Sears and Zemansky’s University Physics. 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.
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).
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.
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.
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).
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.
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;
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.
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
whereqis 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
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:
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.
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 dBthrough this area as
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
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)
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
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.
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.
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)
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.
28-6 APPLICATIONS OF MOTION OF CHARGED PARTICLES
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:
The electrons pass straight through when Eq.(28-13) is satisfied; combining this with Eq.(28-14), we get
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.
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.
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
and contains a number of charges equal to nAl . The total force F on all the moving charges in this segment has magnitude
From Eq. (26-3) the current density is
is the total current I, so we can rewrite Eq. (28-16) as
(magnetic force on a (magnetic force on a straight wire segment). (28-19)
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
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.
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
(vector torque on a current loop). (28-26)
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
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.
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.
This confirms that when q is positive, Ez is positive. The current density Jx is JX = nqvd. Eliminating vd between these equations, we find
Note that this result is valid for both positive and negative q. When q is negative, Ez is positive, and conversely.
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.
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.
Where 0/4 is a proportionality constant.
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)
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,
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.
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
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.
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)
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
The components of the vector dB are
(on the axis of a circular loop). (29-15)
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)
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
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
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,
along the circular arc cd of radius r2, B and dl are antiparallel and
The B field is perpendicular to dl at each point on the straight sections bc and da, so
and these sections contribute zero to the line integral. The total line integral is then
We can also derive these results for more general integration paths, At the position of the line element dl, the angle between dl and B is Φ, and
where d is the angle
subtended by dl at the position of the conductor and r is the distance of dl from the conductor. Thus
is just equal to 2 the total angle swept out by the
radial line from the conductor to dl during a complete trip around the path. So we get
This result doesn't depend on the shape of the path or on the position of the wire inside it. If the current in the wire is opposite to that shown, the integral has the opposite sign. But if the path doesn't enclose the wire .
5. If B is perpendicular to all or some portion of the path, that portion of the path makes no contribution to the integral.
6. The sign of a current enclosed by the integration path is given by a right-hand rule. Curl the fingers of your right hand so that they follow the integration path in the direction that you carry out the integration. Your right thumb then points in the direction of positive current. If B is as described in Step 4 and Iencl is positive, then the direction of B is the same as the direction of the integration path; if instead Iencl is negative, B is in the direction opposite to that of the integration.
7. In the integral , B is always the total magnetic field at each point on the path. This field can be caused partly by currents enclosed by the path and partly by currents outside. If no net current is enclosed by the path. The field at points on the path need not be zero, but the integral is always zero.
In discussing how currents cause magnetic fields, we have assumed that the conductors are surrounded by vacuum. Permanent magnets, magnetic recording tapes, and computer disks depend directly on the magnetic properties of materials. We will discuss three broad classes of magnetic behavior that occur in materials; these are called paramagnetism, diamagnetism, and ferromagnetism.
The Bohr Magneton
The equivalent current is the total charge passing any point on the orbit per unit time, which is just the magnitude e of the electron charge divided by the orbital period T:
The magnetic moment = IA is then
It is useful to express in terms of the angular momentum L of the electron.
Equation (29-26) shows that associated with the fundamental unit of angular momentum is a corresponding fundamental unit of magnetic moment. If L = h/2, then
This quantity is called the Bohr magneton, denoted by B. Its numerical value is
A material showing the behavior just described is said to be paramagnetic. The result is that the magnetic field at any point in such a material is greater by a dimensionless factor Km, called the relative permeability.
In some materials the total magnetic moment of all the atomic current loops is zero when no magnetic field is present. But even these materials have magnetic effects because an external field alters electron motions within the atoms, causing additional current loops and induced magnetic dipoles comparable to the induced electric dipoles we studied in section 25-6. In this case the additional field caused by these current loops is always opposite in direction to that of the external field. Such material are said to be diamagnetic.
There is a third class of materials, called ferromagnetic materials, which includes iron, nickel, cobalt, and many alloys containing these elements. In these materials, strong interactions between atomic magnetic moments cause them to line up parallel to each other in regions called magnetic domains, even when no external field is present.
Ampere's law, in the form stated in Section 29-7, is incomplete. As we learn how it has to be modified for the most general situations, we will uncover some profound and fundamental aspects of the behavior of electric and magnetic fields.
