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

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Forces between magnets Forces between magnets and magnetizable

1 Lecture 13. Magnetic Field, Magnetic Forces on Moving Charges. What are magnetic forces? Forces between magnets Forces between magnets and magnetizable materials Forces between magnets and moving charges (currents) Forces between moving charges (currents) Mediated by a magnetic field B.

2 Characteristic Magnetic Fields Units: tesla, T B 10 4 T rare-earth magnets B up to 1.4T Superconducting solenoids in MRI scanners B 1 2T Superconducting solenoids in the LHC B up to 8T Superconducting solenoids B up to 30T

Sources of Magnetic Field Charges at rest do not generate magnetic fields B.

Have never been observed - though their existence doesn t contradict any fundamental laws of

Time-dependent electric fields: we ll consider this later in electrodynamics.

3 Sources of Magnetic Field Charges at rest do not generate magnetic fields B. What are the sources of the magnetic field? Magnetic point charges (magnetic monopoles)? Have never been observed - though their existence doesn t contradict any fundamental laws of physics. Magnets (ferromagnetic materials): electron spins purely quantum phenomenon. Time-dependent electric fields: we ll consider this later in electrodynamics. Charges in motion: currents, orbital motion of electrons in atoms. In magnetostatics: B field independent of time Steady-state (time-independent) currents.

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

Flux of the Magnetic Field & Gauss Law Just like for

lines start/end at charges Magnetic field lines form

5 Flux of the Magnetic Field & Gauss Law Just like for the electric field, we define the magnetic flux through a surface: Units: Tm 2 =Weber, Wb Φ B = B da Compare with electric flux: Φ E = E da Electric field lines start/end at charges Magnetic field lines form closed loops Gauss law for closed surfaces: E da = Q ε 0 B da = 0

6 Force on a Charge Moving in Magnetic Field Charges moving in a magnetic field experience a transverse force: Lorentz force: Magnitude: F = q v B F = qvbsinφ Right-hand rule (for positive charges)

7 Force on a Charge Moving in Magnetic Field Lorentz force F = q v B F = 0 for charges moving along B F is max when v B Compare with electrostatic force: F = qe (defines the electric field) F = q v B (defines the magnetic field) Units: Tesla T = N C m s = Ns Cm = N Am Another unit: Gauss G = 10 4 T

You bring a permanent magnet toward the CRT with its north pole oriented upward.

8 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 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

10 Work Done by Magnetic Force F = q v B F dl dw = F dl = 0 The work done by the magnetic field on a moving charge = 0 You cannot increase the speed of charged particles using a magnetic field. The B-induced acceleration is not zero, it s perpendicular to the velocity a = dv dt.

11 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

12 Example An electron moves along the z axis. The B field is in the x-y plane. v = 1k m/s B = 2i + 3j T v z k What is the Lorentz force on the electron? x i B y j F = e v B = e 1k 2i + 3j = e 1k 2i + 1k 3j z k i j = k = e 2j 3i = e 3i 2j x i y j j k = i k i = j

13 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. 20iˆ+ 15 ˆj N 20iˆ 15 ˆj N 20iˆ+ 15 ˆj N 20iˆ15 ˆj N E. none of these z k v y j B x i i j = k j k = i k i = j F = q v B

14 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. Nothing much v x i z k F i j = k j k = i k i = j y j F = q v B

Cyclotron Motion in Magnetic Field The Lorentz force causes charged particles to move in a circular trajectory: F = q v B qvb = m v2 R R = mv qb By measuring R

15 Cyclotron Motion in Magnetic Field The Lorentz force causes charged particles to move in a circular trajectory: F = q v B qvb = m v2 R R = mv qb By measuring R and v, one can determine the ratio m/q. T = 2πR v = 2πm qb The period T is independent of v! If a charge has a velocity component along B, get a helical trajectory:

Large Hadron Collider R = mv qb m 1.6 10 27 kg v c B = 8T R = 1.6 10 27 kg 3 10 8 m/s 1.6 10 19 C 8T 0.38m 8.6 km What s wrong?

16 Large Hadron Collider R = mv qb 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 rest mass m with the relativistic mass m/ 1 v/c 2.

17 Mass Spectrometer mv 2 2 = qv qe = qvb v = 2qV v = E m B R = mv qb

Magnetic field and magnetic poles

Magnetic field and magnetic poles Magnetic Field B is analogically similar to Electric Field E Electric charges (+ and -)are in analogy to magnetic poles(north:n and South:S). Paramagnetism, Diamagnetism,