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28_Lec-15 - Physics 241 Lecture 15 Y E Kim Chapter 28...

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Physics 241 Lecture 15 Y. E. Kim October 19, 2010 Chapter 28, Sections 3, 4, 6 October 18, 2010 University Physics, Chapter 25 1
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October 18, 2010 University Physics, Chapter 28 2 $PSHUH·V /DZ LV Where the integral is carried out around an Amperian loop and i enc is the current enclosed by the loop As an example of $PSHUH·V /DZ , consider the five currents shown below The currents are perpendicular to the plane shown An Amperian loop is represented by the red line 0 enc B d s i P ³ ± ² 0 1 2 3 B d s i i i P ´ ´ ³ 0 3 4 id s r dB r P S u Biot-Savart Law $PSHUH¾V /DZ ±²³ ± ² 1 2 3 e n c i i i i ´ ´
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October 18, 2010 University Physics, Chapter 28 3 $PSHUH·V /DZ ±²³ This loop encloses i 1 , i 2 , and i 3 and excludes i 4 and i 5 A direction of integration is shown above along with the resulting magnetic field The sign of the contributing currents can be determined using a right hand rule by pointing your fingers along the direction of integration Your thumb will indicate the positive direction of the contributing currents ± ² e n c i 0 enc B d s i P ³ ± ² e n c i ± ² e n c i d s
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October 18, 2010 University Physics, Chapter 28 4 A Magnetic Field inside Long Wire (1) Consider a current i flowing out of the page in a wire with a circular cross section of radius R This current i is uniformly distributed over the cross sectional area of the wire To find the magnetic field , we use an Amperian loop with radius r A represented by the red circle The magnetic field is tangential to this Amperian loop so ZH FDQ ZULWH WKH OHIW VLGH RI $PSHUH·V /DZ as 2 B d s B d s B r S A ³ ³ 0 enc B d s i P ³ B B
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October 18, 2010 University Physics, Chapter 28 5 Magnetic Field inside Long Wire (2) 7KH ULJKW KDQG VLGH RI $PSHUH·V /DZ FRQWDLQV WKH HQFORVHG current which can be calculated from the ratio of the area of the Amperian loop to the cross sectional area of the wire Equating the left and right sides using $PSHUH·V ODZ we get or and i e n c i A loop A wir e i S r A 2 S R 2 B 2 S r A P 0 i r A 2 R 2 0 2 ( ) , 2 i B r r r R R P S A A A § · d ¨ ¸ © ¹ A e n c i 0 enc B d s i P ³ 0 ( ) , 2 i B r r R r P S A A A § · t ¨ ¸ © ¹
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October 18, 2010 University Physics, Chapter 28 6 Magnetic Field inside Long Wire (3) 0 2 ( ) , 2 i B r r r R R P S A A A § · d ¨ ¸ © ¹ 0 ( ) , 2 i B r r R r P S A A A § · t ¨ ¸ © ¹ 0 enc B d s i P ³
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