The Physics of Energy, page 12
Box 3.4 Electric Motors
Electric motors are used in a wide range of devices, including cell phone vibrators, disk drives, fans, pumps, compressors, power tools, appliances, elevators, and for propulsion of ships and other vehicles – almost anything that moves and is driven by electricity contains a motor of some kind. Motors generally contain two magnets, the rotating rotor and the stationary stator. In the simplest motors the rotor is an electromagnet powered by DC current with a commutator and brushes switching the direction of current flow depending on the angle of the rotor (see figures), while the stator uses a permanent magnet. The rotor often contains three, or sometimes five, sets of windings around an iron core. More complex designs use AC current to power the stator giving a varying magnetic field that drives the rotor (see §38 for further discussion).
Consider as a simple example a small motor, where the stator has a permanent magnet with field 0.02 T, and the rotor has an electromagnet formed by wrapping a wire 600 times around a single cylinder of radius 2 cm.
If the motor uses 300 mA of current and runs at 6000 rpm, what is the average power output of the motor (neglecting frictional losses)?
The average power output is W.
The mechanical power output produced by the motor is equal to the electrical power input needed to keep the charge flowing. From eq. (2.44), the power exerted by the motor is
(3.54)
since . To maintain this power output, an external voltage must drive the current through the wire. To compute the electrical power needed, consider the Lorentz force exerted on the charges in the wire while the loop is in motion, as illustrated in Figure 3.20. A charge q in a section of the wire moving with velocity υ experiences a force . On the two sides of the loop parallel to the -axis this force acts opposite to the direction that the current is flowing. Using , where is the unit vector in the direction of rotation, we find that the magnitude of the force is
(3.55)
Figure 3.20 When the wire in Figure 3.19 rotates with angular velocity , the motion of the wire leads to a force on the charges in both sections of the wire parallel to the rotation axis. In a motor, the force opposes the direction of current flow. In a generator, this force produces an EMF that drives an external current.
To carry the charge around the loop requires work to be done against this force,
(3.56)
Since , the power required to keep the current moving around the loop is
(3.57)
which is precisely the rate (3.54) at which work is done by the motor. This work must be supplied by an external electrical power supply. Integrating the work done in rotating the loop around a half-cycle from to gives
(3.58)
Since the current reverses when changes sign, the average power output when the motor is rotating at frequency ν is then . In most real motors the wire loop is wrapped many times, which multiplies the power by the number of windings. Note that there is no theoretical limit to the efficiency of an electric motor that converts electromagnetic to mechanical energy, though any real device will have losses, for example between the commutator and the brushes (see Box 3.4).
The power (3.57) required to keep the current flowing in the loop can be expressed as
(3.59)
Conservation of energy then leads us to the conclusion that the energy stored in the interaction of the current loop and the magnetic field is
(3.60)
This result is, in fact, quite general (see Box 3.5), and corresponds to the everyday experience that a bar magnet reaches its minimum energy state when it aligns with an external magnetic field, such as when a compass aligns with Earth’s magnetic field.
Box 3.5 Magnetic Dipoles in Magnetic Fields
In general, as in the special case of a current loop analyzed in the text, a magnetic dipole in a homogeneous magnetic field experiences a torque . This torque pushes the dipole into alignment with the field . While analyzed in the text in terms of the Lorentz force, this force can be interpreted in terms of an interaction energy between the dipole and the field
This interaction energy between magnetic dipoles and magnetic fields is relevant in later chapters, particularly in understanding simple quantum systems (§7) and the detailed dynamics of generators (§38).
3.4.2 An Electric Generator
The motor described in the previous section can be reversed and used to transform mechanical into electrical energy. The device is then called a generator. Suppose that a mechanical agent forces the wire loop of Figure 3.20 to turn at an angular velocity in the fixed magnetic field B. This gives rise to a force (3.55) on the charges in the sections of the wire parallel to the y-axis. The negatively charged electrons and the positive ions experience opposite forces, but the ions are held in place by interatomic forces while the electrons are free to move and give rise to a current. Although the origin of the force is motion in a magnetic field, the effect is the same as if an electric field
(3.61)
were present in the wire. The integral of this quantity around the loop is the work that is done on a unit of charge that is moved completely around the loop. This is known as the electromotive force or EMF, denoted ,
(3.62)
The electromotive force generalizes the notion of voltage difference from electrostatics, and is applicable in time-dependent circuits as well as in static situations where a static voltage source like a battery provides an EMF. Note that the force acting on charges moving around the loop in the motor described in the last subsection can also be thought of as an electromotive force, known as the back-EMF.
If a load is connected across the terminals C and D in Figure 3.20, then power equal to is delivered to the load and must be supplied by the mechanical agent that rotates the loop. The wire loop rotating in a magnetic field generates electricity; it transforms mechanical into electric power. Again, there is in principle no theoretical limit to the efficiency of mechanical to electric energy conversion though any real system using this mechanism will have some losses.
