Chapter 19: Magnetism

1. Chapter 19: Magnetism Homework assignment : 18,25,38,45,50 Read Chapter 19 carefully especially examples. • Magnets Magnets

2. Magnets • Magnetic force

3. Magnets • Magnetic field lines

4. Magnets • Magnetic field lines

5. Electric Field Linesof an Electric Dipole Magnetic Field Lines of a bar magnet • Magnetic field lines (cont’d)

6. S N S N S N Magnets • Magnetic monopole? Perhaps there exist magnetic charges, just like electric charges. Such an entity would be called a magnetic monopole (having + or magnetic charge). How can you isolate this magnetic charge? Try cutting a bar magnet in half: Even an individual electron has a magnetic “dipole”! • Many searches for magnetic monopoles—the existence of which would explain (within framework of QM) the quantization of electric charge (argument of Dirac) • No monopoles have ever been found:

7. Orbits of electrons about nuclei Intrinsic “spin” of electrons (more important effect) • Source of magnetic field • What is the source of magnetic fields, if not magnetic charge? • Answer: electric charge in motion! • e.g., current in wire surrounding cylinder (solenoid) produces very similar field to that of bar magnet. • Therefore, understanding source of field generated by bar magnet lies in understanding currents at atomic level within bulk matter. Magnets

8. Magnets • Magnetic field of Earth • The geographic North Pole corresponds a magnetic south pole. • The geographic South Pole corresponds a magnetic north pole. • The angle between the direction of the magnetic field • and the horizontal is • called the dip angle. • The difference between • true north and, defined • as the geographic North • Pole, and north indicated • by a compass varies from • point to point on Earth. • This difference is referred • to as a magnetic declination.

9. Magnetic Fields • Magnetic force: Observations magnitude: vector product

10. Magnetic force (Lorentz force) right-hand rule Magnetic Fields SI unit : tesla (T) = Wb/m2

11. Magnetic force (cont’d) Units of magnetic field Magnetism

12. B B B x x x x x x x x x x x x x x x x x x ® ® ® ® ® ® ® ® ® ® ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ v v v ´ q q q F F = 0 F • Magnetic force (Lorentz force) Magnetic force Magnetic Fields

13. Magnetic Force on a Current-Carrying Conductor • Magnetic force on a current (straight wire)

14. Magnetic Force on a Current-Carrying Conductor • Magnetic force on a current (straight wire) (cont’d)

15. Force and Torque on a Current Loop • Plane of loop is parallel to the magnetic field t=rFsinq

16. Force and Torque on a Current Loop • Plane of loop : general case

17. Force and Torque on a Current Loop • Plane of loop and magnetic moment

18. Force and Torque on a Current Loop • Motor flip the current direction

19. Motion of Charged Particles in a Magnetic Field • Case 1: Velocity perpendicular to magnetic field υ perpendicular to B The particle moves at constant speed υ in a circle in the plane perpendicular to B. F/m = a provides the acceleration to the center, so v R F B x

20. Motion of Charged Particles in a Magnetic Field • Case 1: Velocity perpendicular to magnetic field (cont’d)

21. Motion of Charged Particles in a Magnetic Field • Case 1: Velocity perpendicular to magnetic field (cont’d) Velocityselector

22. Motion of Charged Particles in a Magnetic Field • Case 1: Velocity perpendicular to magnetic field (cont’d) Massspectrometer

23. Motion of Charged Particles in a Magnetic Field • Case 1: Velocity perpendicular to magnetic field (cont’d) Mass spectrometer

24. Motion of Charged Particles in a Magnetic Field • Case 1: Velocity perpendicular to magnetic field (cont’d) Mass spectrometer

25. Motion of Charged Particles in a Magnetic Field • Case 1: Velocity perpendicular to magnetic field (con’t) Mass spectrometer

26. Motion of Charged Particles in a Magnetic Field • Case 2: General case υ at any angle to B. Begin by separating the two components of υ into u// // [(u// u// u u//.

27. Motion of Charged Particles in a Magnetic Field • Case 2: General case (cont’d) Since the magnetic field does not exert force on a charge that travels in its direction, the component of velocity in the magnetic field direction does not change.

28. v 1 x x r x x 2 Exercises • Exercise 1 If a proton moves in a circle of radius 21 cm perpendicular to a B field of 0.4 T, what is the speed of the proton and the frequency of motion?

29. F v N B Exercises • Exercise 2 Example of the force on a fast moving proton due to the earth’s magnetic field. (Already we know we can neglect gravity, but can we neglect magnetism?) Let v = 107 m/s moving North. What is the direction and magnitude of F? Take B = 0.5x10-4 T and v B to get maximum effect. (a very fast-moving proton) vxB is into the paper (west). Check with globe

30. Magnetic Field of a Long Straight Wire and Ampere’s Law • Magnetic field due to a long straight wire right-hand rule 2 Magnetic field by a long wire m0=4px10-7 T m/A Iron filings permeability of free space

31. Ampere’s Law • Ampere’s (circular) law : A circular path • Consider any circular path of radius R centered on the wire carrying current I. • Evaluate the scalar product B·Ds around this path. • Note that B and Ds are parallel at all points along the path. • Also the magnitude of B is constant on this path. So the sum of all the B·Ds terms around the circle is Ds Ds Ds Ds Ampere’s circuital law (valid for any closed path) On substitution for B amount of current that penetrates the loop

32. Force Between Parallel Conductors • Two parallel wires At a distance afrom the wire with current I1the magnetic field due to the wire is given by d Force per unit length

33. Force Between Parallel Conductors • Two parallel wires (cont’d) d d Parallel conductors carrying current in the same direction attract each other. Parallel conductors carrying currents in opposite directions repel each other.

34. Force Between Parallel Conductors • Definition of ampere d The chosen definition is that for d = L = 1m, The ampere is made to be such that F2 = 2×10−7 N when I1=I2=1 ampere  This choice does two things (1) it makes the ampere (and also the volt) have very convenient magnitudes for every day life and (2) it fixes the size of μ0 = 4π×10−7. Note ε0 = 1/(μ0c2). All the other units follow almost automatically.

35. Magnetic Fields of Current Loops and Solenoids • Magnetic field by a current loop Dx1 • The segment Dx1 produces a magnetic field • magnitude B1 at the center of the loop, directed • out of the page. I R 2) The segment Dx2 produces a magnetic field magnitude B2 at the center of the loop, directed out of the page. The magnitude of B1 and B2 are the same. B Dx2 The magnitude of the magnetic field at the center of a circular loop carrying current I The magnitude of the magnetic field at the center of N circular loops carrying current I

36. d Magnetic Fields of Current Loops and Solenoids L • Magnetic field by a solenoid • A solenoid is defined by a current i flowing through a wire that is wrapped nturns per unit length on cylinder of radius d and length L. • If d<< L, the B field is to first order contained within the solenoid, in the axial direction, and of constant magnitude. In this limit, we can calculate the field using Ampere's law. Inside the solenoid, B is constant and outside it is zero in this approximation. Apply Ampere’s law to the rectangular loop represented by blue dashed lines. number of turn per unit length

37. Magnetic field by a solenoid (cont’d) The magnetic field of a solenoid is essentially identical to that of a bar magnet solenoid bar magnet P A mystery of : x B field at point P: In a solenoid, the B field at its axis: R I