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Magnetic Fields Chapter 26 26.2 The force exerted by a magnetic field Definition of B 26.3 Motion of a charged particle in a magnetic field Applications A circulating charged particle Crossed fields: discovery of the electron The cyclotron and mass spectrometer 26.4 Today Magnetic force on currents Using integration The Hall effect, Hall potential 26.4 Magnetic force on a current-carrying wire Wire segment of length L carrying current I. If the wire is in a magnetic field, there will be a force on each charge carrier resulting in a force on the wire. 26.4 Magnetic force on a current-carrying wire Flexible wire passing between pole faces of a magnet. (a) no current in wire (b) upward current (c) downward current CHECKPOINT: the figure shows a current I through a wire in a uniform magnetic field B, as well as the magnetic force FB acting on the wire. The field is oriented so that the force is maximum. In what direction is the field? A: upwards (+y) B: downwards (-y) Answer: downwards (or –y direction) C: -z direction EXERCISE: A wire segment 3 mm long carries a current of 3 A in the +x direction. It lies in a magnetic field of magnitude 0.02 T that is in the xy plane and makes an angle of 30° with the +x direction, as shown. What is the magnetic force exerted on the wire segment? Torque on a current loop The elements of an electric motor. The current-carrying loop is free to rotate in the magnetic field. Magnetic forces produce a torque that rotates the loop. The direction of the current is reversed every half revolution so that the torque always acts in the same direction. If the conductor is not straight, we can divide it into infinitesimal segments dl. The force on each segment is Then we can integrate this expression along the wire to find the total force on a conductor of any shape. EXAMPLE: A wire bent in a semicircular loop of radius R lies in the xy plane. It carries a current I from point a to point b. There is a uniform magnetic field B perpendicular to the plane of the loop. Find the force acting on the semicircular loop part of the wire. Picture the problem: The force dF exerted on a segment of the semicircular wire lies in the xy plane. We find the total force by expressing the x and y components of dF in terms of and integrating them separately from = 0 to = More examples will be done on Wednesday after the Workshop The Hall Effect – the separation of charge in a conducting wire It results in a voltage VH called the Hall voltage. The sign of the charge carriers can be determined by measuring the sign of the Hall voltage, and the number of carriers per unit volume from the magnitude of VH. The Quantum Hall Effects In 1980, while studying the Hall effect in semiconductors at very low temperatures and very large magnetic fields, Klaus von Klitzing discovered that the Hall voltage is quantized. (Nobel Prize in physics, 1985.) Further reading Tipler p 847 Sources of the Magnetic Field Next The magnetic field of moving charges The magnetic field of currents Oersted’s experiment: he showed that a compass needle is deflected by an electric current. no current current flows Magnetic field lines Magnetic field B can be represented by field lines, and as with electric field lines, the direction of the field is indicated by the direction of the field line, and the magnitude of the field is indicated by the density of lines. There are two important differences: 1. Electric field lines are in the direction of the electric force on a positive charge, but magnetic field lines are perpendicular to the magnetic force on a moving charge. 2. Electric field lines begin on positive charges and end on negative charges; magnetic field lines neither begin nor end.