CHAPTER (6) Biot-Savart law Ampere s Circuital Law Magnetic Field Density Magnetic Flux
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1 CAPTE 6 Biot-Svt w Ampee s Ciuit w Mgneti Fied Densit Mgneti Fu
2 Soues of mgneti fied: - Pemnent mgnet - Fow of uent in ondutos -Time ving of eeti fied induing mgneti fied Cuent onfigutions: - Fiment uent d uent eement A.m - Sufe uent: J s A/ m Cuent eement: J s ds A. m, whee: W J s A/m d dw
3 - Voume uent: Cuent eement dv A m J S n A/ m J., dv d S. d S ds d Biot-Svt w: d â d d? d Whee mgneti fied intensit unit veto fom the uent eement to the point whee we wnt to find distne between the uent eement nd the point p t it Simi: J s.ds â d A / m J dv â d A / m Sufe uent eement Voume uent eement
4 The mgneti fied intensit t point P due to wie of finite ength nd uent pssing though it ong the - is Deivtion d d b d d d α C C o o α o d C C d o o [ C C o [ ] o ] ẑ o d d [ ] o d b d o. [ ]
5 [ ]. b o o d d d [ ] { } b Z o o. [ ] [ ]. b b [ ] [ ]. b b [ ] α α sin sin Note: Fo infinite ine: α α o b α α
6 ? d d,,z o Empe: Find t the ente of iu oop ing uent. Soution: â d d d d o d d d d d. d d d
7 Empe: Find the mgneti fied intensit t the ente of soenoid oi of dius nd ength nd the numbe of N tuns ing uent. Soution: ẑ І N d І Cuent pe unit width J s N d d J s ds J s d d So, Cuent in ength d is J s d N d Sine due to iu oop of uent nd dius is given b So
8 d N d. t the ente of the soenoid is given b N d N N N N N f >> : N Notes: Fo the mgneti fied t the end of the soenoid we must integte fom, so t the end:
9 N N N d N * * / Fo << : m A N / This mens tht the mgneti fied intensit t the end of the soenoid is ppoimte hf its vue t its ente.
10 Ampee s iuit w: t sttes tht the ine integ of the tngenti omponent of mgneti fied intensit ound osed pth is equ to the uent enosed b the pth Empe:. d en Find the mgneti fied intensit t point,, due to infinite wie of uent. Soution: - Ampee s iuit w. d en - Choie of Ampein oop - en -.d A/m Conditions fo ppition of Ampee s w: must be onstnt on the oop.,,
11 Empe: Find inside nd outside onduto mgneti mtei of infinite ength ing uent nd of dius. Soution: egion < - Ampee s iuit w. d en - Choie of Ampein oop - en J. S.. -.d J s І * П *
12 egion > -Ampee s iuit w. d en -Choie of Ampein oop - en J. S. -.d 5-. C 6-
13 The mgneti fied intensit bove nd beow sufe uent distibution of infinite etended sheet with sufe uent densit J s Deivtion -Ampee s iuit w. d en -Choie of Ampein oop pne - en J s * W. d. d. d - d.. d
14 bove the sufe : d â d beow : ŷ ẑ d. d. d. d 5- W W J sw J s 6- Gene : J s n whee J s n sufe uent denist unit veto to the sufe
15 Empe: Find in egions fo the foowing uent onfigution shown in figue. J s J s -5 5 Soution: egion : > egion : t t J J 5 J s ẑ n ŷ ẑ ŷ s s 5 < < n ẑ ŷ 5 n ẑ ŷ 5 5
16 egion : t < 5 ẑ ŷ ẑ ŷ eo 5 5 Empe: Find in egions fo the foowing uent onfigution shown in figue. Soution: egion : s J S > n. d.. en 5 J s d d *.
17 m A s t /. 5 n J S s 5 egion : < <. *. m A s t /. 5 n J S s 5 egion : <. *. m A s t /. 5
18 Empe: Find in egion fo oi be of dii, b, ing in inne onduto nd in oute onduto Soution: egion < b - Ampee s iuit w. d en - Choie of Ampein oop - en J. S.. -.d
19 egion < < b -Ampee s iuit w. d en -Choie of Ampein oop - en J. S. -.d 5-. C 6-
20 egion b < < -Ampee s iuit w. d en -Choie of Ampein oop - en J. S b b b b -.d. 5- b. C b 6- b b
21 egion V > -Ampee s iuit w. d en -Choie of Ampein oop - en. -.d
22 Cu w J - A.m Stok s theoem: but :. d en J. d S. d. d S J. d S Empe:. Find J. Soution: h h h h u u h u h u u h u h u u h u
23 J Mgneti fu densit B B µ µ Mgneti fu m : S d B m. Fo osed sufe:. S B d m
24 Apping divegene theoem B. d S. B eo,. B dv But E. d eo Due to the onsevtive popet of the fied. Apping Stoke's theoem E. d s E. d S E Mwe s equtions in eeto nd mgneto stti fieds Diffeenti Fom J E. D ρ. B. J v ρv t nteg Fom. d E. d D. d S B. d S J. d S Q en Q t
25 Empe: Find the tot mgneti fu tht ossing the e shown in figue. Soution: µ µ B [ ] B ŷ ẑ â d d o µ 6 5
26 d S m m m dd 5 µ 6 o dd µ o 6 5 n µ o [n6 n ] Webe
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