To introduce the problem, let's consider the process of charging a capacitor (Fig.29-26). Conducting wires lead current ic into one plate and out of the other; the charge Q increases, and the electric field E between the plates increases. The notation ic indicates conduction current to distinguish it from another kind of current we are about to encounter, called displacement current iD. We use lowercase i’s and v’s to denote instantaneous values of currents and potential differences, respectively, that may vary with time.
is the electric flux through the surface.
As the capacitor charges, the rate of change of q is the conduction current,
Taking the derivative of
Eq. (29-33) with respect to time, we get
(displacement current) (29-35)
Displacement current density
The answer is a phenomenon known as electromagnetic induction: If the magnetic flux through a circuit changes, an emf and a current are induced in the circuit. In a power-generating station, magnets move relative to coils of wire to produce a changing magnetic flux in the coils and hence an emf.
Other key components of electric power systems, such as transformers, also depend on magnetically induced emfs. When electromagnetic induction was discovered in the 1830s, it was a mere laboratory curiosity; today, thanks to its central role in the generation of electric power, it is fundamentally responsible for the nature of our technological society. The central principle of electromagnetic induction, and the keystone of this chapter, is Faraday's law. This law relates induced emf to changing magnetic flux in any loop. including a closed circuit. We also discuss Lenz's law, which helps us to predict the directions of induced emfs and currents. This chapter provides the principles we need to understand electrical energy-conversion devices such as motors, generators, and transformers. We will also present a neat package of formulas, called Maxwell's equations, that describe the behavior of electric and magnetic fields in any situation. These equations pave the way for the analysis of electromagnetic waves in Chapter 33.
In an electric circuit there must be a device somewhere in the loop that acts like the water pump in a water fountain. In this device a charge travels “uphill”, from lower to higher potential energy, even though the electrostatic force is trying to push it from higher to lower potential energy. The direction of current in such a device is from lower to higher potential , just the opposite of what happens in an ordinary conductor. The influence that make currents flow from lower to higher potential is called electromotive force. This is a poor term because emf is not a force but an energy-per-unit-charge quantity.
To explore further the common elements in these observations, let's consider a more detailed series of experiments with the situation shown in Fig. 30-2. We connect a coil of wire to a galvanometer, then place the coil between the poles of an electromagnet whose magnetic field we can vary. Here's what we observe:
1. When there is no current in the electromagnet, so that B=0, the galvanometer shows no current.
2. When the electromagnet is turned on, there is a momentary current through the meter as B increases.
3. When B levels off at a steady value, the current drops to zero, no matter how large B is.
4. With the coil in a horizontal plane, we squeeze it so as to decrease the cross-section area of the coil. The meter detects current only during the deformation, not before or after. When we increase the area to return the coil to its original shape, there is current in the opposite direction, but only while the area of the coil is changing.
5. If we rotate the coil a few degrees about a horizontal axis, the meter detects current during the rotation, in the same direction as when we decreased the area. When we rotate the coil back, there is a current in the opposite direction during this rotation.
6. If we jerk the coil out of the magnetic field, there is a current during the motion, in the same direction as when we decreased the area.
7. If we decrease the number of turns in the coil by unwinding one or more turns, there is a current during the unwinding, in the same direction as when we decreased the area. If we wind more turns onto the coil, there is a current in the opposite direction during the winding.
8. When the magnet is turned off, there is a momentary current in the direction opposite to the current when it was turned on.
9. The faster we carry out any of these changes, the greater the current.
10. If all these experiments are repeated with a coil that has the same shape but different material and different resistance, the current in each case is inversely proportional to the total circuit resistance. This shows that the induced emfs that are causing the current do not depend on the material of the coil but only on its shape and the magnetic field.
The common element in all induction effects is changing magnetic flux through a circuit. Before stating the simple physical law that summarizes all of the kinds of experiments described in Section 30-2, let's first review the concept of magnetic flux B (which we introduced in Section 28-4). For an infinitesimal area element dA in a magnetic field B (Fig. 30-3), the magnetic flux dB through the area is
Faraday's law of induction states:
The induced emf in a closed loop equals the negative of the time rate of change of magnetic flux through the loop.
