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Sears and Zemansky’s University Physics

Sears and Zemansky’s University Physics. 28 Magnetic of Field and Magnetic Forces 28-1 INTRODUCTION

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Sears and Zemansky’s University Physics

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  1. Sears and Zemansky’sUniversity Physics

  2. 28 Magnetic of Field and Magnetic Forces 28-1 INTRODUCTION The most familiar aspects of magnetism are those associated with permanent magnets, which attract unmagnetized iron objects and can also attract or repel other magnets. A compass needle aligning itself with the earth's magnetism is an example of this interaction. As we did for the electric force, we will describe magnetic forces using the concept of a field. A magnetic field is established by a permanent magnet, by an electric current in a conductor, or by other moving charges. This magnetic field, in turn, exerts forces on moving charges and current-carrying conductors. In this chapter we study the magnetic forces and torques exerted on moving charges and currents by magnetic fields.

  3. 28-2 MAGNETISM Permanent magnets were found to exert forces on each other as well as on pieces of iron that were not magnetized. It was discovered that when an iron rod is brought in contact with a natural magnet, the rod also becomes magnetized. When such a rod is floated on water or suspended by a string from its center, it tends to line itself up in a north-south direction. The needle of an ordinary compass is just such a piece of magnetized iron. Before the relation of magnetic interactions to moving charges was understood, the interactions of permanent magnets and compass needles were described in terms of magnetic poles. If a bar-shaped permanent magnet, or bar magnet, is free to rotate, one end points north. This end is called a north pole or N-pole; the other end is a south pole or S-pole. Opposite poles attract each other, and like poles repel each other (Fig. 28-1).

  4. The earth itself is a magnet. Its north geographical pole is close to a magnetic south pole, which is why the north pole of a compass needle points north.

  5. The earth's magnetic axis is not quite parallel to its geographic axis (the axis of rotation), so a compass reading deviates somewhat from geographic north. This deviation, which varies with location, is called magnetic declination or magnetic variation. Also, the magnetic field is not horizontal at most points on the earth's surface; its angle up or down is called magnetic inclination. At the magnetic poles the magnetic field is vertical. The concept of magnetic poles may appear similar to that of electric charge, and north and south poles may seem analogous to positive and negative charge. But the analogy can be misleading. While isolated positive and negative charges exist, there is no experimental evidence that a single isolated magnetic pole exists; poles always appear in pairs. If a bar magnet is broken in two, each broken end becomes a pole.

  6. The existence of an isolated magnetic pole, or magnetic monopole, would have sweeping implications for theoretical physics. Extensive searches for magnetic monopoles have been carried out, but so far without success. The first evidence of the relationship of magnetism to moving charges was discovered in 1819 by the Danish scientist Hans Christian Oersted. He found that a compass needle was deflected by a current-carrying wire, as shown in Fig. 28-4. Similar investigations were carried out in France by Andr6 Ampere. A few years later, Michael Faraday in England and Joseph Henry in the United States discovered that moving a magnet near a conducting loop can cause a current in the loop. We now know that the magnetic forces between two bodies shown in Figs. 28-1 and 28-2 are fundamentally due to interactions between moving electrons in the atoms of the bodies (above and beyond the electric interactions between these charges).

  7. Inside a magnetized body such as a permanent magnet, there is a coordinated motion of certain of the atomic electrons; in an unmagnetized body these motions are not coordinated. e q 

  8. 28-3 MAGNETIC FIELD • To introduce the concept of magnetic field properly, let's review our formulation of electric interactions in Chapter 22, where we introduced the concept of electric field. We represented electric interactions in two steps: • A distribution of electric charge at rest creates an electric field E. in the surrounding space. • 2. The electric field exerts a force F =Eq on any other charge q that is present in the field. • We can describe magnetic interactions in a similar way: • A moving charge or a current creates a magnetic field in the surrounding space (in addition to its electric field). • 2. The 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 presence of a magnetic field, what force does it exert on a moving charge or a current?