Note that a generator of this type naturally produces an electromotive force that varies sinusoidally with time when the loop is rotated at a constant angular velocity. Just as the wire loop in Figure 3.21 rotates at an angular frequency , the EMF between terminals marked a and b is proportional to , where A is the area of the loop. As for the motor, winding a wire N times around the loop will multiply the EMF, and hence the output power at a given rate of rotation, by N.
Figure 3.21 (a) A sketch of the physical arrangement of an AC generator. As the wire loop rotates, each end slides over a slip ring (), maintaining electrical contact. The EMF between terminals a and b oscillates sinusoidally. (b) The orientation of the current loop is shown together with the EMF and the magnetic flux Φ through the loop as functions of time.
3.5Induction and Inductors
We have described how electric and magnetic fields are generated by and exert forces on charges and currents. There are still missing pieces, however, in our description of how these fields are produced. In particular, electric fields are produced when magnetic fields change and vice versa. These relationships underlie the phenomenon of light waves – electromagnetic waves that propagate through the vacuum (§4). In this section we describe how electric fields must be produced when magnetic fields change; this is Faraday’s law of induction, which is closely related to the mechanisms behind the motors and generators we have just studied. Like magnetic fields and forces in general, Faraday’s law is most clearly explained as a simple application of Einstein’s special theory of relativity. Here, however, we introduce induction empirically, as it was first discovered by Faraday; Box 3.6 provides a synopsis of the argument based on relativity.
Box 3.6 Induction and Relativity
While Faraday’s law was discovered empirically before Einstein’s special theory of relativity, the clearest logical explanation for Faraday’s law follows from a simple application of relativity, which we describe here. A more detailed discussion of Einstein’s theory is given in §27.2.
Consider a wire in the shape of a rectangular loop in the xy-plane that is translated horizontally as shown in the figure at right, moving from a region without a magnetic field into a region with a field. (This could be done, for example, by passing the loop between two of the coils of a large solenoid.) While only one side of the rectangle is within the magnetic field , the electrons on that side of the rectangle will experience a Lorentz force, giving a net EMF that drives a current clockwise around the wire loop, where l is the length of the rectangle perpendicular to its motion, and v is the loop’s speed. This is analogous to the force experienced on the charges in the wire loop in the motor and generator described in the previous section.
How does this force arise from the point of view of a physicist who is moving along with the loop (or alternatively, as seen from a stationary loop when the device producing the magnetic field is moved across the loop at a constant velocity υ)? There cannot be any Lorentz force on the charges in a stationary loop since there is no velocity. But Einstein’s principle of special relativity requires the physics in this reference frame to be the same as in the frame in which the loop is moving, so something must generate a force. The only electromagnetic force that acts on stationary charges is that from an electric field. Thus, there must be an electric field along the part of the wire that is within the magnetic field region. The integral of this electric field around the loop is
where the sign arises since the integral is taken counterclockwise around the loop by convention. The right-hand side of this equation is precisely minus the rate of change of the flux of the magnetic field through the planar surface S bounded by the wire loop,
This is precisely Faraday’s law (3.64). While the argument just given relies on a wire loop of a particular shape, the same argument generalizes to an arbitrary curve , and the result does not depend upon the presence of an actual wire loop around the curve . Faraday’s law is valid for any closed loop through which the magnetic flux changes, either in magnitude or direction.
3.5.1 Faraday’s Law of Induction
By means of a series of ingenious experiments in the mid-nineteenth century, the English physicist Michael Faraday discovered that changing the magnetic flux through a wire loop induces an EMF that drives a current in the loop. Figure 3.22 shows two different ways in which this might occur. In case (a), the loop moves through a stationary magnetic field whose strength varies with position. In case (b) the loop is stationary but the strength of the magnetic field increases with time as, for example, when the current in a nearby wire is increased. In both cases the flux through the loop,
(3.63)
increases with time (here S is a surface bounded by the loop as in eq. (3.10)). Faraday observed that the EMF induced in the loop was proportional to the time rate of change of the flux,
(3.64)
Equation (3.64), known as Faraday’s law, describes the induction of an electric field by a time-dependent magnetic flux. Since the integral of the electric field around a closed path is not zero in this situation, it is no longer possible to define an electrostatic potential as we did in eq. (3.7). As mentioned earlier, the EMF generalizes the concept of voltage to include time-dependent phenomena of this type.
Figure 3.22 Two examples of EMF induced by changing magnetic flux. (a) A wire loop in the plane moves into an increasing magnetic field. (b) A fixed wire and a magnetic field that is constant in space, but grows with time. In both cases, the direction of the EMF (shown by the arrows) is determined by Lenz’s law.
Example 3.5 AC Power
Generators provide a compact and efficient means of transforming mechanical to electrical power. Suppose you seek to generate an RMS voltage of 120 V at 60 Hz using a permanent magnet for a rotor, in a roughly constant magnetic field of 1000 gauss. What area (or ) must the current loop span?
