Physics 2020 Lectures and Clicker Quizzes Weeks 9-10

1. Tues, March 29, 2011 Physics 2020 Lectures and Clicker Quizzes Weeks 9-10 M. Goldman Spring, 2011

2. Mid term 2 correct histogram

3. Review of how magnetic fields are produced by currents i • A DC-current-carrying wire creates magnetic field lines and vectors at every point in space, even far away from the wire. • Magnetic field lines around a current-carrying wire are an infinite number of closed loops around the wire • Use right-hand-rule #1 to find the direction of magnetic field, B • Magnitude of B a distance R away from a long current-carrying wire • Magnitude B at center of circular current-carrying wire of radius, R • • X X • X X • • X X •

4. Magnetic fields produced by currents (cont'd) • Magnetic field lines of a bar magnet are equivalent to the magnetic field lines of a current loop. • Bar magnets attract or repel each other the same way that current loops attract or repel each other • Magnetic field lines from Earth are equivalent to those of an upside down bar magnet with its south pole pointing up

5. Clearer explanation of why the B-field of one current loop attracts a second parallel loop N Outward component of force is balanced byequal and opposite outward force on otherside. However down-ward currents are sameon both sides F S

6. What is magnitude B of B-field created by a current loop? Simple expression only along symmetry (z) axis far from loop B Z A = πR2 A Valid only for |z| >> R. B-field drops fast with z: as z3.

7. What is the magnetic dipole moment of a current loop? B-field created by current loop along symmetry axis, zwhen |z| >> R B, µ z µ is called the magnetic dipole moment of the current loop R i Magnitude of magnetic dipole moment, µ A = πR2 Direction of µ is direction of B(known from right hand rule #1) Any current loop looks like a Magnetic Dipole far away.

8. Review of how a magnetic field vector, B, exerts a force, F, on a current-carrying wire segment • A magnetic vector B exactly at the location of the current segment, itest exerts a force on itest (unless it is parallel to itest) • Magnetic field vectors created by a short current segment do not contribute to the force on that segment (but magnetic field from other current segments may contribute to the force). • Magnetic fields in regions of space without currents exert no force! (Just as electric fields in regions without charge) Force B θ • itest B L No Force itest L Magnetic field, B, acts at this point

9. Force on test current, itest exerted by magnetic field, B created from unseen currents (cont'd) • Direction of force,F, on current segment itest due toB is given by right hand rule #2 • Magnitude of force, F on current segment itest due to B is given by F B θ • itest L itest |F| =|i| ·|L| ·|B|·sinθ θ = magnitude of angle < 180° between vectors i and B

10. What is mathematical expression of force on current by B using right-hand rule #2? • What is a vector cross-product? • Cross-product of any two vectors, A and B is another vector, C • Definition: C = A x B has direction of right hand rule #2 and magni-tude |C| = |A||B|sinq, q < 180° • Force, F, due to wire segment L with current i:F = L·i x B C A B

11. Clicker question • A current-carrying wire is in a B-field. The wire is oriented to the B-field as shown. What is the direction of the magnetic force on the wire? • Right • B) Down • C) Out of the Page • D) Into the Page • E) None of these. B i

12. Clicker Question A square loop of wire carrying current I is in a uniform magnetic field B. The loop is perpendicular to B (B out of the page). What is the direction of the net force on the wire? B A: out of the page B: into the page C:  D:  E: None of these

13. Clicker Question The same loop is now in a non-uniform field. where A is a constant. The direction of the net force is? Answer one of following B stronger B A y C D x B weaker E: net force is zero

14. Does no net force on the loop mean no net torque? In a uniform B-field, regardless of the orientation between the B and the Magnetic Moment of the loop m, the net force is always zero. However, that does not mean thenet torqueon the loopis zero!

15. Loop has magnetic moment, µ,even though it is rectangular If µ is not parallel to B, then there is a net torqueabout the dashed green axis in diagram below: Direction of torque is green arrow. Magnitude of torque r m i Simple formula fornet torque on loop(i.e., on magnetic moment) Note angle between r and F is same as angle between µ and B

16. Basic principle behind many motors! r m Torque wants to twist loop so that mfrom loop aligns with B from magnet.

17. Can a magnetic field exert a force on a single moving electron? • Currents are just moving electrons. • Direction of current is opposite to direction of electron velocity, vd B A θ i vd Density of electrons, ne Magnitude of F on Ne electrons in wire length L = [i ·L] ·B ·sinθ Magnitude of force, F, onone moving electron= [i ·L/Ne] ·B ·sinθ = -ev ·B ·sinθ