In symbols, Faraday's law is
(Faraday's law of induction). (30-3)
3. Determine the sign of the induced emf or current. If the flux is increasing, so dB / dt is positive, then the induced emf or current is negative; if the flux is decreasing, dB / dt is negative and the induced emf or current is positive.
4. Finally, determine the direction of the induced emf or current using your right hand. Curl the fingers of your right hand around the S, vector, with your right thumb in the direction of S. If the induced emf or current in the circuit is positive, it is in the same direction as your curled fingers; if the induced emf or current is negative, it is in the opposite direction.
If we have a coil with N identical turns, and if the flux varies at the same rate through each turn, the total rate of change through all the turns is N times as large as for a single turn. If B is the flux through each turn, the total emf in a coil with N turns is
Lenz's law is a convenient alternative method for determining the direction of an induced current or emf. Lenz's law is not an independent principle; it can be derived from Faraday's law. It always gives the same results as the sign rules we introduced in connection with Faraday's law, but it is often easier to use. Lenz's law also helps us gain intuitive understanding of various induction effects and of the role of energy conservation. H. F. E. Lenz (1804-1865) was a German scientist who duplicated independently many of the discoveries of Faraday and Henry. Lenz's law states:
The direction of any magnetic induction effect is such as to oppose the cause of the effect. Lenz's law is also directly related to energy conservation. If the induced current in Example 30-7 were in the direction opposite to that given by Lenz's law, the magnetic force on the rod would accelerate it to ever-increasing speed with no external energy source,
even though electrical energy is being dissipated in the circuit. This would be a clear violation of energy conservation and doesn't happen in nature.
LENZ'S LAW ANDTHE RESPONSE TO FLUX CHANGES
Since an induced current always opposes any change in magnetic flux through a circuit, how is it possible for the flux to change at all? The answer is that Lenz's law gives only the direction of an induced current; the magnitude of the current depends on the resistance of the circuit. The greater the circuit resistance, the less the induced current that appears to oppose any change in flux and the easier it is for a flux change to take effect. If the loop in Fig. 30-12 were made out of wood (an insulator), there would be almost no induced current in response to changes in the flux through the loop.
In situations in which a conductor moves in a magnetic field, as in the generators discussed in Examples 30-4 through 30-7, we can gain additional insight into the origin of the induced emf by considering the magnetic forces on mobile charges in the conductor. Figure 30-13a shows the same moving rod that we discussed in Example 30-6, separated for the moment from the U-shaped conductor. The magnetic field B is uniform and directed into the page, and we move the rod to the right at a constant velocity v . A charged panicle q in the rod then experiences a magnetic force
We'll assume in the following discussion that q is positive; in that case the direction of this force is upward along the rod, from b toward a.
Now suppose the moving rod slides along a stationary U-shaped conductor, forming a complete circuit (Fig. 30-13b). No magnetic force acts on the charges in the stationary U-shaped conductors, but the charge that was near points a and b redistributes itself along the stationary conductors, creating an electric field within them. This field establishes a current in the direction shown. The moving rod has become a source of electromotive force; within it, charge moves from lower to higher potential, and in the remainder of the circuit, charge moves from higher to lower potential. We call this emf a motional electromotive force, denoted by . From the above discussion, the magnitude of this emf is
(motional emf; length and velocity perpendicular to uniform B)
We can generalize the concept of motional emf for a conductor with any shape, moving in any magnetic field, uniform or not (assuming that the magnetic field at each point does not vary with time). For an element dl of conductor, the contribution to the emf is the magnitude dl multiplied by the component of v B (the magnetic force per unit charge) parallel to dl; that is,
For any closed conducting loop, the total emf is
30-6 INDUCED ELECTRIC FIELDS
As an example, let's consider the situation shown in Fig. 30-15. A long, thin solenoid with cross-section area A and n turns per unit length is encircled at its center by a circular conducting loop. The galvanometer G measures the current in the loop.
