Ch 30 - Sources of Magnetic Field! The Biot-Savart Law! = k m. r 2. Example 1! Example 2!

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1 Ch 30 - Souces of Magnetic Field 1.) Example 1 Detemine the magnitude and diection of the magnetic field at the point O in the diagam. (Cuent flows fom top to bottom, adius of cuvatue.) Fo staight segments, 0. Fo cuved segment,, because s and ae. B µ o 4# 2 B µ o 4# 2 d% B µ o % 4# 3.) The Biot-Savat Law magnetic field at a point P pependicula to segment of wie & pependicula to unit vecto fom to P. k m 10-7 T m/a k m µ o /4π, whee d B d s ˆ k m µ o pemeability of fee space µ o 4π x10-7 steady cuent in diection 2 db out ˆ P element of wie detemines diection to point Example 2 Detemine the diection and total magnitude of the magnetic field at the point P shown hee, nea a long, thin, wie. The magnetic field fo an infinitely long staight wie is B µ o 2a a P 2.) 4.)

2 Magnetic Foce between wies F B Bsin B (due to wie) F 1 µ o 2 ' & ), o % 2#a ( F µ o 1 2 2#a µ o 2#a OR, if you find that the diection of the magnetic field lines in between the two souces ae paallel to one anothe, it means the souces will epulse, wheeas if the diection of the magnetic field lines between the two souces ae anit-paallel to one anothe, it means the souces will attact one anothe. Ampèe s Law f we evaluate the dot-poduct # in a cicle aound this wie, and sum this poduct ove the entie path of the cicle, we get B d # s B µ & o % (( 2) µ 2' o Line integal, taken aound a closed path that suoun the cuent. B d s This esult is valid fo any closed path that encloses a wie conducting a steady cuent, and is known as Ampèe s Law: B d s µ o B 5.) 7.) Oested (1820) f the wie is gasped with the ight hand, with the thumb in the diection of cuent flow, the finges cul aound the wie in the diection of the magnetic field. The magnitude of B is the same eveywhee on a cicula path pependicula to the wie and centeed on it. Expeiments eveal that B is popotional to, and invesely popotional to the distance fom the wie. Gauss & Ampèe E d A q in o B d s µ o thu 6.) 8.)

3 Example 3 Use Ampèe s Law to calculate the magnetic field at a distance away fom the cental axis of a cuent-caying wie of coss-sectional aea R and cuent. Do this fo: i.) >R R Example 3 i.) <R The cuent though the face of the Ampeian path will be the faction of cuent though the whole face. That is: So Ampee s Law yiel: i thu 2 % # R 2 & ' 2 R 2 B µ o i thu Ampeian path R ii.) <R B µ o i thu # 2 B(2) µ o R & % 2 ' ( ) B µ o 2R 2 9.) 11.) Example 3 i.) >R B µ o i thu B µ o i thu B(2) µ o B µ o 2 Ampeian path R Example 4 Use Ampèe s Law to calculate to calculate the magnetic field inside a tooid of N loops, at a distance fom the cente. side view: edge view: 10.) 12.)

4 As the wie passes though that Ampeian path N times, the cuent though the face will be Ni. With that, we can wite: side view: Example 5 Find the magnetic field B fo an thin, infinite sheet of cuent, caying cuent of linea density J in the z diection. B µ o B µ o B(2) µ o N B µ o N 2 edge view: imaginay Ampeian path B B µ o µ o (J) B(2) µ o (J) B µ oj 2 L w 13.) 15.) Note that if the path had been outside the tooid, thee would have been as much cuent moving inwad though the face as outwad though the face. n othe wo, the net cuent though the face would have been zeo and the B field outside zeo. cuent into page: cuent out of page imaginay Ampeian path Electomagnets & Solenoi A solenoid is a long wie wound in the fom of a helix. Typically, you ae given the numbe of win pe unit length, o n. f the tuns ae closely space and the solenoid is finite in length, the field lines fom a solenoid look vey simila to those of a ba magnet. 14.) 16.)

5 B Field fo a Solenoid nside stong B field Outside weak B field Between coils 0 field B µ o i thu ( ) B µ o nl BL µ o ( nl) B µ o n This is the magnetic field function fo the magnetic field down the cental axis of a vey long coil. 17.) 19.) Classically, the Ampeian path fo a vey long coil, used to detemine the effective magnetic field in the cental egion of the coil, is as shown to the ight. Below is a side view. Note that the net magnetic field components pependicula to the coil s axis ae zeo, and the magnetic field stength fa away fom the coil is also appoximately zeo, so the only ciculation that occus is down the axis of the coil. With the path length of L and the amount of cuent passing though the Ampeian face being (nl)i, whee n is the numbe of win-peunit-length and nl identifies how many wies thee ae inside the Ampeian suface, we can wite: L fom side L Electic Flux Bief eview of Electic Flux though an open suface is: E A EAcos The net electic flux though any closed suface is equal to the net chage inside the suface divided by. As such, we can wite: o c E d A q in # o # 18.) 20.)

6 Magnetic Flux By the same token, though an open suface we can also wite: B B A B BAcos The net magnetic flux though any closed suface is equal to zeo as thee ae no magnetic monopoles (i.e., situations in which you can find a Noth Pole that isn t attached to a coesponding South Pole. That is: B B d A 0 # B Example 7 Find the total magnetic flux though the loop shown hee. B B da µ o ' % & 2#x( ) ( b dx) µ a+c ob 1 2# x dx c µ ob 2# ln a + c ' % & c ( ) Note that being able to do magnetic flux deivations will be needed in the next chapte on Faaday s Law whee we will be detemine the time ate of change of magnetic flux. c x a dx b 21.) 23.) Example 7 Find the total magnetic flux though the loop shown hee. c a b Gauss s Laws Electic field intensity depen only on the net intenal chage. t is possible to have a single chage whose field lines poduce a net electic flux though a closed suface. As such, we can wite: c E d A q in t isn t possible to have a single Noth Pole without a South Pole attached, which means that magnetic field lines ae continuous and fom closed loops (and magnetic field lines ceated by cuents don t begin o end at any point. Conclusion: the net magnetic flux though any closed suface is always zeo, which means: B B d A 0 # o N S 22.) 24.)

7 Poblems w/ampèe s Law B d s µ o d E dt dq ( dt ) o i displacement o Two poblems: 1.) What if cuent is changing? 2.)What if Ampeian path doesn t enclose cuent? open end # i d o d E dt f we include this displacement cuent in Ampee s Law, we take cae of the pesky poblems. Doing so yiel a moe complete vesion of Ampee s Law which is: B d s µ o ( + d ) d# µ o ( + E o ) dt 25.) 27.) Thee is an electic field between the plates, and accoding to Gauss s Law, thee is an electic flux: E E d A q S # o Futhemoe, the electic flux is changing in time. As such, we can wite: d d E d A dq E ( S ) ( dt ) dt dt gnoing the middle expession and noting that dq/dt is cuent (efeed to as the displacement cuent), we can wite: # o 26.)