Magnetic Fields and Forces February 11 th , 2013

1. Chapter 20 Magnetic Fields and Forces February 11th, 2013

2. REVISED!!!!!room assignments for exam 1, 2, 3: Sections 302, 307, 309, 314, 323, 324 (TAs Lau & Woods) meet in 145 Birge Sections 305, 310, 312, 317, 321 (TAs Dodd & Lee) meet in 2103 Chamberlin Sections 301, 303, 311, 315, 319, 327, 328 (TAs Aggarwal, Osborne & Pankhuri) meet in 204 Education Sciences Sections 304, 306, 313, 316, 320, 322, 326 (TAs Kim, Patel & Ramesh) meet in B102 Van Vleck

3. Magnetism: outline of this week’s topics • Magnetic fields are produced by moving electric charges • Magnetic fields exert forces on other moving charges • Magnetism and electricity are connected • Connections will be discussed later • Focus of this week: • To understand magnetic fields and their sources • To calculate the magnetic force on a moving charged particle

4. Permanent Magnets • Permanent magnets (e.g. magnetite Fe3O4) are used in various instruments: • Compass needles • Microphones and speakers • Computer hard disks • Electric motors • Generators • Rare-earth magnets are much stronger than magnetite (e. g. neodymium magnets and samarium-cobalt magnets). • Electromagnets (e.g. solenoids) are NOT permanent magnets, but they are very widely used • Superconducting magnets (also NOT permanent) are just for an “elite” of physicists

5. Magnetic Field Lines • The magnetic poles are indicated at the ends of the bar magnet • Called north and south • The symbol for the magnetic field is • The magnetic poles are analogous to positive and negative charges • The north poles of the compass needles are attracted to the south pole of the bar magnet • The SI unit of the magnetic field is the tesla (T) • The magnitude of the field decreases as you move farther from a pole • The magnetic field lines form closed loops • A general property of magnetic fields, not just bar magnets

6. demo: compass arrays near permanent magnet Hexagonal pattern of compass needles Square pattern of compass needles Iron filings in sealed water tank, bar magnet inserted at the center

7. The Earth acts like a large magnet

8. Question: Earth’s Magnetic Field As a bird flies, the distance between the Earth’s magnetic north-pole and its geographic north-poleis about: (R) Negligible (Y) 1000 miles (G) twice radius of Earth (B) π times radius of Earth North magnetic pole is about half way around the Earth (πrE) from the North geographic pole

9. Horseshoe Magnet • Can be made by bending a bar magnet • There are poles at the ends of the horseshoe • The field is largest in the horseshoe gap • The field is directed across the gap, and the field lines are parallel to one another in the gap

10. Magnetic Field from Current • Moving charges produce magnetic fields • An electric current consists of moving charges, so it will produce a magnetic field • The iron filings show the magnetic field pattern due to the current • The direction of the field is given by the right-hand rule

11. demo: current through wire and compass needle I = 45 A VDC = 30 V

12. Right-Hand Rule 1 • Direction of the magnetic field: • Point thumb in the direction of the current, so thumb is parallel to wire • Curl fingers around the wire • fingers give the direction of the magnetic field (knuckle to finger tips) • Use right hand! Does not work with left hand!

13. Right-Hand Rule 1 • why not say clockwise and counterclockwise? • because CW and CCW depend on your point of view

14. demo: repulsion of wires I = 45 A VDC = 30 V

15. demo: current in a wire, in the jaws of 2 horseshoe magnets I = 45 A VDC = 30 V

16. Plotting Field Lines • Field lines are three-dimensional • A dot (•) indicates the tip of the vector when it points out of the plane • A cross (×) denotes the tails of the vector when it points into the plane • (imagine an arrow, with a point arrowhead and a feather arrowtail) Claes Oldenburg and Coosjevan Bruggen’s sculpture “Cupid’s Span” Rincon Park in San Francisco

17. Charges and Magnetic Fields • The electric current can be modeled as a collection of positive electric charges • These charges move with a velocity parallel to the current direction • The direction of the magnetic field is given by the right-hand rule • A positive charge moving to the left produces the same magnetic field as a negative charge moving to the right

