Electricity and Magnetism Particle Accelerators

Transcription

1 Electricit and Magnetism Particle Accelerators Lana Sheridan De Anza College Feb 23, 2018

2 Last time charged particle in E and B fields applications of crossed fields discover of the electron Hall effect

3 Overview Hall effect eample cclotrons snchotrons

4 The Hall effect - eample v : question tron has charge q and is moving through a magnetic field with velocit, the magnetic force acting on the electron is given b Eq Because q is negative, the direction of of edge F : is length opposite d = the 1.5 cross cm, product moving in B : B v : the, which is in A solid metal cube, positive direction at a constant velocit v of magnitude 4.0 m/s. The cube moves through a uniform magnetic field B of magnitude T in the positive z direction. v F : B This is the crossproduct result. z B d (a) d d v B (b) This is the resulting electric field. The weak electric field creates a weak electric force. Which cube face is at a lower electric potential and which is at a higher electric potential because of the motion through the field? 1 Hallida, Resnick, Walker, 9th ed, page 743.

5 Reasoning: When the cube first begins to move through the magnetic field, its electrons do also. Because each electron has charge q and is moving through a magnetic field with velocit v :, the magnetic force F : B acting on the electron is given b Eq Because q is negative, the direction of F : is opposite the cross product B : B v :, which is in Free charges in the conductor will feel a force as the move along with the entire conductor through the field. The Hall effect - eample question The free charges are electrons. We have to find the direction of the force on them. v This is the crossproduct result. z B d (a) d d v B (b) This is the resulting electric field. The weak electric field creates a weak electric force.

6 er and lower electric potential? The Hall effect - eample question he electric Free charges field in thecreated conductor b will the feel charge a forceseparation as the move along roduces with the an entire electric conductor force F through : theon field. E qe : each electron E : The free charges are electrons. We have to find the direction of the force on them. is the magnetic e on an electron. KEY IDEAS Electrons are forced to the left face, leaving the right face positive. A v B F B (c) (d)

7 The Hall effect - eample v : question tron has charge q and is moving through a magnetic field with velocit, the magnetic force acting on the electron is given b Eq Because q is negative, the direction of of edge F : is length opposite d = the 1.5 cross cm, product moving in B : B v : the, which is in A solid metal cube, positive direction at a constant velocit v of magnitude 4.0 m/s. The cube moves through a uniform magnetic field B of magnitude T in the positive z direction. v F : B This is the crossproduct result. z B d (a) d d v B (b) This is the resulting electric field. The weak electric field creates a weak electric force. What is the potential difference between the faces of higher and lower electric potential? 1 Hallida, Resnick, Walker, 9th ed, page 743.

8 The Hall effect - eample question When does the potential difference between the faces stabilize?

9 v B The Hall effect - eample question When does the potential (c) difference between (d) the faces stabilize? (b) v B F B The weak electric field creates a weak electric force. More migration creates a greater electric field. The forces now balance. No more electrons move to the left face. E E F B F E (f ) (g) F B F E l cube moves at constant velocit through a uniform magnetic field. (b) F agnetic force acting on an electron forces the electron E = F to the B left face, d leaving the opposite face positive. (e) (f) The resulting weak electric force on the net electron, but it too is forced to the left face. Now (g) the (h) the electric force matches the magnetic force. (h)

10 v B The Hall effect - eample question When does the potential (c) difference between (d) the faces stabilize? (b) v B F B The weak electric field creates a weak electric force. More migration creates a greater electric field. The forces now balance. No more electrons move to the left face. E E F B F E (f ) (g) F B F E l cube moves at constant velocit through a uniform magnetic field. (b) F agnetic force acting on an electron forces the electron E = F to the B left face, d leaving the opposite face positive. (e) (f) The resulting weak electric force on the net electron, but it too is forced ( ee to ) = evb the left face. Now (g) the (h) the electric force matches the magnetic V force. d = vb V = vbd (h)

11 v B The Hall effect - eample question When does the potential (c) difference between (d) the faces stabilize? (b) v B F B The weak electric field creates a weak electric force. More migration creates a greater electric field. The forces now balance. No more electrons move to the left face. E E F B F E (f ) (g) F B F E l cube moves at constant velocit through a uniform magnetic field. (b) F agnetic force acting on an electron forces the electron E = F to the B left face, d leaving the opposite face positive. (e) (f) The resulting weak electric force on the net electron, but it too is forced ( ee to ) = evb the left face. Now (g) the (h) the electric force matches the magnetic V force. d = vb V = vbd V = 3.0 mv (h)

12 Related Effects the Hall effect in semiconductors - can be more comple! Depends on the material. the quantum Hall effect - can observe quantization of the Hall potential difference. Can be used to measure the charge of the electron.

13 The Lorentz Force A charged particle can be affected b both electric and magnetic fields. This means that the total force on a charge is the sum of the electric and magnetic forces: F = qe + qv B This total force is called the Lorentz force. This can alwas be used to deduce the electromagnetic force on a charged particle in E or B fields.

14 Accelerating Charged Particles High speed beams of particles are useful for studing nuclear and particle phsics. The can be trick to create, however. Charged particles can be accelerated with a potential difference.

