1. Lectures 11 & 12: Magnetic Fields and the Motion of Charged Particles Chapters 26-28 (Tipler) Electro magnetism

3. Learning Objectives To introduce the concepts of magnetic fields Current Magnetic force on a moving charge Magnetic field lines Magnetic flux

4. The Nature of Electric Current Recognizable effects of current flow are: heating magnetic electrolytic

5. The Nature of Electric Current Recognizable effects of current flow are: heating magnetic electrolytic

6. Definition of Current Suppose a conductor carries a current I Rate of flow of charge Q past a given cross-section is defined by:

7. The SI unit of current is the ampere: one ampere is defined to be one coulomb per second. André Marie Ampére (1775-1836)

8. The Current in a Conductor nene Area A vdvd In a time  t, volume “swept” out is: Av d  t Charge contained in this volume is: Therefore:

9. The current per unit cross section is called the current density J:

10. Magnetic Field A moving charge (current) creates a magnetic field in the surrounding space The magnetic field exerts a force F m on any other moving charge (or current) that is present in the field

11. The Magnetic Force on a Moving Charge Experimentally: A particle of charge +q moving with velocity v in a magnetic field B, experiences a (magnetic) force F m : F m is  to v & B FmFm  B v

12. In vector form

14. The unit of B is: N C -1 m -1 s N Cms -1 This is given a special name tesla (T), in honour of Nikola Tesla

15. If 1 C of charge moving at 1 m/s perpendicular to a magnetic field experiences a force of 1 Newton, the magnetic field is 1 tesla. Earth’s magnetic field  5  10 -5 T Poles of a large electromagnet  2 T Surface of a neutron star  10 8 T Nikola Tesla (1856- 1943) a Slovenian born American electrical engineer Gauss (G) is another unit in common use: 1G = 10 -4 T

16. In regions where both E and B fields are present, the total force is the vector sum of the electric and magnetic forces: in direction of E  to v and B Lorentz Equation

17. The Earth’s magnetic field at a particular region is represented by A proton is moving in this magnetic field with a velocity Obtain an expression for the direction and magnitude of magnetic force acting on the proton. Worked Exercise

19. Magnetic Field Lines

20. NOTE: unlike electric field lines, magnetic field lines are ALWAYS continuous ( no magnetic monopoles ) and they do not point in the direction of the force on the moving charge in a magnetic field – they ARE NOT lines of force.

21. Magnetic Field Lines the tangent to a field line at a point P gives the direction of B at that point the number of field lines drawn per unit cross sectional area is proportional to the magnitude of B

22. Magnetic Flux The magnetic flux  B passing through the small area  A shown is defined by: AA

23. Gauss’s flux law for magnetism:

24. Review and Summary A magnetic field B is defined in terms of the force F m acting on a test particle with charge q and moving through the field with velocity v: The SI unit for B is the tesla (T)

25. Review and Summary Compare this with the definition of the electric field E F E = qE

26. Review and Summary Gauss’s Law for Magnetism The net magnetic flux through any closed surface is zero As a result, magnetic field lines always close on themselves

27. Class Exercise False 1. True or False: The magnetic force does not accelerate a moving charged particle because the force is perpendicular to the velocity of the particle. 2.What is the force acting on an electron with velocity in a magnetic field

28. F = 0.621 pN 1 pN = 10 -12 N pN

29. Last lecture: Magnetic force on moving charges Magnetic flux and Gauss’s flux law

30. A few special cases: v parallel to B v perpendicular to B v makes an angle  to B Today’s lecture: Magnetic force on a current, torque on a current loop.

31. (a) v parallel to B F = 0 (b) v  BF  to the plane containing B and v

32. x x x x x The direction of B field in sketches Into the paper, away from you Out of the paper, towards you

33. (c) v makes an angle  with B (i) a uniform circular motion in which it has the speed vsin  in a plane perpendicular to the direction of B (ii) a steady speed of magnitude vcos  along the direction of B  Helical Motion B v 

35. The Force on a Current-carrying Conductor Single Charge N, the number of charge carriers in volume Al, is: N = nAl n = charge number density Total force F:

36. In General magnetic force on a straight wire segment The direction of l is defined as the direction of the current I

37. If the conductor is not straight, consider individual segments and use magnetic force on an infinitesimal wire segment x x x x

38. B I b F1F1  The Torque on a Current-Carrying Loop (26-3 Tipler) l -F 1 F2F2 -F 2 b parallel to y l in the x-z plane, makes an angle  to x F 1, points in y direction -F 1, points in -y, net torque from F 1 and -F 1 is zero.

39.  The Torque on a Current-Carrying Loop (26-3 Tipler) O F2F2 -F 2 F 2 passes through O, so torque from F 2 is zero. Torque from -F 2 The direction of the torque?? r Pointing out towards you Will the loop accelerate towards you?

40.  The Torque on a Current-Carrying Coil A l

41. This result is true for loops of any shape Define Magnetic Dipole Moment The torque tends to align  and B

42.   Unstable equilibrium  = 180 degrees Stable equilibrium  = 0

43. I Comparison between Magnetic and Electric Dipole Moments Magnetic Dipole  B -q-q +q+q p Electric Dipole E Electric Dipole Potential EnergyMagnetic Dipole Potential Energy

44. Electric dipole, Molecules behave like electric dipoles Why do we bother with magnetic dipoles?

45. v L Magnetic Dipole Moment of an Electron in an Atom Angular Momentum L = m e r x v

46. Magnetic Dipole Moment of an Electron in an Atom Q.M. L is quantised. Fundamental unit is: Fundamental Unit of Magnetic Moment Bohr Magneton

47. All atoms have orbiting electrons, Why only some have a non-zero magnetic moment?? Next lecture: Connecting B to current I