Review The force that a magnetic field exerts on a charge moving with velocity v is given by FB qv B Physics for Scientists Engineers 2 The magnitude of the force exerted by a magnetic field on a moving charge is Spring Semester 2005 Lecture 21 FB qvBsin If the charge moves perpendicular to the magnetic field then F qvB February 20 2005 Physics for Scientists Engineers 2 1 Review 2 February 20 2005 Physics for Scientists Engineers 2 Orbits in a Constant Magnetic Field The unit of magnetic field strength the tesla T Ns N 1T 1 1 Cm Am Consider the situation in which you tie a string to a rock and twirl it at constant speed in a circle over your head Another unit of magnetic field strength that is often used but is not an SI unit is the gauss G The tension on the string always points to the center of the circle and creates a centripetal acceleration 1 G 10 4 T The tension of the string provides the centripetal force that keeps the rock moving in a circle A similar physical situation occurs when a particle with charge q and mass m moves with velocity v perpendicular to a uniform magnetic field B 10 kG 1 T Typically the Earth s magnetic field is about 0 5 G at the surface In this case 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 The NSCL K1200 superconducting cyclotron has a magnetic field of 5 5 T February 20 2005 Physics for Scientists Engineers 2 2 3 February 20 2005 Physics for Scientists Engineers 2 4 1 Orbits in a Constant Magnetic Field 2 Example Moving Electrons For motion perpendicular to the magnetic field the force required to keep the particle moving in a circle with radius r is the centripetal force F In this photo an electron beam is accelerated by an electric field After acceleration the electrons move in a circle perpendicular to the constant magnetic field created by a pair of Helmholtz coils mv 2 r Setting this centripetal force equal to the magnetic force we obtain vBq F mv 2 r v Rearranging we get an expression for the radius of the circle in which the particle is traveling r Is the magnetic field into the page or out of the page mv qB February 20 2005 The magnetic field is out of the page Physics for Scientists Engineers 2 5 Mass Spectrometer In a magnetic spectrometer the magnetic field is fixed and the radius of curvature of the trajectory of the particles is measured by various means such as a photographic plate or electronic detectors Physics for Scientists Engineers 2 6 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 Remembering that the momentum of a particle is p mv we can write In these collisions many particles are created that stream away from the interaction point at high speeds p q The fixed magnetic field implies that all particles with the same radius of curvature have the same ratio of momentum to charge A simple particle detector is not sufficient to measure and identify these particles Thus by measuring the charge and the radius of the trajectory of a charged particle moving in a constant magnetic field one can measure the momentum or energy of the particle February 20 2005 February 20 2005 Example The Time Projection Chamber One common application of moving charged particles in magnetic field is a mass spectrometer Br Remember that the magnetic force on an electron is opposite that on a proton Physics for Scientists Engineers 2 A device that can help physicists study these collisions is the time projection chamber TPC 7 February 20 2005 Physics for Scientists Engineers 2 8 2 Example The Time Projection Chamber 2 Example The Time Projection Chamber 3 The STAR TPC consists of a large cylinder filled 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 The produced charged particles have a component of their velocity that is perpendicular to the magnetic field and thus have circular trajectories when viewed endon 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 endcaps of the TPC where they are recorded electronically From the radius of curvature one can extract the particles momenta Using the drift time and the recording positions the computer software reconstructs the trajectories that the produced particles took through the TPC February 20 2005 Physics for Scientists Engineers 2 The TPC sits inside a giant solenoid magnet shown to the right with the magnetic field pointing along the beam direction 9 February 20 2005 Events in the TPC Physics for Scientists Engineers 2 10 Example Momentum of a Track Here are two events in the STAR TPC at Brookhaven National Lab Calculate the momentum of this track p q p qBr r 2 3 m Br Magnetic field is directed into the screen r 2 3 m B 0 50 T q 1 6 10 19 C p 1 8 10 19 kg m s Let s analyze this track 1 MeV 1 602 10 13 J pc 1 8 10 19 kg m s 3 0 10 8 m s pc 5 53 10 11 J p 345 MeV c nuclear high energy physics units February 20 2005 Physics for Scientists Engineers 2 11 February 20 2005 Physics for Scientists Engineers 2 12 3 Orbits in a Constant Magnetic Field Cyclotrons 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 T A cyclotron is a particle accelerator The golden D shaped pieces of metal in the upcoming video 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 2 r 2 m v qB From the period we can get the frequency and angular frequency f 1 qB T 2 m 2 f The resulting electric field accelerates the particle qB m Because the cyclotron sits in a strong magnetic field the trajectory is curved The frequency of the rotation is independent of the speed of the particle The radius of the trajectory is proportional to the momentum so the accelerated particle spirals outward Isochronous orbits Basis for cyclotrons February 20 2005 Physics for
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