As an example, let's consider the situation shown in Fig. 30-15. A long, thin solenoid with cross-section area A and n turns per unit length is encircled at its center by a circular conducting loop. The galvanometer G measures the current in the loop. A current I in the winding of the solenoid sets up a magnetic field B along the solenoid axis, as shown, with magnitude B as calculated in Example 29-10 (Section 29-8): B = 0 n I, along the solenoid axis, as shown, with magnitude B as calculated in Example 29-10 (Section 29-8):
When the solenoid current I changes with time, the magnetic flux Balso changes, and according to Faraday's law the induced emf in the loop is given by
But what force makes the charges move around the loop? It can't be a magnetic force because the conductor isn't moving in a magnetic field and in fact isn't even in a magnetic field. We are forced to conclude that there has to be an induced electric field in the conductor caused by the changing magnetic flux. This may be a little jarring; we are accustomed to thinking about electric field as being caused by electric charges, and now we are saying that a changing magnetic field somehow acts as a source of electric field. Furthermore, it's a strange sort of electric field. When a charge q goes once around the loop, the total work done on it by the electric field must be equal to q times the emf .
From Faraday's law the emf is also the negative of the rate of change of magnetic flux through the loop. Thus for this case we can restate Faraday's law as
30-7 Eddy Currents
In the examples of induction effects that we have studied, the induced currents have been confined to well-defined paths in conductors and other components forming a circuit. However, many pieces of electrical equipment contain masses of metal moving in magnetic fields or located in changing magnetic fields. In situations like these we can have induced currents that circulate throughout the volume of a material. Because their flow patterns resemble swirling eddies in a river. We call these eddy currents.
30-8 MAXWELL’S EQUATIONS
We are now in a position to wrap up in a wonderfully neat package all the relationships between electric and magnetic fields and their sources that we have studied in the past several chapters. The package consists of four equations, called Maxwell’s equations. You may remember Maxwell as the discoverer of the concept of displacement current, which we studied in Section 29-10. Maxwell didn't discover all of these equations single-handedly, but he put them together
and recognized their significance, particularly in predicting the existence of electromagnetic waves.
For now we'll state Maxwell's equations in their simplest form, for the case in which we have charges and currents in otherwise empty space. In Chapter 33 we'll discuss how to modify these equations if a dielectric or a magnetic material is present.
( Gauss's law for E)
( Gauss's law for B)
( Ampere's law). (30-14)
(Faraday's law). (30-15)
How can a 12-volt car battery provide the thousands of volts needed to produce sparks across the gaps of the spark plugs in the engine? If you turn off a vacuum cleaner by yanking its plug out of the wall socket, what's the source of the high voltage that causes breakdown in the air so that sparks fly in the socket? The answers to these questions and many others concerned with varying currents in circuits involve the induction effects that we studied in Chapter 30. A changing current in a coil induces an emf in an adjacent coil. The coupling between the coils is described by their mutual inductance. A changing current in a coil also induces an emf in that same coil. Such a coil is called an inductor; the relationship of current to emf is described by the inductance (also called self-inductance) of the coil. If a coil is initially
carrying a current, energy is released when the current decreases; this principle is used in automotive ignition systems. We'll find that this released energy was stored in the magnetic field caused by the current that was initially in the coil, and we'll look at some of the practical applications of magnetic-field energy.
31-2 MUTUAL INDUCTANCE
In Section 29-5 we considered the magnetic interaction between two wires carrying steady currents; the current in one wire causes a magnetic field, which exerts a force on the current in the second wire. But an additional interaction arises between two circuits when there is a changing current in one of the circuits. Consider two neighboring coils of wire, as in Fig.31-1. A current flowing in coil I produces a magnetic field B and hence a magnetic flux through coil 2. If the current in coil 1 changes, the flux through coil 2 changes as well; according to Faraday's law, this induces an emf in coil 2. In
this way, a change in the current in one circuit can induce a current in a second circuit. Let's analyze the situation shown in Fig. 31-1 in more detail. We will use lowercase letters to represent quantities that vary with time; for example, a time-varying current is i, often with a subscript to identify the circuit. In Fig. 31–1 a current i1 in coil 1 sets up a magnetic field (as indicated by the blue lines), and some of these field lines pass through coil 2. We denote
the magnetic flux through each turn of coil 2, caused by the current i1 in coil l, as B2 . (If the flux is different through different turns of the coil, then B2 denotes the average flux.) The magnetic field is proportional to i1, so B2 is also proportional to i1 . When i1 changes, B2 changes; this changing flux induces an emf 2 in coil 2, given by
We could represent the proportionality of B2 and i1 in the form , but instead it is more convenient to include the number of turns N2 in the relation. Introducing a proportionality constant M21, called the mutual inductance of the two coils, we write
Where B2 is the flux through a single turn of coil 2. From this,
That is, a change in the current i1 in coil I induces an emf in coil 2 that is directly proportional to the rate of change of i1 . We may also write the definition of mutual inductance, Eq.(31-2), as
We can repeat our discussion for the opposite case in which a changing current i2 in coil 2 causes a changing flux B1 and an emf 1 in Coil 1. We might expect that the corresponding constant M12 would be different from M12 because in general the two coils are not identical and the flux through them is not the same. It turns out, however, that M12 is always equal
to M12 , even when the two coils are not symmetric. We call this common value simply the mutual inductance, denoted by the symbol M without subscripts; it characterizes completely the induced-emf interaction of two coils. Then we can write
(mutually induced emf’s), (31-4)
where the mutual inductance M is
(mutual inductance). (31-5)
The negative signs in Eq.(31-4) are a reflection of Lenz's law. The first equation says that a change in current in coil I causes a change in flux through coil 2, inducing an emf in coil 2 that opposes the flux change; in the second equation the roles of the two coils are interchanged.
