1 Exam 2 covers Ch , Lecture, Discussion, HW, Lab Chapter 27: The Electric Field Chapter 29: Electric potential & work Chapter 30: Electric potential & field (exclude 30.7) Chapter 31: Current & Resistance Chapter 32: Fundamentals of Circuits (exclude 32.8) Chapter 33: The Magnetic Field (exclude , , & Hall effect) Exam 2 is Tue. Oct. 27, 5:30-7 pm, 145 Birge

Tue. Oct. 27, 2009Physics 208 Lecture 162 Law of Biot-Savart Each short length of current produces contribution to magnetic field. r  I in plane of page dsds B out of page dsds dBdB r = permeability of free space r = distance from current element Field from very short section of current

Vector cross product Tue. Oct. 27, 2009Physics 208 Lecture 163 Short length of current Unit vector toward point at which field is evaluated Dist. to point at which field is evaluated

Tue. Oct. 27, 2009Physics 208 Lecture 164 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, 2009Physics 208 Lecture 165 Magnetic field from loop of current Looks like magnetic dipole

Tue. Oct. 27, 2009Physics 208 Lecture 166 Building a solenoid

Tue. Oct. 27, 2009Physics 208 Lecture 167 Solenoid: many current loops

Tue. Oct. 27, 2009Physics 208 Lecture 168 Magnetic Force on a Current S N I Current Magnetic field Magnetic force Force on each charge Force on length of wire Force on straight section of wire, length L

Tue. Oct. 27, 2009Physics 208 Lecture 169 Quick Quiz A current I flows in a square loop of wire 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? x y I I I I L A.4LB B.2LB C.LB D.0

No force, but torque Torque is Net torque can be nonzero even when net force is zero. Tue. Oct. 27, 2009Physics 208 Lecture 1610 Lever arm Force

12/09/2002U. Wisconsin, Physics 208, Fall Which of these loop orientations has the largest magnitude torque? Loops are identical apart from orientation. (A) a (B) b (C) c Question on torque a b c

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. Tue. Oct. 27, 2009Physics 208 Lecture 1612 L W L/2 2W 2L W/2 A. B. C. D. All same

Tue. Oct. 27, 2009Physics 208 Lecture 1613 Torque on current loop =loop area B F B I  I F Torque proportional to Loop area Current sin θ

Tue. Oct. 27, 2009Physics 208 Lecture 1614 Current loops & magnetic dipoles Current loop produces magnetic dipole field. Magnetic dipole moment: current Area of loop magnitude direction In a uniform magnetic field Magnetic field exerts torque Torque rotates loop to align with

Works for any shape planar loop Tue. Oct. 27, 2009Physics 208 Lecture 1615 I perpendicular to loop Torque in uniform magnetic field Potential energy of rotation: Lowest energy aligned w/ magnetic field Highest energy perpendicular to magnetic field

Tue. Oct. 27, 2009Physics 208 Lecture 1616 Magnetic flux Magnetic flux is defined 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-m 2 )

Tue. Oct. 27, 2009Physics 208 Lecture 1617 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, 2009Physics 208 Lecture 1618 Why perpendicular component? Suppose surface make angle  surface normal  B = BA cos   B =0 if B parallel A  B = BA (max) if B  A Flux SI units are T·m 2 =Weber Component || surface Component  surface Only  component ‘goes through’ surface

Tue. Oct. 27, 2009Physics 208 Lecture 1619 Total flux E not constant add up small areas where it is constant Surface not flat add up small areas where it is ~ flat Add them all up:

Tue. Oct. 27, 2009Physics 208 Lecture 1620 Magnetic flux What is that magnetic flux through this surface? A.Positive B.Negative C.Zero

Tue. Oct. 27, 2009Physics 208 Lecture 1621 Properties of flux lines Net magnetic flux through any closed surface is always zero: No magnetic ‘charge’, so right-hand side=0 for mag. Basic magnetic element is the dipole For electric charges, and electric flux

Tue. Oct. 27, 2009Physics 208 Lecture 1622 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, 2009Physics 208 Lecture 1623 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: