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

Transcription

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 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? 2

3 Sources of Magnetic Field Charges at rest do not generate B. What would be our best bet for the source of the magnetic field? magnetic point charge (a magnetic monopole): have not been observed yet (though its existence doesn t contradict anything) steady (time-independent) current: our best bet _ a moving single charge: the field is non-stationary, not good for magnetostatics + _ Sources of B: Charges in motion: currents, orbital motion of electrons in atoms. Units: tesla, T Ferromagnetic materials (electron spins purely quantum phenomenon). Time-dependent electric fields (we ll consider this source later in Electrodynamics). 3

4 Characteristic Magnetic Fields 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 4

5 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 5

6 Magnetic Field Lines Magnetic field is a non-conservative vector field. In general B dl 0 to be specified later Magnetic field lines are closed loops (no mag. monopoles): 6

7 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! 7

8 No Work Done by Magnetic Force! F = q v B The work done by the magnetic field on a moving charge = 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. An accelerated charge (e.g. moving with constant dd F dl dd = F dl = 0 speed along a curved trajectory) loses its energy by radiating electromagnetic waves. 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 12

9 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 13

10 Cyclotron Motion in Magnetic Field Motion along a circular orbit: F = q v B R = mv qq = qqq = m v2 R 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: 20

11 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. 21

12 Mass Spectrometer mv 2 2 = qq qq = qqq v = 2qq v = E m B R = mv qq 22

13 Conclusion Magnetostatics: B B t Sources of B: motion of charges (currents), orbital motion of electrons in atoms, electron spins, time-dependent electric fields. 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

14 Structure of the Course Electrostatics Electric fields generated by charges at rest Ch d dd = 0 0 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. 24

15 Cosmic Rays in the Earth s Magnetic Field 27

16 Experiment with the vacuum electron tube 28