  9. In this chapter we'll concentrate on the second aspect of the interaction: Given the presence of a magnetic field, what force does it exert on a moving charge or a current? In Chapter 29 we will come back to the problem of how magnetic fields are created by moving charges and currents. Magnetic field is a vector field, that is, a vector quantity associated with each point in space. We will use the symbol B for magnetic field. At any position the direction of B is defined as that in which the north pole of a compass needle tends to point. What are the characteristics of the magnetic force on a moving charge? First, its magnitude is proportional to the magnitude of the charge. If a 1- C charge and a 2- C charge move through a given magnetic field with the same velocity, the force on the 2- C charge is twice as great as that on the 1- C charge. The magnitude of the force is also proportional to the magnitude, or "strength," of the field;

  10. If we double the magnitude of the field (for example, by using two identical bar magnets instead of one) without changing the charge or its velocity, the force doubles. The magnetic force also depends on the particle's velocity. This is quite different from the electric-field force, which is the same whether the charge is moving or not. A charged particle at rest experiences no magnetic force. Furthermore, the magnetic force F does not have the same direction as the magnetic field B, but instead is always perpendicular to both B and the velocity V.

  11. Figure 28-5 shows these relationships. The direction of F is always perpendicular to the plane containing V and B. Its magnitude is given by (28-1) whereqis the magnitude of the charge and Φis the angle measured from the direction of V to the direction of B, 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 V and B. 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 V and B with their tails together, as in Fig. 28-5b. Imagine turning V until it points in the direction of B. Wrap the fingers of your right hand around the line perpendicular to the plane of V and B so that they curl around with the sense of rotation from V and B. Your thumb then points in the direction of the force F on a

  12. 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 V in a magnetic field B is given, both in magnitude and in direction, by (magnetic force on a moving charged particle). (28-2) This is the first of several vector products we will encounter in our study of magnetic field relationships. It's important to note that Eq. (28-2) was not deduced theoretically; it is an observation based on experiment. When a charged particle moves through a region of space where both electric and magnetic fields are present, both fields exert forces on the particle. The total forceis the vector sum of the electric and magnetic forces: (28-4)

  13. 28-4 MAGNETIC FIELD LINES AND MAGNETIC FLUX We draw the lines so that the line through any point is tangent to the magnetic field vector B at that point. Just as with electric field lines, we draw only a few representative lines; otherwise, the lines would fill up all of space. Where adjacent field lines are close together, the field magnitude is large; where these field lines are far apart, the field magnitude is small. Also, because the direction of B at each point is unique, field lines never intersect.

  14. MAGNETIC FLUX AND GAUSS'S LAW FOR MAGNETISM We define the magnetic flux B through a surface just as we defined electric flux in connection with Gauss's law in Section 23-3. We can divide any surface into elements of area dA For each element we determine B , the component of B normal to the surface at the position of that element, as shown. From the figure, where is the angle between the direction of B and a line perpendicular to the surface. We define the magnetic flux dBthrough this area as (28-5)

  15. The total magnetic flux through the surface is the sum of the contributions from the individual area elements: (magnetic flux through a surface). (28-6) Magnetic flux is a scalar quantity. In the special case in which B is uniform over a plane surface with total area A, B and Φare the same at all points on the surface, and (28-7) If B happens to be perpendicular to the surface, then cos =1 and Eq. (28-7) reduces to B = BA. We conclude that the total magnetic flux through a closed surface is always zero. Symbolically, (magnetic flux through closed surface). (28-8)

  16. This equation is sometimes called Gauss's law for magnetism. 28-5 MOTION OF CHARGED PARTICLES IN A MAGNETIC FIELD When a charged particle moves in a magnetic field, it is acted on by the magnetic force given by Eq. (28-2), and the motion is determined by Newton's laws. Figure 28-13 shows a simple example. A panicle with positive charge q is at point O, moving with velocity V in a uniform magnetic field B directed into the plane of the figure. The electors V and B are perpendicular, so the magnetic force has magnitude and a direction as shown in the figure. The force is always perpendicular to V, so it can-not change the magnitude of the velocity, only its direction. To put it differently, the magnetic force never has a component parallel to the particle's motion, so the magnetic force can never do work on the particle.

  17. This is true even if the magnetic field is not uniform. Motion of a charged particle under the action of a magnetic field alone is always motion with constant speed. (28-10)

  18. where m is the mass of the particle. Solving Eq. (28-10) for the radius R of the circular path, we find (radius of a circular orbit in a magnetic field). (28-11) (28-12) The number of revolutions per unit time is This frequency f is independent of the radius R of the path. It is called the cyclotron frequency; Motion of a charged particle in a non-uniform magnetic field is more complex. Figure 28-15 shows a field produced by two circular coils separated by some distance. Particles near either coil experience a magnetic force toward the center of the region; particles with appropriate speeds spiral repeatedly from one end of the region to the other and back. Because charged particles can be trapped in such a magnetic field, it is called a magnetic bottle.