An important related effect occurs even if we consider only a single isolated circuit. When a current is present in a circuit, it sets up a magnetic field that causes a magnetic flux through the same circuit; this flux changes when the current changes. Thus any circuit that carries a varying current will have an emf induced in it by the variation in its own magnetic field. Such an emf is called a self-induced emf. By Lenz's law a self-induced emf always opposes the change in the current that caused the emf and so tends to make it more difficult for variations in current to occur. For this reason, self-induced emfs can be of great importance whenever there is a varying current. Self-induced emf's can occur in any circuit, since there will always be some magnetic flux through the closed loop of a current-carrying circuit. But the effect is greatly enhanced if the circuit includes a coil with N turns of wire (Fig. 31-3). As a result of the current i, there is an average magnetic flux B through each turn of the coil.
In analogy to Eq. (31-5) we define the self-inductance L of the circuit as follows:
When there is no danger of confusion with mutual inductance, the self-inductance is also called simply the inductance. Comparing Eqs.(31-5) and (31-6), we see that the units of self-inductance are the same as those of mutual inductance; the SI unit of self-inductance is one henry. If the current i in the circuit changes, so does the flux B ; from rearranging Eq.(31-6) and taking the derivative with respect to time, the rates of change are related by
From Faraday's law for a coil with N turns. Eq.(30-4), the self-induced emf is
, so it follows that
(self-induced emf). (31-7)
The minus sign in Eq. (31-7) is a reflection of Lenz's law; it says that the self-induced emf in a circuit opposes any change in the current in that circuit. (Later in this section we'll explore the significance of this minus sign in greater depth.) Equation (31-7) also states that the self-inductance of a circuit is the magnitude of the self-induced emf per unit rate of change of current.
A circuit device that is designed to have a particular inductance is called an inductor, or a choke. The usual circuit symbol for an inductor is
31-4 MAGNETIC-FIELD ENERGY
Establishing a current in an inductor requires an input of energy, and an inductor carrying a current has energy stored in it. Let's see how this comes about. In Fig. 31-4, an increasing current in the inductor causes an emf between its terminals, and a corresponding potential difference Vab between the terminals of the source, with point a at higher potential than point b. Thus the source must be adding
energy to the inductor, and the instantaneous power P (rate of transfer of energy into the inductor) is
We can calculate the total energy input U needed to establish a final current I in an inductor with inductance L if the initial current is zero. We assume that the inductor has zero resistance, so no energy is dissipated within the inductor. Let the current at some instant be i , and let its rate of change be di/dt; the current is increasing, so di/dt 0. The voltage between the terminals a and b of the inductor at this instant is Vab = L di/dt , and the rate P at which energy is being delivered to the inductor (equal to the instantaneous power supplied by the external source) is
The energy dU supplied to the inductor during an infinitesimal time interval dt is dU = Pdt, so The total energy U supplied while the current increases from zero to a final value I is
(magnetic energy density in a vacuum). (31-10)
( magnetic energy density in a material). (31-11)