With 0.1 T, s1, and V, we find from eq. (3.62), 4.5 m2. A wire wound times around a sturdy armature of radius 12 cm would satisfy these specs. This is a fairly compact and simple device: a magnet of modest strength and a mechanical system of modest dimensions. Note, however, that there is no free lunch: you must supply enough mechanical power to turn the armature. If, for example, you demand an RMS current of 10 A from this generator, you would have to supply an RMS power of 1200 W, which is nearly two horsepower.
The minus sign in eq. (3.64) indicates that a current produced by the EMF would act to reduce the magnetic flux through the loop (Question 3.4). If it were the other way, the EMF would produce a current that would further increase the EMF, leading to a runaway solution that violates conservation of energy. This sign, and the general fact that an induced current acts to reduce any imposed magnetic field, is known as Lenz’s law.
Faraday’s law is closely related to the mechanism underlying motors and generators. In these devices, including generators of the type described above, an EMF develops in a loop rotating in a constant magnetic field. While the flux through the loop changes with time, eq. (3.64) does not apply in this situation since there is no appropriate reference frame in which the loop is stationary.8 Nonetheless, generators that work on this principle are often spoken of as working by induction, in part because Faraday’s law was understood before Einstein’s theory of relativity. Other types of generators that do use induction as described by Faraday’s law (3.64) are discussed in §38.
Faraday’s law can be rewritten in a local form that emphasizes the direct relationship between electric and magnetic fields. First, note that the surface integral is the same no matter what surface is chosen to span the curve (Problem 3.21), so the flux Φ can be associated with the curve independent of the choice of S. Next, apply Stokes’ theorem (B.27) to the left-hand side of eq. (3.64) giving
(3.65)
for any surface S spanning the current loop . Since S is arbitrary, the integrands must be equal, giving the local form
(3.66)
which replaces eq. (3.14) if time-varying magnetic fields are present.
An important feature of Faraday’s law is that a wire wound N times around the curve captures N times the flux, enhancing the EMF by a factor of N. So, a succinct summary of Faraday’s law for a wire with N windings is
(3.67)
Faraday’s Law of Induction
A time-varying magnetic field gives rise to a circulating electric field according to Faraday’s law,
or in integral form,
3.5.2 Inductance and Energy in Magnetic Fields
In an electrical circuit, time-varying currents produce time-varying magnetic fields, . Time-varying magnetic fields in turn produce electromotive force, . Combining these two effects, we see that a time-varying current through a circuit or a part of a circuit leads to an EMF proportional to . The coefficient of proportionality, denoted by L, is called the inductance of the circuit or circuit element, so that there is a contribution to the voltage across a circuit component with inductance L given by
(3.68)
The minus sign implements Lenz’s law: with L positive, the voltage acts to oppose the increase in current. Inductance is measured in henrys (H), defined by 1 H = 1 V s/A, and named in honor of the American physicist Joseph Henry, the co-discoverer of induction. The electric circuit symbol for an inductor is .
Any circuit will have some inductance by virtue of the magnetic fields created by the current flowing through its wires. An inductor is a circuit element designed principally to provide inductance. Typically, an inductor consists of a winding of insulated wire surrounding a core of magnetizable material like iron. Some examples of inductors are shown in Figure 3.23.
Figure 3.23 A sample of simple inductors, basically loops of wire wound around an iron core with to enhance the inductance by a factor . (Credit: Manutech, Haiti)
A classic example of an inductor is a long solenoid of length l with a wire winding N times around a core made of material with permeability μ. Ignoring end effects, the magnetic field in the solenoid is given by eq. (3.51), so , and
(3.69)
with , so the inductance of a solenoid is , where is the volume.
Because they generate an EMF opposite to , inductors act to slow the rate of change of the current in a circuit. They also store energy. The power expended to move a current I through a voltage V is . Since the voltage across an inductor is , the power needed to increase the current through an inductor is , where the sign reflects the fact that work must be done to increase the current. This work is stored in the magnetic fields in the inductor. Starting with zero current at and increasing the current to I, the energy stored in the inductor is
(3.70)
can be expressed in terms of the integral of the magnetic field strength over all space, in a form analogous to eq. (3.20),
(3.71)
This is a general result for the energy contained in a magnetic field. Although the derivation of this result in a completely general context would require further work, the fact that it holds for the specific case of a solenoid is easy to demonstrate (Problem 3.22). As indicated by eq. (3.71), if B is held fixed, less energy is stored in a magnetic material when the permeability μ is larger. It is more natural, however, to compare stored energy with currents held fixed, since they can be controlled externally. Then, as can be seen from eq. (3.70) and the fact that , for the same current, more energy is stored in a system with larger permeability. In the following chapter (§4), we give a more detailed description of the storage and propagation of energy in electric and magnetic fields.
