Physics 123A,B Waves and Modern Physics John G. Cramer Professor of Physics E&M Review 1 (T&M: Ch. 26) Magnetic Fields and Forces March 31, 2008 Physics 123C - Lecture 1 2 Ampere?s Experiment André Marie Ampère (1775 ? 1836) When Ampere heard of Oersted?s results showing that a current produced a magnetic effect, he reasoned that a current might also respond to a magnetic effect. Therefore, he measured the force between two parallel current-carrying wires. He found that parallel currents create an attraction between the wires, while anti-parallel currents create repulsion. March 31, 2008 Physics 123C - Lecture 1 3 The Magnetic Force A current consists of moving charges. Ampere?s experiment implies that a magnetic field exerts a force on a moving charge. This is true, although the exact form of the force relation was not discovered until later in the 19 th century. The force depends on the relative directions of the magnetic field and the velocity of the moving charge, and is perpendicular to both.. March 31, 2008 Physics 123C - Lecture 1 4 Magnetic Force on Moving Charges Properties of the magnetic force: 1.? Only moving charges experience the magnetic force. There is no magnetic force on a charge at rest (v=0) in a magnetic field. 2.? There is no magnetic force on a charge moving parallel (?=0°) or anti-parallel (?=180°) to a magnetic field. 3.? When there is a magnetic force, it is perpendicular to both v and B. 4.? The force on a negative charge is in the direction opposite to vxB. 5.? For a charge moving perpendicular to B (?=90°) , the magnitude of the force is F=|q|vB. March 31, 2008 Physics 123C - Lecture 1 5 Motion of a Point Charge in a Magnetic Field March 31, 2008 Physics 123C - Lecture 1 6 Magnetic Forces on Current-Carrying Wires When a wire carries a current that is parallel or anti-parallel to a magnetic field, there is no force (because the charges move along field lines). When a wire carries a current that is perpendicular to a magnetic field, there is a force on the wire perpendicular to the current and field. March 31, 2008 Physics 123C - Lecture 1 7 Example: Force on a Straight Wire A 3.0 mm long segment of wire carries a current of 3.0 A in the +x direction. It lies in a 0.020 T magnetic field that is in the xy plane and makes an angle of 30° with the +x axis. What is the magnetic force exerted on the wire segment? March 31, 2008 Physics 123C - Lecture 1 8 Example: Force on a Bent Wire A wire bend in a semicircular loop of radius R lies in the xy plane and carries a current I from point a to point b. Throughout the region there is a uniform magnetic field B along the z axis and perpendicular to the loop. Find the magnetic force acting on the semicircular loop section of the wire. March 31, 2008 Physics 123C - Lecture 1 9 Circular Motion in a B Field Consider a positive charged particle with mass m and charge q moving at velocity v perpendicular to a uniform magnetic field B. The particle will move in a circular path of radius r cyc because of the force F on the particle, which is: March 31, 2008 Physics 123C - Lecture 1 10 Example: The Radius of Circular Motion An electron is accelerated from rest through a potential of 500 V, then injected into a uniform magnetic field B. Once in the magnetic field, it completes a half revolution in 2.0 ns. What is the radius of the orbit? March 31, 2008 Physics 123C - Lecture 1 11 Example: Circular Period of a Proton A proton (m p = 1.67x10 -27 kg, q = 1.60x10 -19 C) moves in a circle of radius r = 21.0 cm perpendicular to a magnetic field B = 4000 G. (a) What is the speed of the proton? (b) What is the period of motion? March 31, 2008 Physics 123C - Lecture 1 12 Motion of Particles in Uniform B Fields If B is along z, then in general the x-y motion of a charged particle is a circle and the z motion linear translation. Therefore, the trajectory is a helix. March 31, 2008 Physics 123C - Lecture 1 13 Motion of Particles in Non-Uniform B Fields When a non-uniform B ?pinches?, there is force component on a charged particle (independent of charge) in the direction away from the pinch. This produces a ?magnetic bottle? that can trap and contain charged particles. The pinches in the Earth?s field near the poles trap charged particles from the Sun and form the Van Allen radiation belts. March 31, 2008 Physics 123C - Lecture 1 14 A Velocity Selector Consider a charged particle moving to the right in a region containing a downward electric field E and an inward magnetic field B. There is a velocity such that the electric and magnetic forces on the particle are equal and opposite and the particle moves in an undeflected straight line. Thus, crossed E and B fields form a ?velocity selector?. March 31, 2008 Physics 123C - Lecture 1 15 Thopmson?s e/m Measurement March 31, 2008 Physics 123C - Lecture 1 16 Example: Electron Beam Deflection Electrons pass undeflected through the plates of Thompson?s apparatus then the electric field is 3000 V/m and there is a crossed magnetic field of 0.140 mT. The plates are 4.00 cm long and the ends of the plates are 30.0 cm from the screen. What is the deflection ?y on the screen when the magnetic field is turned off? March 31, 2008 Physics 123C - Lecture 1 17 The Mass Spectrometer March 31, 2008 Physics 123C - Lecture 1 18 Example: Separating Isotopes of Nickel A 58 Ni ion with a charge q = +e and a mass M = 9.62x10 -26 kg is accelerated through a potential drop of 3.00 kV and deflected in a 0.120 T magnetic field. (a) What is the radius of curvature of the ion?s orbit? (b) What is the difference in radii between 58 Ni ions and 60 Ni ions (assuming a mass ratio of 58:60)? March 31, 2008 Physics 123C - Lecture 1 19 The Cyclotron Accelerator Because when a charged particle moves in a uniform field, f cyc is independent of both radius and energy, one an ?pump? energy into the particle as it cycles, using an electric field that varies with the frequency f cyc . This is the basic principle of the cyclotron, a particle accelerator that can increase the energies of charged particles. The ultimate energy of a cyclotron is determined by the radius r max of the magnet. March 31, 2008 Physics 123C - Lecture 1 20 Example: Energy of Accelerated Protons A cyclotron for accelerating protons (m p = 1.67x10 -27 kg, q = 1.60x10 -19 C) has a magnetic field B = 0.150 T and a maximum radius r max = 0.500 m. (a) What is the cyclotron frequency? (b) What is kinetic energy of the protons when they emerge? March 31, 2008 Torques on Current Loops Consider the forces on a current loop carrying current I that is a square of length L on a side that is in a uniform magnetic field B. Its area vector makes an angle ? with B. The Hall Effect When a charged particle moving in a vacuum, it is deflected perpendicular to its velocity by a magnetic field. In 1879, Edwin Hall, a graduate student at Johns Hopkins Univ., discovered that the same behavior is true of charged particles moving in a conductor. Edwin Herbert Hall (1855 ? 1938) Example: Hall Probe Measurement of Magnetic Field A Hall probe consists of a strip of metallic bismuth that is 0.15 mm thick and 5.0 mm wide. Bismuth is a poor conductor with a charge carrier density of n = 1.35x10 25 m -3 . The Hall voltage on the probe is measured to be 2.5 mV when the current is 1.5 A. What is the magnetic field, and what is the electric field inside the bismuth? Paul
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