18. Superposition • The Principle of Superposition states that the total magnetic field produced by two or more different sources is equal to the sum of the fields produced by each source individually • This principle can be used to find the pattern of magnetic field lines in virtually all situations • Most cases can be analyzed as a collection of bar magnets, straight wires, or current loops

19. Magnetic Field and Current Loop • Treat the loop as many small pieces of wire • Use the right-hand rule 1to find the field from each piece of wire • Applying superposition gives the overall pattern shown in this figure

20. Magnetic Forces & Bar Magnets • To determine the total force on the bar magnet, you must look at the forces on each pole • The total force is zero • Thetorque is non-zero • The torque acts to align the bar magnet along the magnetic field

21. Like and Unlike Poles

22. Comparing Electric and Magnetic Fields and Forces • There are many similarities in the behavior of electric charges and magnetic poles • Unlike charges and unlike poles attract • Like charges and like poles repel • BUT: North and south magnetic poles always occur in pairs: “magnetic dipole” • an isolated magnetic “monopole” has never been observed, although in principle it is possible that it could exist. Some particle physics theories assume that it exists.

23. Lorentz force on moving charges In the presence of a magnetic field B: • A magnetic force acts on individual charges • The force depends on the velocity of the charge • If the charge is not moving, there is no magnetic force • If a positive charge qis moving with a given velocity in an external magnetic field, then the force on the charge has magnitude: • FB = qvB sin θ θis the angle between the velocityv and the field B Lorentz Force

24. Direction of Lorentz Force:Right Hand Rule 2 • Direction of force FB: • Point hand and fingers in the direction of v • Curl fingers towards the field (smallest angle btwn v and B) • The thumb points in the direction of the magnetic force FB • This is for a positive charge • For a negative charge, the force goes in the opposite direction • Note: The magnetic force is always perpendicular to both the magnetic field and the particle’s velocity

25. Question:Direction of magnetic force on moving charges (R) Up (Y) Down (G) Right (B) Left (W) Zero VelocityB Force out of screen right up out of screen left down out of screen up ? Right Hand Rule 2: Point fingers along out of pageCurl fingers towards up Thumb points along left

26. Question:Direction of magnetic force on moving charges (R) Up (Y) Down (G) Right (B) Left (W) Zero VelocityB Force out of page right up out of page left down out of page up left out of page down ? Right Hand Rule 2: Point fingers along out of pageCurl fingers towards down Thumb points along right

27. Motion of a Charged Particle • If the charged particle moves parallel to the magnetic field • The angle between the velocity and the field is zero • Therefore, the force is also zero • Since sin θ = 0

28. Motion of a Charged Particle, 2 • If a charged particle moves perpendicular to the magnetic field • The angle between the velocity and the field is 90o • Therefore, the force is F=qvB • The particle will move in a circle

29. Motion of a Charged Particle, 3 • If a charged particle moves neither parallel nor perpendicular to the magnetic field • The angle between the velocity and the field varies • The path of the particle is helical • The charged particle will spiral around the magnetic field lines

30. Right Hand Rules, Summary • Right-hand rule number 1: to find the direction of the magnetic field from an electric current • Place the thumb of your right hand along the direction of the current • Curl your fingers; they will then give the direction of the magnetic field as the field lines encircle the current • Right-hand rule number 2: to find the direction of the magnetic force (Lorentz force) on a moving charge, q • Point the fingers of your right hand along the direction of the velocity • Curl your fingers towards the direction of the field • Curl your fingers through the smallest angle that connects the velocity and the field • If q is positive, the magnetic force is parallel to your thumb. If q is negative, the magnetic force is in the opposite direction

31. Magnetic Force on a wire • An electric current is a collection of moving charges, a force acts on a current • From the equation of the force on a moving charge, the force on a current-carrying wire is Fon wire = I L B sin θ • The direction of the force is given by the right-hand rule 2

32. Problem 20.47 A long, straight wire of length 0.75 m carries a current I = 1.5 A in a region where B = 2.3 T. If the force on the wire is 1.4 N, what is the angle between the field and the wire? up to here

