Magnetic Field of a Wie Fundamental Laws fo Calculating B-field Biot-Savat Law (long method, but woks always) Ampee s Law (high symmety) B-Field of a Staight Wie Fo a thin staight conducto caying cuent the magnetic field is: We epect B fom wie to be popotional to /R. Chapte 28 1
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B-Field of a Staight Wie Whee B 2 Magnetic Field of a Wie 7. 4 1 Tm A is the Pemeability of fee space Use the ight hand ule to detemine the diection. (Place thumb in diection of cuent and B Field is in diection of finges gabbing the wie.) dl 3
Ampee s Law dl 4
Today Eample B-field Calculations (Application of Ampee s Law) nside a Long Staight Wie nfinite Cuent Sheet Solenoid Tooid Biot-Savat Law Cicula Loop 5
Ampee s Law A) Evaluate the integal dl aound each loop R dl 1) 3) 2R 2) R 2 4) not enough infomation a b 1) 3) ab 2) 2(a + b) 4) Not enough info dl 6
Ampee s Law - Eamples Two identical loops ae placed in poimity to two identical cuent caying wies. 3) Fo which loop is B dl the geatest? 1) A 2) B 3) Same A B 7
Two identical loops ae placed in poimity to two identical cuent caying wies. C B D) Now compae loops B and C. Fo which loop is B dl the geatest? 1) B 2) C 3) Same 8
Suppose a total cuent flows though the wie of adius a into the sceen as shown. B Field nside a Long Wie To calculate B field as a function of, fom cente of the wie: Take an ampeian loop of adius outside the wie, using Ampee s Law: Bdl The enclosed cuent is all of cuent though wie: The B-field diminishes as 1/ outside the wie Chapte 28 enc B d B 2 enc R B 2 9
Now B-field inside the wie: We choose an ampeean loop of adius inside the wie: B dl B Field nside a Long Wie B d 2 B enc But the enclosed cuent is a faction of total; since cuent is unifom: enc R 2 2 nside the wie the B-field is linea with. enc theefoe B 2 R 2 R Chapte 28 1
nside the wie: ( < R) μ B 2 π R Outside the wie:( >R ) 2 B Field nside a Long Wie y = B b1();b2() 1 B = 2 R 4 = 11
B Field of Cuent Sheet Conside an sheet of cuent descibed by n wies/length each caying cuent into the sceen as shown. Calculate the B field. What is the diection of the field? Fom the Symmety +/ y diection Calculate using Ampee's law fo a squae of side w B dl Bw Bw 2Bw nwi theefoe, B dl μ B μ ni 2 w constant y constant 12
B-Field of A Solenoid A unifom magnetic field can be poduced by a solenoid A solenoid is defined by a cuent flowing though a wie that is wapped n tuns pe unit length on a cylinde of adius a and length L. Bdl Bl N N B n l dl l 13
B-Field of A Tooid Tooid defined by a solenoid of N total tuns with cuent connected at both ends. t becomes a donut shape with a coil wapped aound it. outside Tooid B = integating B on cicle outside tooid, enclosed cuent = +(- ) = nside the Tooid Bdl B2 N B 2 N Get a concentated field inside the tooid. dl 14
Eamples Two cylindical conductos, one solid and the othe hallow in middle, each cay cuent into the sceen as shown. The conducto has adius R=4a. The conducto on the ight has a hole in the middle and caies cuent only between R=a and R=4a. a 4a 4a At R = 5a which conducto poduces stonge B-field? 1) Left conducto 3) Both ae the same 2) Right Conducto 4) both ae zeo 15
Eamples Two cylindical conductos, one solid and the othe hallow in middle, each cay cuent into the sceen as shown. The conducto has adius R=4a. The conducto on the ight has a hole in the middle and caies cuent only between R=a and R=4a. a 4a 4a At R = 2a which conducto poduces stonge B-field? 1) Left conducto 3) Both ae the same 2) Right Conducto 4) both ae zeo 16
Eamples Use Ampee s Law in both cases by dawing a loop in the plane of the sceen at R = 5a and R = 2a. Both fields have cylindical symmety, so the fields ae tangent to the loop at all points, thus the field at R=5a only depends on cuent enclosed enc = in both cases Field only depends on enclosed cuent 4a a 4a 17
Eamples A cuent caying wie is wapped aound an ion coe, foming an electo-magnet. Which diection does the magnetic field point inside the ion coe? 1) left 4) ight 2) up 5) down 3) out of the sceen 6) into the sceen Which side of the solenoid should be labeled as the magnetic noth pole? 1) left 3) ight 2) up 4) down 18
nside the wie: ( < R) μ B 2 π R 2 B Field nside a Long Wie y = 1 R b1();b2() Outside the wie:( >R ) B = 2 4 = 19
B Field of Cuent Sheet Conside an sheet of cuent descibed by n wies/length each caying cuent into the sceen as shown. Calculate the B field. What is the diection of the field? Fom the Symmety +/ y diection Calculate using Ampee's law fo a squae of side w B dl Bw Bw 2Bw nwi theefoe, B dl μ B μ ni 2 w constant y constant Chapte 28 2