  19. 线圈 线圈 hot plasmas 28-6 APPLICATIONS OF MOTION OF CHARGED PARTICLES VELOCITY SELECTOR (28-13)

  20. THOMSON'S e/m EXPERIMENT In one of the landmark experiments in physics at the end of the nineteenth century, J. J. Thomson (1856-1940) used the idea just described to measure the ratio of charge to mass for the electron. For this experiment, carried out in 1897 at the Cavendish Laboratory in Cambridge, England, Thomson used the apparatus shown in Fig. 28-19. The speed v of the electrons is determined by the accelerating potential V, just as in the derivation of Eq.(24-24). The kinetic energy 1/2mv2 equals the loss of electric potential energy eV , where e is the magnitude of the electron charge: (28-14) or The electrons pass straight through when Eq.(28-13) is satisfied; combining this with Eq.(28-14), we get (28-15)

  21. All the quantities on the right side can be measured, so the ratio e/m of charge to mass can be determined. It is not possible to measure e or m separately by this method, only their ratio. MASS SPECTROMETERS 28-7 MAGNETIC FORCE ON A CURRENT-CARRYING CONDUCTOR We can compute the force on a current-carrying conductor starting with the magnetic force on a single moving charge. B Fm j q Figure 28-21

  22. Figure 28-21 shows a straight segment of a conducting wire, with length l and cross-section area A; the current is from bottom to top. The wire is in a uniform magnetic field B, perpendicular to the plane of the diagram and directed into the plane. Let's assume first that the moving charges are positive. Later we'll see what happens when they are negative. The drift velocity vd is upward, perpendicular to B. The average force on each charge is directed to the left as shown in the figure; since vd and B are perpendicular, the magnitude of the force is We can derive an expression for the total force on all the moving charges in a length l of conductor with cross-section area A, using the same language we used in Eqs.(26-2) and (26-3) of Section 26-2. The number of charges per unit volume is n; a segment of conductor with length l has volume Al and

  23. and contains a number of charges equal to nAl . The total force F on all the moving charges in this segment has magnitude (28-16) From Eq. (26-3) the current density is The product is the total current I, so we can rewrite Eq. (28-16) as (28-17) (28-18) (magnetic force on a (magnetic force on a straight wire segment). (28-19)

  24. If the conductor is not straight, we can divide it into infinitesimal segments dl. The force dF on each segment is (magnetic force on an infinitesimal wire segment). (28-20) Then we can integrate this along the wire to find the total force on a conductor of any shape. 28-8 FORCE AND TORQUE ON A CURRENT LOOP fab b B fbc a n Φ c fad fcd d

  25. The total force on the loop is zero because the forces on opposite sides cancel out in pairs. The net force on a current loop in a uniform magnetic field is zero, however, the net torque is not in general equal to zero. (28-22) (28-23) The product IA is called the magnetic dipole moment or magnetic moment of the loop, for which we use the symbol =IA (28-24) It is analogous to the electric dipole moment introduced in Section 22-9. In terms of  the magnitude of the torque on a current loop is (28-25) (vector torque on a current loop). (28-26)

  26. An arrangement of particular interest is thesolenoid, a helical winding of wire, such as a coil wound on a circular cylinder (Fig. 28-30). If the windings are closely spaced, the solenoid can be approximated by a number of circular loops lying in planes at right angles to its long axis. The total torque on a solenoid in a magnetic field is simply the sum of the torques on the individual turns. For a solenoid with N turns in a uniform field B, the magnetic moment is  = NIA and (28-28) Where  is the angle between the axis of the solenoid and the direction of the field. 28-10 The Hall Effect The reality of the forces acting on the moving charges in a conductor in a magnitude field is striking demonstrated by the Hall effect, an effect analogues to the transverse deflection of an electron beam in a magnetic field in vacuum.

  27. + + + + – – – – Z d b I I l X a Y

  28. To describe this effect, let’s consider a conductor in the form of a flat strip, as shown in Fig. The current is in the direction of the +x-axis, and there is a uniform magnetic field B perpendicular to the plane of the strip, in the y-axis. The drift velocity of the moving charges has magnitude vd. A moving charge is driven toward the lower edge of the strip by the magnetic force If the charge carriers are positive charge, as in Fig, an excess positive charge accumulates at the lower edge of the strip, leaving an excess negative charge at its upper edge. In terms of the coordinate axes in fig, the electrostatic field Ee for the positive charge q case is in the +z-direction; its z-component Ez is positive. The magnetic field is in the +y-direction, and we write it as By. The magnetic force is qvdBy. The current density Jx is in the +x-direction. In the steady state, when the forces qEz and qvdBy are equal in magnitude and opppsite in direction.

