Lecture 13. Magnetic Field, Magnetic Forces on Moving Charges. Outline:

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1 Lecture 13. Magnetic Field, Magnetic Forces on Moving Charges. Outline: Intro to Magnetostatics. Magnetic Field Flux, Absence of Magnetic Monopoles. Force on charges moving in magnetic field. 1

2 Structure of the Course Electrostatics d dd = 0 0 Electric fields generated by charges at rest Ch Magnetostatics Electromagnetism Magnetic fields generated by time-independent currents Ch E - electric fields B magnetic fields Q charges I currents Φ - fluxes Electromagnetic Waves d dd 0 Ch covered in detail in more advanced courses on electrodynamics and optics. 2

3 Intro to Magnetostatics Our goal: to describe the magnetostatic field the same way we ve described the electrostatic field. E da = Q eeee ε 0 - always true B da =? E dl = 0 F = qe - true only if the fields are static B dl =? F =? Do you know a vector field a that has both a da = 0 and a dl = 0? 3

4 Sources of Magnetic Field Charges at rest do not generate B. What are the sources of the magnetic field? magnetic point charge (a magnetic monopole): has not been observed yet (though its existence doesn t contradict anything) ferromagnetic materials (electron spins purely quantum phenomenon). Time-dependent electric fields (we ll consider this source later in Electrodynamics). Charges in motion: currents, orbital motion of electrons in atoms. In magnetostatics: The B field a moving single charge is non-stationary, not good for magnetostatics + Steady (time-independent) current: our best bet. 4

5 Characteristic Magnetic Fields Units: tesla, T B 10 4 T rare-earth magnets B uu tt 1.4T B 1 2T Superconducting solenoids in LHC B uu tt 8T Superconducting solenoids B uu tt 30T 5

6 Flux of the Magnetic Field Flux of the magnetic field: Φ B = B da Units: T m 2 =Weber, Wb compare with Φ E = E da No magnetic point charges (magnetic monopoles): have not been observed yet. Magnetic dipole Electric dipole Consequence: for ANY closed surface B da = 0 - always true, not only in electrostatics but also in electrodynamics 6

7 Magnetic Field Lines Magnetic field lines are closed loops (no mag. monopoles): Magnetic field is a non-conservative vector field. B dl 0 to be specified later 7

8 Force on a Charge Moving in Magnetic Field Lorentz force F = q v B F = qqqqqqφ F = 0 for charges moving along B Heaviside F is max when v B Magnetic field lines are not lines of force! demonstration 8

9 Iclicker Question Imagine that you are looking at the face of a CRT. The bright spot indicating where the electron beam hits the face. You bring a permanent magnet toward the CRT with its north pole oriented upward. Which direction will the spot be deflected across the screen? A. up F = q v B B. down C. the spot does not deflect D. right E. left 9

10 Iclicker Question Imagine that you are looking at the face of a CRT. The bright spot indicating where the electron beam hits the face. You bring a permanent magnet toward the CRT with its north pole oriented upward. Which direction will the spot be deflected across the screen? A. up F = q v B B. down C. the spot does not deflect D. right E. left 10

11 Iclicker Question When does a magnetic field exert a force on a charged particle? A. Always. B. Only when the particle moves exactly perpendicular to the magnetic field lines. C. When the particle is moving at a non-zero angle with respect to the magnetic field lines. D. When the particle is moving along the magnetic field lines. E. When the particle is moving. F = q v B 11

12 Iclicker Question When does a magnetic field exert a force on a charged particle? A. Always. B. Only when the particle moves exactly perpendicular to the magnetic field lines. C. When the particle is moving at a non-zero angle with respect to the magnetic field lines. D. When the particle is moving along the magnetic field lines. E. When the particle is moving. F = q v B 12

13 No Work Done by Magnetic Force! F = q v B F dl The work done by the magnetic field on a moving charge = 0 dd = F dl = 0 You cannot increase the speed of charged particles using magnetic field. However, the B-induced acceleration is non-zero, it s just perpendicular to the velocity a dv dd. However, there are many situations in which this statement appears to be false: in a non-uniform magnetic field, a current-carrying wire loop would accelerate, two permanent magnets would accelerate towards one another, etc. In these cases, the kinetic energy of objects increases. Who does the work? For discussion, see 13