33. Torque on a Current Loop • A magnetic field can produce a torque on a current loop • Assume a square loop with sides of length L carrying a current I in a constant magnetic field • The directions of the forces can be found from right-hand rule 2 Section 20.5

34. Torque, cont. • On two sides, the current is parallel or antiparallel to the field, so the force is zero on those sides • The forces on sides 1 and 3 are in opposite directions and produce a torque on the loop • When the angle between the loop and the field is θ, the torque is τ = I L2B sin θ • For different shapes, this becomes τ = I A B sin θ Section 20.5

35. Magnetic Moment • For a current loop, the magnetic moment is I A • The direction of the magnetic moment is either along the axis of the bar magnet or perpendicular to the current loop • The strength of the torque depends on the magnitude of the magnetic moment Section 20.5

36. Mass Spectrometer • Allows for the separation of ions according to their mass or charge • The ions enter with some speed v • They pass into a region where the magnetic field is perpendicular to the velocity Section 20.6

37. Mass Spectrometer, cont. • The ions travel in a circle in the mass spectrometer • The radius of the circle is mv/qB • Ions with different masses will travel in arcs with different radii • Mass spectrometer can also be used to find the composition of a material • Measure the values of v, B and r • Calculate q/m • Charge to mass ratio Section 20.6

38. Hall Effect • An electric current is produced by moving electric charges • A particular value of current can be produced by positive charges moving to the right or negative charges to the left • The Hall Effect can distinguish between the two options Section 20.6

39. Hall Effect, cont. • Place a current-carrying wire in a magnetic field directed perpendicular to the current • With positive charge carriers, an excess positive charge accumulates on the top edge • With negative charge carriers, an excess negative charge accumulates on the top edge • Measuring the potential difference distinguishes between positive or negative charge carriers producing the current Section 20.6

40. Ampère’s Law • Ampère’s Law is useful when the magnetic field lines have a simple symmetry • Similar in approach to Gauss’s Law for electric fields • Relates the magnetic field along a path to the electric current enclosed by the path Section 20.7

41. Ampère’s Law, cont. • For the path shown in fig. 20.29, Ampère’s Law states that • μo is the permeability of free space • μo = 4 πx 10-7T.m / A • If B varies along the path, Ampère’s Law cannot be used = Section 20.7

42. Magnetic Field near a Long Straight Wire • Ampère’s Law can be used to find the magnetic field near a long, straight wire • B|| is the same all along the path • If the circular path has a radius r, then the total path length is 2 πr Section 20.7

43. Field from a Current Loop • It is not possible to find a path along which the magnetic field is constant • So Ampère’s Law cannot be easily applied • From other techniques, the field at the center of the loop is Section 20.7

44. Field Inside a Solenoid • By stacking many loops close together, the field along the axis is much larger than for a single loop • A helical winding of wire is called a solenoid • More practical than stacking single loops Section 20.7

45. Solenoid, cont. • For a very long solenoid, it is a good approximation to assume the field is constant inside the solenoid and zero outside • Use the path shown in the figure • Only side 1 contributes to the magnetic field Section 20.7

46. Solenoid, final • The magnetic field inside the solenoid is given by • For a solenoid with a length much greater than the diameter Section 20.7

47. Magnetic Materials • Magnetic poles always come in pairs • It is not possible to separate the poles of a magnet • To understand why, the atomic origin of permanent magnetism must be considered Section 20.8

48. Motion of Electrons • The motion of an electron around a nucleus can be pictured as a tiny current loop • The radius is approximately the radius of the atom • The direction of the resulting magnetic field is determined by the orientation of the current loop • Using right-hand rule 1 Section 20.8

49. Electron Spin • The electron also produces a magnetic field due to an effect called electron spin • The spinning charge acts as a circulating electric current • The electron has a spin magnetic moment • When an electron is placed in a magnetic field, it will tend to align its spin magnetic moment with the magnetic field Section 20.8

50. Magnetic Field in an Atom • The correct explanation of electron spin requires quantum mechanics • Confirms an atom can produce a magnetic field in two ways • Through the electron’s orbital current loop • Through the electron’s spin • The total magnetic field produced by a single atom is the sum of these two fields Section 20.8