Magnetism

1. Magnetism Magnetic Force

2. Magnetic Force Outline • Lorentz Force • Charged particles in a crossed field • Hall Effect • Circulating charged particles • Motors • Bio-Savart Law

3. Class Objectives • Define the Lorentz Force equation. • Show it can be used to find the magnitude and direction of the force. • Quickly review field lines. • Define cross fields. • Hall effect produced by a crossed field. • Derive the equation for the Hall voltage.

4. Magnetic Force • The magnetic field is defined from the Lorentz Force Law,

5. Magnetic Force • The magnetic field is defined from the Lorentz Force Law, • Specifically, for a particle with charge q moving through a field B with a velocity v, • That is q times the cross product of v and B.

6. Magnetic Force • The cross product may be rewritten so that, • The angle is measured from the direction of the velocity to the magnetic field . • NB: the smallest angle between the vectors! v x B B v

7. Magnetic Force

8. Magnetic Force • The diagrams show the direction of the force acting on a positive charge. • The force acting on a negative charge is in the opposite direction. B F - v B + F v

9. Magnetic Force • The direction of the force F acting on a charged particle moving with velocity v through a magnetic field B is always perpendicular to v and B.

10. Magnetic Force • The SI unit for B is the tesla (T) newton per coulomb-meter per second and follows from the before mentioned equation . • 1 tesla = 1 N/(Cm/s)

11. Magnetic Field Lines Review

12. Magnetic Field Lines • Magnetic field lines are used to represent the magnetic field, similar to electric field lines to represent the electric field. • The magnetic field for various magnets are shown on the next slide.

15. Magnetic Field Lines Crossed Fields

16. Crossed Fields • Both an electric field E and a magnetic field B can act on a charged particle. When they act perpendicular to each other they are said to be ‘crossed fields’.

17. Crossed Fields • Examples of crossed fields are: cathode ray tube, velocity selector, mass spectrometer.

18. Crossed Fields Hall Effect

19. Hall Effect • An interesting property of a conductor in a crossed field is the Hall effect.

20. B Hall Effect • An interesting property of a conductor in a crossed field is the Hall effect. • Consider a conductor of width d carrying a current i in a magnetic fieldB as shown. x x x x Dimensions: Cross sectional area: A Length: x d x x x x i i x x x x x x x x

21. FB B Hall Effect • Electrons drift with a drift velocity vd as shown. • When the magnetic field is turned on the electrons are deflected upwards. x x x x d x x x x vd - i i x x x x x x x x FB

22. FB B Hall Effect • As time goes on electrons build up making on side –ve and the other +ve. x x x x Low - - - - - d x x x x vd - i i x x x x + + + + + High x x x x

23. FB FE B E Hall Effect • As time goes on electrons build up making on side –ve and the other +ve. • This creates an electric field from +ve to –ve. x x x x Low - - - - - x x x x vd - i i x x x x + + + + + High x x x x

24. FB FE B E Hall Effect • The electric field pushed the electrons downwards. • The continues until equilibrium where the electric force just cancels the magnetic force. x x x x Low - - - - - x x x x vd - i i x x x x + + + + + High x x x x

25. FB FE B E Hall Effect • At this point the electrons move along the conductor with no further collection at the top of the conductor and increase in E. x x x x Low - - - - - x x x x vd - i i x x x x + + + + + High x x x x

26. Hall Effect • The hall potential V is given by, V=Ed

27. Hall Effect • When in balance,

28. Hall Effect • When in balance, • Recall, dx A A wire

29. Hall Effect • Substituting for E, vd into we get,

30. A circulating charged particle

31. Magnetic Force • A charged particle moving in a plane perpendicular to a magnetic field will move in a circular orbit. • The magnetic force acts as a centripetal force. • Its direction is given by the right hand rule.

32. Magnetic Force

33. Magnetic Force • Recall: for a charged particle moving in a circle of radius R, • As so we can show that,

34. Magnetic Force on a current carrying wire

35. B x x x x I x x x x Magnetic Force • Consider a wire of length L, in a magnetic field, through which a current I passes.

36. B x x x x I x x x x Magnetic Force • Consider a wire of length L, in a magnetic field, through which a current I passes. • The force acting on an element of the wire dl is given by,

37. Magnetic Force • Thus we can write the force acting on the wire,

38. Magnetic Force • Thus we can write the force acting on the wire, • In general,

39. Magnetic Force • The force on a wire can be extended to that on a current loop.

40. Magnetic Force • The force on a wire can be extended to that on a current loop. • An example of which is a motor.

41. Interlude Next…. The Biot-Savart Law

42. Biot-Savart Law

43. Objective • Investigate the magnetic field due to a current carrying conductor. • Define the Biot-Savart Law • Use the law of Biot-Savart to find the magnetic field due to a wire.

44. Biot-Savart Law • So far we have only considered a wire in an external field B. Using Biot-Savart law we find the field at a point due to the wire.

45. Biot-Savart Law • We will illustrate the Biot-Savart Law.

46. Biot-Savart Law • Biot-Savart law:

47. Biot-Savart Law • Where is the permeability of free space. • And is the vector from dl to the point P.

48. l Biot-Savart Law • Example: Find B at a point P from a long straight wire.

49. l Biot-Savart Law • Sol:

50. l Biot-Savart Law • We rewrite the equation in terms of the angle the line extrapolated from makes with x-axis at the point P. • Why? • Because it’s more useful.