14 Iclicker Question When a charged particle moves through a magnetic field, the trajectory of the particle at a given point is A. parallel to the magnetic field line that passes through that point. B. perpendicular to the magnetic field line that passes through that point. C. neither parallel nor perpendicular to the magnetic field line that passes through that point. D. any of the above, depending on circumstances 14

15 Iclicker Question When a charged particle moves through a magnetic field, the trajectory of the particle at a given point is A. parallel to the magnetic field line that passes through that point. B. perpendicular to the magnetic field line that passes through that point. C. neither parallel nor perpendicular to the magnetic field line that passes through that point. D. any of the above, depending on circumstances 15

16 Cross Product F = e v B An electron moves along the z axis, the B field is in the x-y plane. v = 1k m/s B = 2ı + 3ȷ T v z k y ȷ F = e v B = e 1k 2ı + 3ȷ x ı B = e 1k 2ı + 1k 3ȷ z k ı ȷ = k = e 2ȷ 3ı = e 3ı 2ȷ x ı y ȷ ȷ k = ı k ı = ȷ 16

17 Iclicker Question A particle with charge q = 1 C is moving in the positive z- direction at 5 m/s. The magnetic field at its position is B= 3iˆ 4 ˆj T ( ) What is the magnetic force on the particle? A. B. C. D. ( iˆ+ ˆj) N ( iˆ ˆj) ( iˆ+ ˆj) ( iˆ ˆj) N N N E. none of these v z k y ȷ B x ı ı ȷ = k ȷ k = ı k ı = ȷ F = q v B 17

18 Iclicker Question A particle with charge q = 1 C is moving in the positive z- direction at 5 m/s. The magnetic field at its position is B= 3iˆ 4 ˆj T ( ) What is the magnetic force on the particle? A. B. C. D. ( iˆ+ ˆj) N ( iˆ ˆj) ( iˆ+ ˆj) ( iˆ ˆj) N N N E. none of these v z k y ȷ B x ı ı ȷ = k ȷ k = ı k ı = ȷ F = q v B F = 1 5k 3ı 4ȷ = 15ȷ 20ı 18

19 Iclicker Question A positively charged particle moves in the positive z-direction. The magnetic force on the particle is in the positive y-direction. What can you conclude about the z-component of the magnetic field at the particle s position? A. B z > 0 B. B z = 0 C. B z < 0 D. not enough information given to decide v x ı z k F ı ȷ = k ȷ k = ı k ı = ȷ y ȷ F = q v B 19

20 Iclicker Question A positively charged particle moves in the positive z-direction. The magnetic force on the particle is in the positive y-direction. What can you conclude about the z-component of the magnetic field at the particle s position? A. B z > 0 B. B z = 0 C. B z < 0 D. not enough information given to decide v x ı z k F ı ȷ = k ȷ k = ı k ı = ȷ y ȷ F = q v B 20

21 Cyclotron Motion in Magnetic Field Motion along a circular orbit: F = q v B qqq = m v2 R R = mv qq alternatively, by measuring R, one can determine the ratio m/q if v (the kinetic energy) is known. T = 2πR v = 2πm qq - the period T is independent of v If a charge has a velocity component along B: 21

22 Large Hadron Collider R = mv qq m kg v = c B = 8T R = kg m/s C 8T 0.38m 8.6 km What s wrong? In this form, the equation works only for nonrelativistic motion. To correct, you need to replace the classical momentum mm with the relativistic one mm/ 1 v/c 2. 22

23 Mass Spectrometer mv 2 2 = qq qq = qqq v = 2qq v = E m B R = mv qq 23

24 Conclusion Magnetostatics: B B t Sources of B: motion of charges (currents), orbital motion of electrons in atoms, electron spins, time-dependent electric fields. Sources of B in classical magnetostatics: steady currents. Absence of Magnetic Monopoles: B da = 0 Force on charges moving in magnetic field: F = q v B. Next time: Lecture 14: Magnetic Forces on Currents

Chapter 27 Magnetic Field and Magnetic Forces

Chapter 27 Magnetic Field and Magnetic Forces - Magnetism - Magnetic Field - Magnetic Field Lines and Magnetic Flux - Motion of Charged Particles in a Magnetic Field - Applications of Motion of Charged