18. How do we use the right hand rule for the magnetic force on an electron moving with v ? Vector force, F, onone electron moving with instantaneous velocity, v, in a magnetic field, B:F= -ev x B Since the effective current is opposite in directionto the electron velocity, v, use current rule for directionof force with four fingers in direction of –v. B B After finding direction of Fon electron, find magnitude, F = |e|·|v|·|B|sinθ θ v i F -v This force is general!! The electron doesn't have to be in a wire!! • • θ -v F

19. B q v General magnetic force on a + or - charge, q, moving with velocity v Magnetic force on negative charge, q = -|q| Magnetic force on positive charge q = +|q| Magnetic force vector eqn for either sign charge Direction: -v or +v Magnitude:

20. B Clicker Question A negative particle and a positive particle are moving in a constant, uniform magnetic field, as shown. The direction of the B-field is to the right. The (+) particle is moving directly left; the (–) particle is moving directly up. The force on the positive particle due to the B-field is (in = into page, out = out of page). A: in B: out C: zero D: right E: left Answer: The (+) particle is moving anti-parallel to the B-field. The angle  is 180 and the force is FB=qvB sin = 0.

21. B Clicker Question A negative particle and a positive particle are moving in a constant, uniform magnetic field, as shown. The direction of the B-field is to the right. The (+) particle is moving directly left; the (–) particle is moving directly up. The force on the negative particle due to the B-field is A: in B: out C: zero D: right E: left Answer: The (–) particle is moving at right angles to the field. By the right-hand rule for - particles, the direction "-v cross B" is out of the page so the force is out of the page.

22. What is the force on an electron with velocity, v, in a uniform magnetic field, B pointing out of page? • What is resulting motion of electron? Circular motion around B-field line: F = ma = mv2/R = evB centripetal force R = mv/eB = electron cyclotron radius speed x time = vT = 2πR (circumference) T = (2πR)/v = period (time to go around) f = 1/T = v/(2πR) = eB/(2πm) = electron cyclotron frequency • • • F B • • • v • • •

23. What about positively charged particles (e.g., ions in van Allen belt) • Velocity, V, of + charge is in same direction as current, so right hand rule #2 for force, F, due to B on +charge is as below. After finding direction of Fon +charge, find magnitude, F, using F = +eVBsinθ • • • F • • V • • • V Circular motion in uniform B is reversed Cyclotron frequency and radius now con-tain the mass, M of the +charge: • B f = eB/(2πM) R = MV/eB

24. Temperatures are high enough in sun and Earth's magnetic environment to make a plasma • A plasma is an ionized gas. • It consists of both light electrons and heavy ions. • The temperature is so high that electrons are free — not bound to nuclei • Don't need wire or even conductor to guide moving charges. • Unlike in a conductor, both ions and electrons are mobile. • Both electrons and ions feel magnetic forces and are guided by magnetic field lines • Most of universe is composed of this kind of plasma

25. Magnetic field lines • Magnetic field lines are tangent to magnetic vector fields, just as electric field lines are tangent to electric vector fields • Charged particles can move in spirals around magnetic field lines (due to magnetic force on those particles). • Tremendous energy can be released in space! Field vectors

26. Why do particles spiral rather than just go in circles • No magnetic force on charged particle with v purely along lines of B because q = 0 in RH rule for force. • Charged particles can move along lines of B with constant speed (Newton's first law) while they are gyrating in a circle • Velocity of electron or proton is superposition of constant speed along line of B with motion in circle with different constant speed, resulting in spiral! • • • F • • • • • vperp to B B • • vparallel to B

27. Another visualization of charged particles spiraling around magnetic field lines

28. What are electron and ion cyclotron radii as they spiral around Earth's B-field For particle of mass, m and charge magnitude |e| R = mv/|e|B =electron cyclotron radius Estimate v as RMS thermal velocity Te = 1 keV, Ti = 5 keV, B = 30 nT Re = 2.5 kmRi = 240 km

29. Spiraling particles can make lines of B visible • Sun's magnetic field lines • Field lines bubble up from subsurface. Solar flares. • See them because associated spiraling charged particles collide and radiate light • Field lines twist and reconnect, sending magnetic and particle energy towards Earth.

31. C B A E B(out) D: it will not move Clicker Question A positive particle is released from rest in a region of space where there is constant, uniform, electric field and a constant, uniform magnetic field. The electric field points up and the magnetic field points out of the page in the diagram at right. Which path will the positive particle follow? (All paths shown are in plane of the page.) . Answer: The (+) particle will feel a force FE = qE due to the E-field along the direction of the E-field. As it starts moving along the E-field direction, it will acquire a velocity, and it will start to feel a force FB=qvB, due to the B-field. The direction of the force is to the right, by the right-hand-rule.

