The Physics of Energy, page 13
Inductance, Inductors, and Energy
An inductor is a circuit element that produces a voltage proportional to the time rate of change of the current passing through it,
where L is the inductance, measured in henrys, 1 H = 1 V s/A.
The inductance of a solenoid with n turns per unit length, volume , and core permeability μ is approximately
The energy stored in an inductor is This is a special case of the general expression for energy stored in magnetic fields,
3.5.3 Mutual Inductance and Transformers
If two independent circuits or circuit components are in close spatial proximity, then when the current through one changes, the magnetic flux through the other changes proportionately. By Faraday’s law this induces an EMF in the second circuit, proportional to the rate of change of the current in the first,
(3.72)
Similarly, a time-dependent current in circuit two leads to an EMF in circuit one: . It is possible to show using conservation of energy (Problem 3.23) that this relationship is symmetric: , where M is known as the mutual inductance of circuits one and two. The sign of M depends on the relative orientation of the two circuits. In general , and the ideal limit is reached only when all the magnetic flux loops both circuits – as it would for two coincident ideal solenoids of the same area and length. A conceptual model of such a system is shown in Figure 3.24. The symbol for a mutual inductor is , two inductors with lines denoting an iron core that channels the magnetic flux between them.
Figure 3.24 A simple transformer. Two coils are wound around a core with high magnetic permeability (typically iron), which acts as a magnetic circuit, channelling the magnetic flux through both coils. In the ideal limit where all magnetic flux goes through both coils, the ratio of secondary to primary voltages is given by the ratio of the number of turns, in this case roughly 1:2.
Mutual inductance enables the construction of one of the most common and useful electrical devices: the transformer. Consider the AC circuit shown in Figure 3.25. The left-hand circuit (the primary) is coupled by mutual inductance to the right-hand circuit (the secondary). In the ideal limit where , all magnetic flux lines go through both inductors, so and,
(3.73)
(see Problem 3.25). Power is transmitted from the generator on the left to the load on the right.
Figure 3.25 Two circuits, the primary (left) and the secondary (right), are coupled by mutual inductance. The resulting device, a transformer, transforms the AC voltage on the left to on the right.
Transformers play an essential role in the manipulation of AC power in electric power distribution networks. In the absence of imperfections such as resistance in the coils and incomplete containment of the magnetic flux, the efficiency of a transformer would be 100%. Those used in electric power distribution reach 98% or higher. We describe the role of transformers in electrical grids further in §38.
3.6Maxwell’s Equations
Up to this point, we have introduced four equations that describe the way that electric and magnetic fields interact with matter and with one another. Gauss’s law (3.12) and Ampere’s law (3.49) prescribe the way that electric and magnetic fields are produced by charges and currents. Two other equations – eq. (3.47), associated with the absence of magnetic charge, and Faraday’s law (3.66) – constrain E and B. Although we stated these laws under special conditions – for example, Gauss’s law was derived only for static charges – all but one of these equations continue to hold under all known conditions.
The one equation that needs further modification is Ampere’s law, which we have described only in the conditions of steady currents and constant charge distribution. As discussed earlier, Ampere’s law as stated in eq. (3.49) is not consistent with charge conservation. To see this, take the divergence of eq. (3.49). Since (B.21), eq. (3.49) implies , which only holds if (see eq. (3.33)). When current and charge densities change with time, this is not true. Around 1860, the English physicist James Clerk Maxwell recognized this problem, and solved it by subtracting a term proportional to from the left-hand side of Ampere’s law. Maxwell’s alteration of Ampere’s law reads,
(3.74)
The demonstration that this equation is consistent with current conservation is left to Problem 3.26. Note that this modification of Ampere’s law, which states that a time-varying electric field gives rise to a magnetic field, is closely parallel to Faraday’s law of induction stating that a time-varying magnetic field gives rise to an electric field. Indeed, in the absence of electric charges and currents, the laws of electromagnetism are invariant under a symmetry exchanging and .
Maxwell’s Equations
Maxwell’s modification of Ampere’s law completed the unification of electricity and magnetism that began with Faraday’s discovery of induction. Together with the Lorentz force law (3.45), which describes the way that electric and magnetic fields act on charges and currents, Maxwell’s equations give a complete description of electromagnetic phenomena. These equations paved the way for Einstein’s special theory of relativity, which further illuminated the unified structure of electromagnetism. Maxwell’s achievement initiated great advances in understanding and application of electromagnetic theory, including the realization that light is an electromagnetic wave and that other forms of electromagnetic radiation could be produced from oscillating charges and currents.
Discussion/Investigation Questions
3.1 Think of some devices that generally are not run on electrical power. Discuss the reasons why other power sources are favored for these devices.
3.2 Can you identify the largest component of your personal electric power consumption?
3.3 Most countries provide AC electricity at 220–240 V to residential customers. In the US and parts of South America 110–120 V is provided. How do resistive losses in household wires compare between the two? Given the result, why do you think even higher voltages are not used?
3.4 Discuss the statement that Lenz’s law follows from conservation of energy. Consider, for example, the scenario of Figure 3.22(b) if the sign of eq. (3.64) were reversed.
3.5 Currents produce magnetic fields, and magnetic fields exert forces on currents. When a current flows through a wire, does the magnetic field it produces act to make the wire expand or contract? What about the force on the wires in a solenoid? Does it act to make the solenoid expand or contract? These are significant effects for wires carrying very large currents.
Problems
3.1 The electric field outside a charged conducting sphere is the same as if the charge were centered at its origin. Use this fact to calculate the capacitance of a sphere of radius R, taking the second conductor to be located at infinity. What is the most charge you can store on an otherwise isolated spherical conductor of radius R without a breakdown of the surrounding air as discussed in Example 3.2? What is the maximum energy you can store on a conducting sphere of radius 1 mm?