  29. or This confirms that when q is positive, Ez is positive. The current density Jx is JX = nqvd. Eliminating vd between these equations, we find 28-30 Note that this result is valid for both positive and negative q. When q is negative, Ez is positive, and conversely.

  30. 29 Sources of Magnetic 29-1 INTRODUCTION In a word, yes. Our analysis will begin with the magnetic field created by a single moving point charge. We can use this analysis to determine the field created by a small segment of a current-carrying conductor. Once we can do that, we can in principle find the magnetic field produced by any shape of conductor. Then we will introduce Ampere's law, the magnetic analog of Gauss's law in electrostatics. Ampere's law lets us exploit symmetry properties in relating magnetic fields to their sources. Moving charged particles within atoms respond to magnetic fields and can also act as sources of magnetic field. We'll use these ideas to understand how cer-tain magnetic materials can be used to intensify magnetic fields as well as why some materials such as iron act as permanent magnets. Finally, we will study how a time-varying electric field, which we will describe in terms of a quantity called displacement current, can act as a source of magnetic field.

  31. 29-2 MAGNETIC FIELD OF A MOVING CHARGE Let's start with the basics, the magnetic field of a single point charge q moving with a constant velocity v, As we did for electric fields, we call the location of the charge the source point and the point P where we want to find the field the field point. In Section 22-6 we found that at a field point a distance r from a point charge q, the magnitude of the electric field E caused by the charge is proportional to the charge magnitude q and 1/r2, and the direction of E (for positive q) is along the line from source point to field point. The corresponding relationship for the magnetic field B of a point charge q moving with constant velocity has some similarities and some interesting differences. Experiments show that the magnitude of B is also proportional to q and 1/r2. But the direction of B is not along the line from source point to field point. Instead, B is perpendicular to the plane containing this line and the particle's velocity vector v, as shown in Fig. 29-1.

  32. P B v Φ q (29-1) Where 0/4 is a proportionality constant.

  33. (magnetic field of a point charge with constant velocity). (29-2) (29-4)

  34. 29-3 MAGNETIC FIELD OF A CURRENT ELEMENT Just as for the electric field, there is a principle of superposition of magnetic fields: The total magnetic field caused by several moving charges is the vector sum of the fields caused by the individual charges. (magnetic field of a current element), (29-6)

  35. Equations (29-5) and (29-6) are called the law of Biot and Savart (pronounced "Bee-oh" and "Suh-var"). We can use this law to find the total magnetic field B at any point in space due to the current in a complete circuit. To do this, we integrate Eq. (29-6) over all segments dl that carry current; symbolically, P (29-7) B I Φ

  36. Problem-Solving Strategy • MAGNETIC FIELD CALCULATIONS • Be careful about the directions of vector quantities. The current element dl always points in the direction of the current. The unit vector r is always directed from the current element (the source point) toward the point P at which the field is to be determined (the field point). • In some situations the dB at point P have the same direction for all the current elements; then the magnitude of the total B field is the sum of the magnitudes of the dB. But often the dB have different directions for different current elements. Then you have to set up a coordinate system and represent each dB in terms of its components. The integral for the total B is then expressed in terms of an integral for each component. Sometimes you can use the symmetry of the situations to prove that one component must vanish. Always be alert for ways to use symmetry to simplify the problem.

  37. 3. Look for ways to use the principle of superposition of magnetic fields. Later in this chapter we'll determine the fields produced by certain simple conductor shapes. If you encounter a conductor of a complex shape that can be represented as a combination of these simple shapes, you can use superposition to find the field of the complex shape. Examples include a rectangular loop and a semicircle with straight-line segments on both sides. 29-4 MAGNETIC FIELD OF A STRAIGHT CURRENT-CARRYING CONDUCTOR An important application of the law of Blot and Savart is finding the magnetic field produced by a straight current-carrying conductor. This result is useful because straight conducting wires are found in essentially all electric and electronic devices. Figure 29-5 shows such a conductor with length 2a carrying a current I.

