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PHY 184. Week 6 Spring 2007 Lecture 22. Title: The Lorentz Force = q v x B. Announcements. Homework Set 5 is due next Tuesday at 8:00 am. The correction set to Midterm 1 is due next Tuesday at 6:00pm. Review.

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phy 184

PHY 184

Week 6 Spring 2007

Lecture 22

Title: The Lorentz Force = q v x B

184 Lecture 22

announcements
Announcements
  • Homework Set 5 is due next Tuesday at 8:00 am.
  • The correction set to Midterm 1 is due next Tuesday at 6:00pm.

184 Lecture 22

review
Review
  • The force that a magnetic field exerts on a charge moving with velocity v is given by
  • The direction is sideways (RH rule).
  • The magnitude of the force
  • If the charge moves perpendicular to the magnetic field then

184 Lecture 22

review 2
Review (2)
  • The unit of magnetic field strength the tesla (T)
  • Another unit of magnetic field strength that is often used but is not an SI unit is the gauss (G)
  • Typically the Earth’s magnetic field is about 0.5 G at the surface.

184 Lecture 22

example cathode ray tube
Example: Cathode Ray Tube
  • Consider the cathode-ray tube used for lecture demonstrations.
  • In this tube electrons form an electron beam when accelerated horizontally by a voltage of 136 V in an electron gun.
  • The mass of an electron is 9.109410-31 kg while the elementary charge is 1.602210-19 C.
  • (a) Calculate the velocity of the electrons in the beam after leaving the electron gun.

184 Lecture 22

cathode ray tube 2
Cathode Ray Tube (2)
  • (b) If the tube is placed in a uniform magnetic field, in what direction is the electron beam deflected?
  • (c) Calculate the magnitude of acceleration of an electron if the field strength is 3.65×10-4T.

downward

184 Lecture 22

particle orbits in a uniform b
Particle Orbits in a Uniform B
  • Tie a string to a rock and twirl it at constant speed in a circle over your head.
  • The tension of the string provides the centripetal force that keeps the rock moving in a circle.
  • The tension on the string always points to the center of the circle and creates a centripetal acceleration.
  • A particle with charge q and mass m moves with velocity v perpendicular to a uniform magnetic field B.
  • The particle will move in a circle with a constant speed v and the magnetic force F = qvB will keep the particle moving in a circle.

184 Lecture 22

particle orbits in uniform b 2
Particle Orbits in Uniform B (2)
  • Recall centripetal acceleration
  • Newton;’s second law
  • So, for charged particle q in circular motion in magnetic field B

a = v2/r

F = m a

184 Lecture 22

example moving electrons
Example - Moving Electrons

F

v

  • In this photo, an electron beam is initially accelerated by an electric field.
  • Then the electrons move in a circle perpendicular to the constant magnetic field created by a pair of Helmholtz coils.

Is the magnetic field into the page or out of the page?

(Remember that the magnetic force on an electron is opposite that on a proton.)

The magnetic field is out of the page.

184 Lecture 22

clicker question
Clicker Question
  • The figure shows the circular paths of two particles that travel at the same speed in a uniform magnetic field (directed into the page). One particle is a proton and the other is an electron. Which particle follows the smaller circle?

A) the electron

B) the proton

C) Both, proton and electron travel along on the same circle

184 Lecture 22

clicker question11
Clicker Question
  • The figure shows the circular paths of two particles that travel at the same speed in a uniform magnetic field (directed into the page). One particle is a proton and the other is an electron. Which particle follows the smaller circle?

A) the electron

… for same speeds, r is proportional to m

184 Lecture 22

example mass spectrometer 1
Example: Mass Spectrometer (1)
  • Suppose that B=80mT, V=1000V. A charged ion (1.6022 10-19C) enters the chamber and strikes the detector at a distance x=1.6254m. What is the mass of the ion?
  • Key Idea: The uniform magnetic field forces the ion on a circular path and the ion’s mass can be related to the radius of the circular trajectory.

