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introduce the concept of magnetic field properly, let’s review our formulation of interactions in Chapter 22, where we introduced the concept of electric field. Wanted electric interactions in two steps: A distribution of electric charge at rest creates an electric field E in the surrounding space. The electric field exerts a force F – q E on any other charge q that is present in the field. can describe magnetic interactions in a similar way: • moving charge or a current creates a magnetic field in the surrounding space (in addition to its electric field). be magnetic field exerts ‘a force F on any other moving charge or current that is present in the field. In this chapter we’ll concentrate on the second aspect of the interaction: Given the of a magnetic field, what force does it exert on a moving charge or a current? ter 29 we will come back to the problem of how magnetic fields are created by – g charges and currents. – e electric field, magnetic field is a vector field, that is, a vector quantity associate with each point in we will use the symbol B for magnetic field. At any ‘on the direction of B is defined as that in which the north pole of a compass nee-tends to point. The arrows in Fig. 28-3 suggest the direction of the earth’s magnetic .• for any magnet, Points out of its north pole and into its south pole. ‘hat are the characteristics of the magnetic force on a moving charge? First, its is proportional to the magnitude of the charge. If a 1-IlC charge and a 2-IlC _ move through a given magnetic field with the same velocity, the force on the charge is twice as great as that on the 1-IlC charge. The magnitude of the force is  the magnitude, or “strength,” of the field; if we double the magnitude field (for example, by using two identical bar magnets instead of one) without . g the charge or its velocity, the force doubles. magnetic force also depends on the particle’s velocity. This is quite different the electric-field force, which is the same whether the charge is moving or not. A particle at rest experiences no magnetic force. Furthermore, the magnetic force not have the same direction as the magnetic field B but instead is always per to both B and the velocity v.The magnitude F of the force is found to be. to the component of Ii perpendicular to the field; when that component is that is, when v and Bare,the force is zero.



Figure 28-5 shows these relationships. The direction of F is always perpendicular to the plane containing ii and 8. Its magnitude is given by where if is the magnitude of the charge and ¢ is the angle measured from the direction of ii to the direction of 8, as shown in the figure. This description does not specify the direction of F completely; there are always two directions, opposite to each other, that are both perpendicular to the plane of viand 8. To complete the description, we use the same right-hand rule that we used to define the vector

product in Section 1-11. (It would be a good idea to review that section before you go on.) Draw the vectors ii and with their tails together. Imagine turning ii until it points in the direction of 8(turning through the smaller of the two possible angles). Wrap the fingers of your right hand around the line perpendicular to the plane of ii and 8, as shown in Fig. 28-5, so that they curl around with the sense of rotation from ii to 8. Your thumb then points in the direction of the force F on a positive charge. (Alternatively, the direction of the force F on a positive charge is the direction in which a right-hand-thread screw would advance if turned the same way.) This discussion shows that the force on a charge q moving with velocity ii in a magnetic field 8is given, both in magnitude and in direction, by field relationships. It’s important to note that Eq. (28-2) was not deduced theoretically; it is an observation based on experiment. . Equation (28-2) is valid for both positive and negative charges. When q is negative, the direction of the force F is opposite to that of iiX . If two charges with equal magnitude and opposite sign move in the same field with the same velocity (Fig. 28-6), the forces have equal magnitude and opposite direction. Figures 28-5 and 28-6  how several examples of the relationships of the directions of F, ii,and for both positive and negative charges. Be sure you understand the relationships shown in these figures. Equation (28-1) gives the magnitude of the magnetic force Fin Eq. (28-2). We can express this magnitude in a different but equivalent way. Since ¢ is the angle between the directions of vectors ii and 8, we may interpret B sin ¢ as the component of 8 perpendicular to ii,that is, B.1.. With this notation the force magnitude is

This form is sometimes more convenient, especially in problems involving currents rather than individual particles. We will discuss forces on currents later in this chapter. From Eq. (28-1) the units of B must be the same as the units of Flu. Therefore the SI unit of B is equivalent to 1N  siC  m, or, since one ampere is one coulomb per second (1 A = 1CIs), 1 N/A· m. This unit is called the teals (abbreviated T), in honor of Nikolai Tesla (1857-1943), the prominent Serbian-American scientist and inventor:

The unit of B, the gauss (1 G = 10-4T), is also in common use. Instruments for measuring magnetic field are sometimes called gausses meters.
The magnetic field of the earth is of the order of 10-4T or 1 G. Magnetic fields of the order of 10 T occur in the interior of atoms and are important in the analysis of atomic spectra. The largest steady magnetic field that can be produced at present in the laboratory is 45 T. Some pulsed-current electromagnets can produce fields of the order of 120T for short time intervals of the order of a millisecond. The magnetic field at the surface of a neutron star is believed to be of the order of 108 T.


When a charged particle moves through a region of space where both electric and electric fields are present. both fields exert forces on the particle. The total force F is vector sum of the electric and magnetic forces:



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