notes27 2317 - Prof David R Jackson ECE Dept Spring 2011...

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Unformatted text preview: Prof. David R. Jackson ECE Dept. Spring 2011 Notes 27 ECE 2317 ECE 2317 Applied Electricity and Magnetism Applied Electricity and Magnetism Biot-Savart Law Biot-Savart Law ( 29 µ 2 4 V J r R H dV R π ′ × ′ = ∫ ( 29 µ 2 4 s S J r R H dS R π ′ × ′ = ∫ a) Volume current V J R r r ′ b) Surface current S J s R r r ′ Please see the textbooks for a derivation. Biot-Savart Law (cont.) Biot-Savart Law (cont.) µ 2 4 C Id R H R π ′ × = ∫ l Rule: The contour C is in direction of the I arrow (i.e., the reference direction for the current). c) Line current (wire) (This determines the starting point A and the ending point B .) Note: d dr ′ ′ = l I R r C r ′ A B d ′ l Example Example ( 29 ( 29 µ 2 2 2 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ cos sin ˆ ˆ ˆ ˆ d z dz R r r x x y y z z x y z z z z R z z z R z ρ φ ρ φ ρρ ρ ρρ ρ ′ ′ = ′ ′ =- = +- ′ = +- ′ =- ′ = + ′- = ′ + l µ 2 4 C Id R H R π ′ × = ∫ l Infinite line current Find H x y z r R ρ I r ´ d ′ l z = 0 Example (cont.) Example (cont.) ( 29 ( 29 ( 29 $ ( 29 2 2 2 2 3 2 2 2 3 2 2 2 ˆ ˆ ˆ 4 ˆ 4 4 C z z I z dz z H z dz I z I dz z ρρ ρ π ρ ρφ π ρ ρ φ π ρ +∞-∞ +∞-∞ ′- ′ × ′ + = ′ + ′ = ′ + ′ = ′ + ∫ ∫ ∫ µ 2 4 C Id R H R π ′ × = ∫ l x y z r R ρ I r ´ d ′ l Example (cont.) Example (cont.) ( 29 ( 29 ( 29 ( 29 ( 29 3 2 2 2 3 2 2 2 2 2 2 2 ˆ 4 1 ˆ ˆ ˆ 2 2 2 4 4 4 I dz H z I dz I z I z z ρ φ π ρ ρ ρ ρ φ φ φ π π π ρ ρ ρ ρ +∞-∞ ∞ +∞ ′ = ′ + ′ ′ = = = ′ + ′ + ∫ ∫ [ ] ˆ A/m 2 I H φ πρ = Hence we have ( 29 2 1 ˆ 2 4 I H ρ φ π ρ = Note: This agrees with the answer obtained by Ampere’s law. or ( 29 3/2 2 2 2 2 2 dx x a x a x a = + + ∫ Note: Example Example ( 29 µ ( 29 ( 29 2 2 2 2 3 2 2 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 4 C d a d R z z a R z a z z a R z a a z z a I H d z a φ φ ρ ρ φ ρ φ π ′ ′ ′ = ′ = +- = + ′- = + ′ ′ ×- ′ = + ∫ l Find H x y z a R r = (0 , ,z ) I d ′ l r ´ µ 2 4 C Id R H R π ′ × = ∫ l Ñ Note: This problem cannot be solved using Ampere’s law! Example (cont.) Example (cont.) ( 29 [ ] 2 3 2 2 2 ˆ A/m 2...
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notes27 2317 - Prof David R Jackson ECE Dept Spring 2011...

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