32. Cathode ray tube uses force of magnetic field to aim and deflect an electron stream hitting a phosphorous screen http://www.colorado.edu/physics/2000/applets/tubeB.html

33. Go back and look at magnetic materials in whichthe "current loops" are microscopic Atoms have moving electric charges.

34. What is the effective current,i,of a single electron moving in a circular orbit around the nucleus "a" = any small x-section area of "tube" around electron orbit = 1 What is the B-field produced by this current at the center of the electron's orbit (at the nucleus)? Electron in orbit, but current in opposite direction.

35. In most materials all the electrons in orbit have random orientations. Superposition of B-field vectors arising frommany atoms gives B=0

36. . In ferromagnetic materials* the atomic currents can all orient the same way (domains), making a net B-field B B B B *Fe, Ni, Co, and some alloys containing these metals too

37. Sometimes material is fragmented into many domains (top) (Net B = 0). If domains align (bottom) there is a net B (magnetized).

38. Atoms have a Magnetic Dipole Moments and associated B-fields due to the orbit of electrons. But, amazingly electrons themselves also have an intrinsic Magnetic Dipole Moment which has nothing to do with their orbit around the nucleus

39. Atoms in permanent magnets have many electrons in orbits with different orientations. Thus, it is often the intrinsic magnetic dipole moment of the electrons that is important and tends to be aligned into domains. i

40. Why do bar magnets stick to many fridges and attract some metal objects?

41. Remember back to Induced Electric Dipoles…. Wooden 2x4 Demonstration Induced charge always produces an attractive force! The opposite charge is always closest, thus resulting in attraction. + + + + - - - - - - - - + + - -

42. Now think about induced Magnetic Dipole Moment Alignment Randomly Oriented Magnetic Dipoles m Magnetic Dipoles tend to align with external B-field due to torque on them. Once dipoles line up material has temporary net B of its own and is attracted to bar magnet

43. Just like two loops of wire with current going in the same direction – attractive force! i i

44. What if we flip the magnet around? i i Then induced magnetic dipoles in metal are rotated the opposite way. Both current loops are opposite to before, thus still attractive!

45. Many fridges acquire a temporary dipole moment on the door when a bar magnetic is brought near, but not all Some materials, like stainless steel, do not have very large induced Magnetic Dipole Moments. Thus, there is not enough attractive force to hold the magnet to the fridge  To save on cost, the sides of such fridges are not stainless, and thus support magnets 

46. Can charges in nuclei of atoms also have spins that respond to external magnetic field?

47. Review of time-independent magnetism • Time-independent (source) currents, i, produce time-independent magnetic fields. Right-hand rule #1: put thumb in direction of current. Four other fingers curl in direction of B. • Time independent magnetic fields, B, exert constant force, F, on other (test) currents, F = Li x B Right hand rule #2 gives F-direction • Charges moving with constant velocity produce magnetic fields (e.g., magnetic moment of electron orbit in atom) • Time-independent magnetic fields exert constant force on (test) charges. F = qv x B (for electrons q = -e. Use right-hand rule #2 with –v x B giving direction of force. Electrons, ions spiral around B • Next: Time-dependent processes - • Above processes also work when currents. magnetic fields, velocities etc are time-varying • Faraday's Law only works when flux is time-varying

48. Next: Faraday's law, a new fundamental law of electricity and magnetism B E • Review: moving charges, currents create closed magnetic field line loops (RH rule #1). Ampere's law • Earlier: electric charges produce electrostatic electric field lines which begin or end on charge. Coulomb's law • Next: new kind of electric field line can be produced — closed loop of Erather than E lines beginning or ending on charge. • Created by time-varying magnetic fields! • Faraday's law • Includes this effect and more -DB/Dt i E

49. Can Faraday's Law be written in a way that does not involve vectors? Faraday’s Law of Induction • Two new quantities need to be explained • . • Magnetic Flux = FB • Electro Motive Force (EMF) = e

50. ^ n = direction perpendicular (normal) to surface Areas and normals A Imagine a flat surface of area A [m2] • Vectors can lie entirely in the surface. • Other vectors can have components both parallel and perpendicular to the surface. • Still other vectors can be entirely perpendicular to any line parallel to the surface. • One such perpendicular vector is called the normal to the surface. • It has no units and magnitude 1. It is just a direction associated with the surface. A (Convention: Boldface letters represent vectors)