  38. Y a Φ Idl r B X O P I -a

  39. When the length 2a of the conductor is very great in comparison to its distance x from point P, we can consider it to be infinitely long. When a is much larger than x, is approximately equal to a; hence in the limit (a long, straight, current-caning conductor). (29-9) 29-5 FORCE BETWEEN PARALLEL CONDUCTORS In Example 29-4 (Section 29-4) we showed how to use the principle of superposition of magnetic fields to find the total field due to two long current-carrying conductors. Another important aspect of this configuration is the interaction force between the conductors. This force plays a role in many practical situations in which current-carrying wires are close to each other, and it also has fundamental significance in connection with the definition of the ampere. Figure 29-8 shows segments of two long, straight parallel conductors

  40. respectively, in the same direction. Each conductor lies in the magnetic field set up by the other, so each experiences a force. The diagram shows some of the field lines set up by the current in the lower conductor.

  41. From Eq. (29-9) the lower conductor produces a B field that, at the position of the upper conductor, has magnitude From Eq. (28-19) the force that this field exerts on a length L of the upper conductor is where the vector L is in the direction of the current I‘ and has magnitude L. Since B is perpendicular to the length of the conductor and hence to L the magnitude of this force is , and the force per unit length F/L is (two long, parallel, current-carrying conductors). (29-11)

  42. Applying the right-hand rule to shows that the force on the upper conductor is directed downward The attraction or repulsion between two straight, parallel, current-carrying conductors is the basis of the official SI definition of the ampere: One ampere is that unvarying current that, if present in each of two parallel conductors of infinite length and one meter apart in empty space, causes each conductor to experience a force of exactly 2  10-7newtons per meter of length. 29-6 MAGNETIC FIELD OF A CIRCULAR CURRENT LOOP We can use the law of Blot and Savart, Eq. (29-5) or (29-6), to find the magnetic field at a point P on the axis of the loop, at a distance x from the center. As the figure shows, dl and r are perpendicular, and the direction of the field B caused by this particular element dl lies in the xy-plane. Since the magnitudedB of the field due to element dl is

  43. (29-12) The components of the vector dB are (29-13) (29-14) (on the axis of a circular loop). (29-15)

  44. Now suppose that instead of the single loop in Fig. 29-10 we have a coil consisting of N loops, all with the same radius. The loops are closely spaced so that the plane of each loop is essentially the same distance x from the field point P. Each loop contributes equally to the field, and the total field is N times the field of a single loop: (on the axis of N circular loops). (29-16) (at the center of N circular loops). (29-17) (on the axis of any number of circular loops). (29-18)

  45. 29-7 AMPERE'S LAW So far our calculations of the magnetic field due to a current have involved finding the infinitesimal field dB due to a current element, then summing all the dB’s to find the total field. Ampere's law is formulated not in terms of magnetic flux, but rather in terms of the line integral of B around a closed path, denoted by We used line integrals to define work in Chapter 6 and to calculate electric potential in Chapter 24.To evaluate this integral. we divide the path into infinitesimal segments dl, calculate the scalar product of B·dl for each segment, and sum these products. In general, B varies from point to point, and we must use the value of B at the location of each dl. An alternative notation is

  46. where BІІs the component of B parallel to dl at each point. The circle on the integral sign indicates that this integral is always computed for a closed path, one whose beginning and end points are the same. To introduce the basic idea of Ampere's law, let's consider again the magnetic field caused by a long, straight conductor carrying a current I. We found in Section 29-4 that the field at a distance r from the conductor has magnitude and that the magnetic field lines are circles centered on the conductor. Let's take the line integral of B around one such circle with radius r, as in Fig. 29-13a. At every point on the circle, B and dl are parallel, and so B·dl = Bdl, since r is constant around the circle, B is constant as well. Alternatively, we can say that BІІ is constant and equal to B at every point on the circle. Hence we can take B outside of the integral. The remaining integral is just the circumference of the circle, so

  47. The line integral is thus independent of the radius of the circle and is equal to 0 multiplied by the current passing through the area bounded by the circle. In Fig. 29-13b the situation is the same, but the integration path now goes around the circle in the opposite direction. Now B and dl are antiparallel, so , and the line integral equals , We get the same result if the integration path is the same as in Fig. 29-13a, but the direction of the current is reversed. Thus the line integral equals multiplied by the current passing through the area bounded by the integration path, with a positive or negative sign depending on the direction of the current relative to the direction of integration. An integration path that does not enclose the conductor is used in Fig. 29-13c. Along the circular arc ab of radius r1,

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