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Exam 2 covers Ch. 27-33, Lecture, Discussion, HW, Lab Exam 2 is Tue. Oct. 27, 5:30-7 pm, 145 Birge Chapter 27: The Electric Field Chapter 29: Electric potential & work Chapter 30: Electric potential & field Chapter 31: Current & Resistance Chapter 32: Fundamentals of Circuits (exclude 30.7) (exclude 32.8) Chapter 33: The Magnetic Field (exclude 33.5-33.6, 33.9-10, & Hall effect) 1 Law of Biot-Savart B out of page r ds Each short length of current produces contribution to magnetic field. I in plane of page dB r ds r r o Ids rˆ dB 4 r 2 Field from very short r section of current ds r = distance from current element o 4 107 N / A 2 = permeability of free space Tue. Oct. 27, 2009 Physics 208 Lecture 16 2 r r CD Vector cross product r r o Ids rˆ dB 4 r 2 r D r dB r oI r dB ds rˆ 2 4 r Dist. to point at which field is evaluated Short length of current r r Unit vector toward point at which field is evaluated r C r ds Tue. Oct. 27, 2009 Physics 208 Lecture 16 3 Field from a circular loop Each current element produce dB All contributions add as vectors Along axis, all components cancel except for x-comp Tue. Oct. 27, 2009 Physics 208 Lecture 16 4 Magnetic field from loop of current Looks like magnetic dipole Tue. Oct. 27, 2009 Physics 208 Lecture 16 5 Building a solenoid Tue. Oct. 27, 2009 Physics 208 Lecture 16 6 Solenoid: many current loops Bsolenoid Tue. Oct. 27, 2009 oNI L onI Physics 208 Lecture 16 7 Magnetic Force on a Current I Force on each charge qv B Force on length ds of wire Ids B section of Force on straight wire, length L r r r F I BL N Current S Magnetic force Magnetic field Tue. Oct. 27, 2009 Physics 208 Lecture 16 8 Quick Quiz A current I flows in a square loop of wire y with side length L. A constant B field points in the x-direction, perpendicular to the plane of the loop. What is the net force on the wire loop? L I I I A. 4LB I B. 2LB x C. LB D. 0 Tue. Oct. 27, 2009 Physics 208 Lecture 16 9 No force, but torque Lever arm Force r r Torque is r F Net torque can be nonzero even when net force is zero. Tue. Oct. 27, 2009 Physics 208 Lecture 16 10 Question on torque Which of these loop orientations has the largest magnitude torque? Loops are identical apart from orientation. (A) a (B) b (C) c a b c 12/09/2002 U. Wisconsin, Physics 208, Fall 2006 11 Quick Quiz Which of these different sized current loops has the greatest torque from a uniform magnetic field to the right? All have same current. A. L B. 2L W W/2 C. L/2 D. All same r B 2W Tue. Oct. 27, 2009 Physics 208 Lecture 16 12 Torque on current loop r F F 2 F sin 2 I F IB AIBsin A 2 B r =loop area I Torque proportional to • Loop area • Current F B • sinθ Tue. Oct. 27, 2009 Physics 208 Lecture 16 13 Current loops & magnetic dipoles Current loop produces magnetic dipole field. Magnetic dipole moment: IA current Area of loop direction magnitude In a uniform magnetic field Magnetic field exerts torque B, B sin Torque rotates loop to align with B Tue. Oct. 27, 2009 Physics 208 Lecture 16 14 Works for any shape planar loop r IA r perpendicular to loop Torque in uniform magnetic field I B, B sin r r Potential energy of rotation: U B Bcos Lowest energy aligned w/ magnetic field Highest energy perpendicular to magnetic field Tue. Oct. 27, 2009 Physics 208 Lecture 16 15 Magnetic flux Magnetic flux is defined B B dA exactly as electric flux (Component of B surface) x (Area element) zero flux Maximum flux SI unit of magnetic flux is the Weber ( = 1 T-m2 ) Tue. Oct. 27, 2009 Physics 208 Lecture 16 16 Magnetic Flux Magnetic flux through a surface: (component of B-field surface) X (surface area) Proportional to # B- field lines penetrating surface Tue. Oct. 27, 2009 Physics 208 Lecture 16 17 Why perpendicular component? Suppose surface make angle surface normal r B B|| sˆ B nˆ A Anˆ nˆ Component || surface Component surface Only component ‘goes through’ surface sˆ B = BA cos B =0 if B parallel A B = BA (max) if B A Flux SI units are T·m2=Weber Tue. Oct. 27, 2009 Physics 208 Lecture 16 r r M B A 18 Total flux E not constant add up small areas where it is constant Surface not flat add up small areas where it is ~ flat r r BiA i cos Bi Ai i B r r Add them all up: B B dA surface Tue. Oct. 27, 2009 Physics 208 Lecture 16 19 Magnetic flux What is that magnetic flux through this surface? A. Positive B. Negative C. Zero Tue. Oct. 27, 2009 Physics 208 Lecture 16 20 Properties of flux lines Net magnetic flux through any closed surface is always zero: magnetic 0 For electric charges, and electric flux electric Qenclosed o No magnetic ‘charge’, so right-hand side=0 for mag. Tue. Oct. 27, 2009 Basic magnetic element is the dipole Physics 208 Lecture 16 21 Time-dependent fields Up to this point, have discussed only magnetic and electric fields constant in time. E-fields arise from charges B-fields arise from moving charges (currents) Faraday’s discovery Another source of electric field Time-varying magnetic field creates electric field Tue. Oct. 27, 2009 Physics 208 Lecture 16 22 Measuring the induced field A changing magnetic flux produces an EMF around the closed path. How to measure this? Use a real loop of wire for the closed path. The EMF corresponds to a current flow: IR Tue. Oct. 27, 2009 Physics 208 Lecture 16 23