15 The electron-volt (again) One convenient unit of energ for particles is the electron-volt, written ev. This is the amount of energ that an electron accelerated through a potential difference of 1 Volt has. U = qv 1 ev = ( C)(1 V) = J.

16 Accelerating Charged Particles The acceleration of a particle from a potential difference depends on its mass. Suppose there is a practical limit on how strong an electric field can be created, E. The force on the particle is: F = qe The acceleration can be deduced from Newton s second law: a = F m = qe m The final velocit of a particle with this acceleration will be: v 2 f = 2ad = qed m If m is large than the accelerating distance d must be also. For protons the value of d necessar becomes impractical.

17 Cclotrons One wa around this is to have the particles move in a circle. The acceleration can take place in a limited space. Magnetic fields cause the protons to follow circular arcs. The protons are directed repeatedl through a potential difference that accelerates them. The time period of the orbit does not depend on the velocit of the particles! T = 2πm q B

18 Cclotrons Beam The protons spiral outward in a cclotron, picking up energ in the gap. Dee Deflector plate Oscillator Dee Fig The elements of a cclotron, showing the particle source S and the dees. A uniform magnetic field is directed f = up f osc from = q B 2πm the plane of the page. Circulating protons spiral S accelerated to th differences. Beca be done in a reas particles) have g long. A clever solu particles move th amount of energ and move throu repeated thousan Here we disc bring particles ba energ until the The Cclotron Figure is a (protons, sa) ci straight edge) are an electrical oscil gap between the

19 Cclotrons High V Beam The protons spiral outward in a cclotron, picking up energ in the gap. Dee Deflector plate Oscillator Dee Fig The elements of a cclotron, showing the particle source S f = f and the dees. A uniform osc = q B 2πm magnetic field is directed up from the plane of the page. Circulating protons spiral S Low V make and contro accelerated to th differences. Beca be done in a reas particles) have g long. A clever solu particles move th amount of energ and move throu repeated thousan Here we disc bring particles ba energ until the The Cclotron Figure is a (protons, sa) ci straight edge) are an electrical oscil

20 Cclotrons Low V Beam The protons spiral outward in a cclotron, picking up energ in the gap. Dee Deflector plate Oscillator Dee Fig The elements of a cclotron, showing the particle source S f = f and the dees. A uniform osc = q B 2πm magnetic field is directed up from the plane of the page. Circulating protons spiral S High V make and contro accelerated to th differences. Beca be done in a reas particles) have g long. A clever solu particles move th amount of energ and move throu repeated thousan Here we disc bring particles ba energ until the The Cclotron Figure is a (protons, sa) ci straight edge) are an electrical oscil

21 Cclotrons High V Beam The protons spiral outward in a cclotron, picking up energ in the gap. Dee Deflector plate Oscillator Dee Fig The elements of a cclotron, showing the particle source S f = f and the dees. A uniform osc = q B 2πm magnetic field is directed up from the plane of the page. Circulating protons spiral S Low V make and contro accelerated to th differences. Beca be done in a reas particles) have g long. A clever solu particles move th amount of energ and move throu repeated thousan Here we disc bring particles ba energ until the The Cclotron Figure is a (protons, sa) ci straight edge) are an electrical oscil

22 Circular Motion of a Charge To find the radius: F net = F c = F B r = mv q B So, v = qbr/m. When the ion eits the cclotron, it will have kinetic energ: K = 1 2 mv 2 K = (qbr)2 2m

23 Cclotron ating V The first cclotron was built in ter being celerated, the rticles eit here. h pole of magnet Lawrence Berkele National Lab b The world s largest cclotron has a maimum beam radius of at P, two dees D 1 and D 2 across which an alternating potential differth pole of the magnet is not shown.) (b) The first cclotron, invented b 1 Photo from Lawrence Berkele National 7.9 m. Laborator.

24 Snchrotron Once the charged particles reach 10% of the speed of light this stops working. This is because the effective mass of the particles is increasing, so f osc = 2πm q B is no longer a constant. Also, at these speeds the area of the magnetic field for a cclotron must be quite big as the radius of the path becomes large. A solution to this is the snchrotron.

25 Snchrotron Snchrotrons operate similarl to cclotrons, but the frequenc of the potential switching can var. 1 Figure from schoolphsics.co.uk.

26 Snchrotron This also means that the particles can be kept on a single loop, even as their velocit increases. The magnetic field onl has to cover the ring itself. (Not the area in the middle of the ring.)

27 Snchrotron This also means that the particles can be kept on a single loop, even as their velocit increases. The magnetic field onl has to cover the ring itself. (Not the area in the middle of the ring.) The LHC (Large Hadron Collider) at CERN is a tpe of snchrotron. The Tevatron at Fermilab was also one, but it has been shutdown due to Congressional budget cuts.

28 Snchrotron 1 Photo copright c Snchrotron Soleil, used with permission

29 Summar Hall effect tpe question cclotrons snchotrons Homework Serwa & Jewett: Ch 29, Problems: 27 (in part (a), the tetbook answer corresponds to ω, not f ), 33, 35