184 Lecture 22

example mass spectrometer 2
Example: Mass Spectrometer (2)
  • Suppose that B=80mT, V=1000V. A charged ion (1.6022 10-19C) enters the chamber and strikes the detector at a distance x=1.6254m. What is the mass of the ion?
  • From the figure: r=0.5x
  • Also need the velocity v after the ion is accelerated by the potential difference V

Now solve for m

3.4x10-25 kg

184 Lecture 22

example the time projection chamber
Example: The Time Projection Chamber
  • In high-energy nuclear physics, new forms of matter are studied by colliding gold nuclei at very high energies
  • In particle physics, new elementary particles are created and studied by colliding protons and anti-protons at the highest energies
  • In these collisions, many particles are created that stream away from the interaction point at high speeds
  • A simple particle detector is not sufficient to measure and identify these particles
  • A device that can help physicists study these collisions is the time projection chamber (TPC)

184 Lecture 22

example the time projection chamber 2
Example: The Time Projection Chamber (2)
  • The STAR TPC consists of a large cylinder filled with a carefully chosen gas (90% argon, 10% methane) that allows free electrons to drift without recombining
  • As created charged particles pass through the gas, the particles ionize the atoms of the gas, releasing free electrons
  • An electric field is applied between the center of the TPC and the caps of the cylinder that exerts an electric force on these freed electrons, making them drift to the end-caps of the TPC, where they are recorded electronically
  • Using the drift time and the recording positions, the computer software reconstructs the trajectories that the produced particles took through the TPC

184 Lecture 22

example the time projection chamber 3
Example: The Time Projection Chamber (3)
  • The TPC sits inside a giant solenoid magnet shown to the right with the magnetic field pointing along the beam direction
  • The produced charged particles have a component of their velocity that is perpendicular to the magnetic field and thus have circular trajectories when viewed end-on
  • From the radius of curvature one can extract the particle momentum

184 Lecture 22

events in the tpc
Events in the TPC
  • Here are two events in the STAR TPC at Brookhaven National Lab
  • Magnetic field is directed into the screen

Let’s analyze this track

184 Lecture 22

example momentum of a track
Example - Momentum of a Track
  • Calculate the momentum of this track

r = 2.3 m

184 Lecture 22

orbits in a constant magnetic field
Orbits in a Constant Magnetic Field
  • If a particle performs a complete circular orbit inside a constant magnetic field, then the period of revolution of the particle is just the circumference of the circle divided by the speed
  • From the period we can get the frequency and angular frequency
  • The frequency of the rotation is independent of the speed of the particle.

Isochronous orbits.

Basis for the cyclotron.

184 Lecture 22

cyclotrons
Cyclotrons
  • A cyclotron is a particle accelerator
  • The D-shaped pieces (descriptively called “dees”) have alternating electric potentials applied to them such that a positively charged particle always sees a negatively charged dee ahead when it emerges from under the previous dee, which is now positively charged
  • The resulting electric field accelerates the particle
  • Because the cyclotron sits in a strong magnetic field, the trajectory is curved
  • The radius of the trajectory is proportional to the momentum, so the accelerated particle spirals outward

184 Lecture 22

example deuteron in cyclotron
Example: Deuteron in Cyclotron
  • Suppose a cyclotron is operated at frequency f=12 MHz and has a dee radius of R=53cm. What is the magnitude of the magnetic field needed for deuterons to be accelerated in the cyclotron (m=3.34 10-27kg)?
  • Key Idea: For a given frequency f, the magnetic field strength, B, required to accelerate the particle depends on the ratio m/q (or mass to charge):

184 Lecture 22

special clicker
Special Clicker
  • Suppose a cyclotron is operated at frequency f=12 MHz and has a dee radius of R=53cm. What is the kinetic energy of the deuterons in this cyclotron when they travel on a circular trajectory with radius R (m=3.34 10-27kg, B=1.57 T)?

A) 0.9 10-14 J

B) 8.47 10-13 J

C) 2.7 10-12 J

D) 3.74 10-13 J

184 Lecture 22

k500 superconducting cyclotron
K500 Superconducting Cyclotron
  • Movie from Nova program “The Nucleus Factory”

184 Lecture 22