Chapter 28Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Chapter 28Magnetic Fields28.2: What Produces Magnetic Field?:One way that magnetic fields are produced is to use moving electrically charged particles, such as a current in a wire, to make an electromagnet. The current produces a magnetic field that is utilizable. The other way to produce a magnetic field is by means of elementary particles such as electrons, because these particles have an intrinsic magnetic field around them.The magnetic fields of the electrons in certain materials add together to give a net magnetic field around the material. Such addition is thereason why a permanent magnet, has a permanent magnetic field. In other materials, the magnetic fields of the electrons cancel out, giving no net magnetic field surrounding the material.28.3: The Definition of B:We can define a magnetic field, B, by firing a charged particle through the point at which is to be defined, using various directions and speeds for the particle and determining the force that acts on the particle at that point. B is then defined to be a vector quantity that is directed along the zero-force axis. The magnetic force on the charged particle, FB, is defined to be:Here q is the charge of the particle, v is its velocity, and B the magnetic field in the region. The magnitude of this force is then:Here  is the angle between vectors v and B.Right Hand Rule No. 1. Extend the right hand so the fingers pointalong the direction of the magnetic field and the thumb points alongthe velocity of the charge. The palm of the hand then faces in the direction of the magnetic force that acts on a positive charge.If the moving charge is negative,the direction of the force is oppositeto that predicted by RHR-1.28.3: Finding the Magnetic Force on a Particle:28.3: The Definition of B:The SI unit for B that follows is newton per coulomb-meter per second. For convenience, this is called the tesla (T):An earlier (non-SI) unit for B is the gauss (G), and28.3: Magnetic Field Lines:The direction of the tangent to a magnetic field line at any point gives the direction of B at that point.The spacing of the lines represents the magnitude of B —the magnetic field is stronger where the lines are closer together, and conversely.Example, Magnetic Force on a Moving Charged Particle :28.4: Crossed Fields, Discovery of an Electron:When the two fields in Fig. 28-7 are adjusted so that the two deflecting forces acting on the charged particle cancel, we haveThus, the crossed fields allow us to measure the speed of the charged particles passingthrough them. The deflection of a charged particle, moving through an electric field, E, between two plates, at the far end of the plates (in the previous problem) isHere, v is the particle’s speed, m its mass, q its charge, and L is the length of the plates.28.6: A Circulating Charged Particle:Consider a particle of charge magnitude |q| and mass m moving perpendicular to a uniform magnetic field B, at speed v.The magnetic force continuously deflects the particle, and since B and v are always perpendicular to each other, this deflection causes the particle to follow a circular path. The magnetic force acting on the particle has a magnitude of |q|vB.For uniform circular motionFig. 28-10 Electrons circulating in a chamber containing gas at low pressure (their path is the glowing circle). A uniform magnetic field, B, pointing directly out of the plane of the page, fills the chamber. Note the radially directed magnetic force FB ; for circular motion to occur, FB must point toward the center of the circle, (Courtesy John Le P.Webb, Sussex University, England)28.6: Helical Paths:Fig. 28-11 (a) A charged particle moves in a uniform magnetic field , the particle’s velocity v making an angle f with the field direction. (b) The particle follows a helical path of radius r and pitch p. (c) A charged particle spiraling in a nonuniform magnetic field. (The particle can become trapped, spiraling back and forth between the strong field regions at either end.) Note that the magnetic force vectors at the left and right sides have a component pointing toward the center of the figure.The velocity vector, v, of such a particle resolved into two components, one parallel to and one perpendicular to it:The parallel component determines the pitch p of the helix (the distance between adjacent turns (Fig. 28-11b)). The perpendicular component determines the radius of the helix. The more closely spaced field lines at the left and right sides indicate that the magnetic field is stronger there. When the field at an end is strong enough, the particle “reflects” from that end. If the particle reflects from both ends, it is said to be trapped in a magnetic bottle.Example, Helical Motion of a Charged Particle in a Magnetic Field:Example, Uniform Circular Motion of a Charged Particle in a Magnetic Field:28.8: Magnetic Force on a Current-Carrying Wire:28.8: Magnetic Force on a Current-Carrying Wire:Consider a length L of the wire in the figure. All the conduction electrons in this section of wire will drift past plane xx in a time t =L/vd.Thus, in that time a charge will pass through that plane that is given byHere L is a length vector that has magnitude L and is directed along the wire segment in the direction of the (conventional) current.If a wire is not straight or the field is not uniform, we can imagine the wire broken up into small straight segments . The force on the wire as a whole is then the vector sum of all the forces on the segments that make it up. In the differential limit, we can write and we can find the resultant force on any given arrangement of currents by integrating Eq. 28-28 over that arrangement.Example, Magnetic Force on a Wire Carrying Current:28.9: Torque on a Current Loop:The two magnetic forces F and –F produce a torque on the loop, tending to rotate it about its central axis.28.9: Torque on a Current Loop:To define the orientation of the loop in the magnetic field, we use a normal vector n that is perpendicular to the plane of the loop. Figure 28-19b shows a right-hand rule for finding the direction of n. In Fig. 28-19c, the normal vector of the loop is shown at an arbitrary angle  to the direction of the magnetic field.For side 2 the magnitude of the force acting on this side is F2=ibB sin(90°-)=ibB cos =F4.F2 and