Chapter 8

1. Chapter 8 Charges in Magnetic Fields

2. Introduction • In the previous chapter it was observed that a current carrying wire observed a force when in a magnetic field • This force is experienced by any moving charge in a magnetic field

3. Introduction • In applications where this interaction is used, the charges are moving through near vacuum so that relatively free motion can occur across that space (low electrical resistance)

4. 8.1 Forces on a charged particle in a magnetic field

5. Factors Affecting the Force When a charged particle is in a magnetic field, the force on the charged particle depends on the following factors: • The magnitude and direction of the velocity of the particle • The magnitude and sign of the charge on the particle • The magnetic field strength

6. Factors Affecting the Force • There is no interaction between a magnetic field and a stationary particle • Stationary charges do not generate a magnetic field to interact with the magnetic field they are in • The electric field created by the charged particle does not interact with the magnetic field

7. The force on a Charged Particle Moving in a Magnetic field • An electric current is a flow of electric charges • The magnitude of the current is defined as the rate of flow of electric charge: I = Where Δq is the charge and Δt is the time

8. The force on a Charged Particle Moving in a Magnetic field • The rate of flow of charge is taken from a point: e.g. if a current of 2A is flowing through a circuit, 2 coulombs of charge passes any point in the circuit each second

9. The force on a Charged Particle Moving in a Magnetic field • This idea can be extended to point charges: • If one alpha particle (q = 3.2x10-19C) passes a point in one second, then the average current is 3.2x10-19A past that point • If one alpha particle (q = 3.2x10-19C) passes a point in two seconds, then the average current is 1.6x10-19A past that point

10. The force on a Charged Particle Moving in a Magnetic field • The force on a current carrying wire in a magnetic field from the formula: F = IΔlB sinθ • However to apply this to a charged particle, we need to consider how to define IΔl

11. The force on a Charged Particle Moving in a Magnetic field • As discussed before, current is given by: Iavg= • In this time, the particle has moved a distance of vtmetres, this can be taken as the length, Δl, of the current element

12. The force on a Charged Particle Moving in a Magnetic field • Substituting the expressions for current and element length gives:

13. The force on a Charged Particle Moving in a Magnetic field As with a current carrying wire in a magnetic field • the force on a charge moving in a magnetic field is maximum when it is travelling perpendicular to the field • the force in a charge moving parallel or anti-parallel to the field is zero

14. The direction of the magnetic force • The direction of magnetic force on a moving charge in a magnetic field can be found using the right-hand palm rule • However, the thumb points in the direction of conventional current (positive charge flow) • This means that the thumb points in the opposite direction to the motion of a negative charge

15. Class problems Conceptual questions: 1-4 Descriptive questions: 2 Analytical questions: 2

16. 8.2motion at right angles to the field

17. Motion at Directions other than 90° to the Magnetic Field • Charged particles moving parallel to a magnetic field experience no magnetic force, and therefore move with constant velocity • Motion at angles θ to the magnetic field are more complex and are not included in the syllabus • Only charges moving perpendicular to the field are considered in this course

18. Motion at Directions other than 90° to the Magnetic Field • Example of motion at an angle to the magnetic field http://www.youtube.com/watch?v=a2_wUDBl-g8

19. Motion of Charged Particles at Right Angles to the Magnetic Field • In the diagram shown, a charged particle enters a uniform magnetic field directed into the page • Using the right hand rule, the force is acting towards the top of the page

20. Motion of Charged Particles at Right Angles to the Magnetic Field • As the particle changes direction, so does the direction of the magnetic force acting on it • Since the magnetic force is always perpendicular to the velocity, the speed of the particle does not change

21. Motion of Charged Particles at Right Angles to the Magnetic Field • This motion is uniform circular motion • Charged particles moving at right angles to a magnetic field always follow a circular path

22. Determination of the Radius of the Circular Path • Centripetal acceleration is given by:

23. Determination of the Radius of the Circular Path • The force is also given by: Hence:

24. Class problems Conceptual questions: 4, 8, 10 Descriptive questions: 4 Analytical questions: 1, 3-4, 6-9

25. 8.3application:The cyclotron

26. Introduction • The acceleration of charged particles to very high speeds, and hence very high energies, is essential in many fields • It is particularly useful in atomic and nuclear physics, and in medical research, diagnosis and treatment

27. Introduction • The most obvious way to do this is to pass the charged particle though a potential difference • Passing a proton through a potential difference of 1000V will result in a gain of 1000eV in kinetic energy • However we often require energies of MeV (106 eV) to GeV (109 eV)

28. Introduction • We can accomplish higher energies by passing particles through a series of potential differences • Passing an electron 100 times in succession through 1000V is equivalent to passing it through 100,000V

29. Introduction • To accelerate particles to energies in a linear accelerator to GeV energies requires a series of thousands of potential differences • This is impractical due to the sheer size of accelerator needed • Use of a cyclotron reduces the size of the accelerator considerably

30. Components of a cyclotron Ion source: A source of protons to be accelerated Semi-circular metal containers (Dees): Two terminals of alternating potential difference between which the protons are accelerated Ion source

31. Components of a cyclotron Vacuum chamber: The interior of the cyclotron is housed in an evacuated chamber High frequency input: The source of alternating potential difference Ion source

32. Components of a cyclotron Electromagnets: The South pole of an electromagnet is below the Dees, and the North pole of another electromagnet is above, this generates a uniform magnetic field for the circular motion

33. Principles of Operation • The protons are accelerated towards the negatively charged Dee • Within the Dee they experience circular acceleration due to the magnetic field

34. Principles of Operation • The electric field does not exist in the Dees because they are effectively hollow conductors

35. Principles of Operation • When the proton leaves the Dee, the potential difference is reversed, accelerating the proton towards the other Dee

36. Principles of Operation • This process repeats many times, each time the proton is accelerated across the gap between the Dees, the radius gets larger

37. Principles of Operation • The proton is eventually removed from the cyclotron using electrodes

38. Computational Considerations • The radius of the proton’s circular orbit at any time in the Dees is given by:

39. Computational Considerations • The period of the proton’s motion is independent of its speed: Derivation on p. 172 of Key Ideas textbook

40. Computational Considerations • Kinetic energy of the particle: Derivation on p. 173 of Key Ideas textbook

41. Some Uses of Cyclotrons • The plutonium used to make the first atomic bomb was made by bombarding Uranium 238 with deuterons • Production of isotopes to use in nuclear medicine • Injecting radioactive isotopes into organs and detecting them with gamma ray detectors • Positron decay from Nitrogen-13 used in Positron Emission Tomography (PET)

42. Class problems Conceptual questions: 10-13, 15 Descriptive questions: 12, 14, 18 Analytical questions: 8, 10, 11