the ight, to the magnitude of the foce on chage b moving up and to the ight at the instant shown (i.e. same distance fom the wie). v v (a) (b) A) a > b ) a = b C) a < b Clicke = qv = qvsin Same q, v, and (=90) ut, oces ae in diffeent diections
15 Clicke Two long wies cay opposite cuent x What is the diection of the magnetic field above, and midway between the two wies caying cuent at the point maked X? A) Left ) Right C) Up D) Down E) Zeo x
16 CheckPoint 2 & 4 X What is the diection of the foce on wie 2 due to wie 1? A) Up ) Down C) nto Sceen D) Out of sceen E) Zeo What is the diection of the toque on wie 2 due to wie 1? A) Up ) Down C) nto Sceen D) Out of sceen E) Zeo Unifom foce at evey segment of wie No toque about any axis
17 Clicke A cuent flows in the positive y diection in an infinite wie;; a cuent also flows in the loop as shown in the diagam. What is x, net foce on the loop in the x- diection? y left X top ight (a) x < 0 (b) x = 0 (c) x > 0 bottom x Recall fom a pevious CLCKER that the net foce on a cuent loop in a constant magnetic field was zeo. Howeve, the magnetic field hee is not a constant field The diection of the magnetic field at the cuent loop is in the -z diection. The foces on the top and bottom segments of the loop DO indeed cancel The foces on the left and ight segments of the loop DO NOT cancel The left segment of the loop is in a lage magnetic field. Theefoe, left > ight
18 CheckPoint 7: What is the diection of the net foce on wie 2 due to wie 1? A) Up ) Down C) nto Sceen D) Out of sceen E) Zeo WHY? DRAW PCTURE
19 Conside oce on Symmetic Segments out of sceen into sceen Net oce is Zeo
20 CheckPoint 9: we ae basically guessing What about the toque? same pictue What is the diection of the toque on wie 2 due to wie A) 1? Up ) Down C) nto Sceen D) Out of sceen E) Zeo One Response: do not undestand toque on wies in scenaios such as this one. O eally at all. out of sceen into sceen Anothe: While the magnetic foces on futhe pats of wie 2 (those which ae not diectly pependicula to wie 1) add up to zeo fo tanslational puposes, they point in opposite diections about a single axis of otation (diectly above wie 1), and thus thee will be a net toque on wie 2. The toque will point upwads due to the diection of the foces elative to the distance they ae fom the axis.
21 Right Hand Rule Review 1. ANY CROSS PRODUCT = qv = L τ = = µ d = τ ˆ 0 d s 2 µ 4π 2. Diection of Magnetic Moment inges: Cuent in Loop Thumb: Magnetic Moment 3. Diection of Magnetic ield fom Wie inges: Magnetic ield Thumb: Cuent
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