3.2 [T] Prove Gauss’s law from Coulomb’s law for static charge distributions by showing that the electric field of a single charge satisfies the integral form of Gauss’s law and then invoking linearity.
3.3 How much energy can you store on a parallel plate capacitor with m, cm2, and , assuming that the breakdown field of the dielectric is the same as for air?
3.4 [T] Starting from Gauss’s law and ignoring edge effects (i.e. assume that the plates are very large and the electric field is uniform and perpendicular to the plates), derive the formula for the capacitance of a parallel plate capacitor, . Refer to Figure 3.11. You can assume that the electric field vanishes outside the plates defining the capacitor. Show that the energy stored in the capacitor, , can be written as the integral of over the region within the capacitor (as asserted in eq. (3.20)).
3.5 Suppose that a capacitor with capacitance C is charged to some voltage V and then allowed to discharge through a resistance R. Write an equation governing the rate at which energy in the capacitor decreases with time due to resistive heating. Show that the solution of this equation is . You can ignore the internal resistance of the capacitor. Show that the heat produced in the resistor equals the energy originally stored in the capacitor.
3.6 The dielectrics in capacitors allow some leakage current to pass from one plate to the other. The leakage can be parameterized in terms of a leakage resistance . This limits the amount of time a capacitor can be used to store electromagnetic energy. The circuit diagram describing an isolated capacitor, slowly leaking charge, is therefore similar to the one analyzed in Problem 3.5. The Maxwell BCAP0310 ultracapacitor (see §37.4.2) is listed as having a capacitance of 310 F with a voltage up to 2.85 V, and a maximum leakage current of 0.45 mA when fully charged. Take this to be the current at . What is the leakage resistance of the BCAP0310? Estimate the time scale over which the charge on the capacitor falls to of its initial value.
3.7 A cloud-to-ground lightning bolt can be modeled as a parallel plate capacitor discharge, with Earth’s surface and the bottom of the cloud forming the two plates (see Example 3.2). A particular bolt of lightning passes to the ground from a cloud bottom at a height of 300 m. The bolt transfers a total charge of 5 C and a total energy of 500 MJ to the ground, with an average current of 50 kA. How long did the lightning bolt last? What was the electric field strength in the cloud–earth capacitor just before it discharged? How does this electric field compare with the breakdown field of air (3 MV/m)? (It is now known that various effects cause lightning to begin at fields that are considerably smaller than the breakdown field.)
3.8 [T] Consider an electric dipole composed of two charges at positions . Write the exact electric field from the two charges and show that the leading term in an expansion in matches the E-field quoted in Box 3.2. [Hint: consider the binomial expansion, eq. (B.66).]
3.9 If each of the batteries used in the flashlight in Example 3.3 has an internal resistance of 0.5 Ω (in series with the circuit), what fraction of power is lost to Joule heating within the batteries?
3.10 [T] Consider two resistors placed in series, one after the other, in an electric circuit connected to a battery with voltage V. Show that the effective resistance of the pair is by using the fact that the current through both resistors is the same while voltages add. Now connect the resistors in parallel, so that the voltage across both resistors is V. Compute the total current and show that the effective resistance satisfies .
3.11 An appliance that uses 1000 W of power is connected by 12 gauge (diameter 2.053 mm) copper wire to a 120 V (RMS) AC outlet. Estimate the power lost per meter, , (in W/m) as resistive heating in the wire. (Remember that the wire to the appliance has two separate current-carrying wires in a single sheath.)
3.12 Electrical power is often used to boil water for cooking. Here are the results of an experiment: a liter of water initially at 30℃ was boiled on an electric stove top burner. The burner is rated at 6.67 A (maximum instantaneous current) and 240 V. It took 7 minutes 40 seconds to reach the boiling point. The experiment was repeated using an “electric kettle.” The kettle is rated at 15 A (maximum instantaneous current again) and uses ordinary (US) line voltage of 120 V. This time it took 4 minutes and 40 seconds to reach boiling. What are the power outputs and resistances of the burner and the kettle? Compare the efficiency for boiling water of the stove top burner and the kettle. To what do you attribute the difference?
3.13 [T] Use the magnetic force law (3.42) and the definition of work to show that magnetic forces do no work. [Hint: Consider the vector identity (B.6).]
3.14 [T] Show that a charged particle moving in the xy-plane in the presence of a magnetic field will move in a circle. Compute the radius of the circle and frequency of rotation in terms of the speed υ of the particle. Show that adding a constant velocity in the z-direction still gives a solution with no further acceleration.
3.15 [T] Review how magnetic fields are calculated from Ampere’s law by computing (a) the magnetic field due to a straight wire and (b) the magnetic field in the interior of a very long solenoid. The contours shown in red in Figure 3.18 will help. You may use the fact that the magnetic field vanishes just outside the outer boundary of the solenoid in (b).
3.16 [T] Show that the force per unit area on the windings of an air-core solenoid from the magnetic field of the solenoid itself is of order . Check that the dimensions of this expression are correct and estimate in pascals if T.
3.17 [T] Derive eq. (3.52) from eq. (3.43) and eq. (3.50). Make sure you get the both the direction and magnitude.
3.18 An electric motor operates at 1000 rpm with an average torque of Nm. What is its power output? If it is running on 1.2 A of current, estimate the back-EMF from the rotor.
3.19 Consider the motor described in Box 3.4. If the resistance in the wire wrapping the rotor is 1 Ω, compute the energy lost under the conditions described. What fraction of energy is lost to Joule heating in this situation? If the current is doubled but the rotation rate is kept fixed, how do the output power, Joule heating losses, and fraction of lost energy change?
3.20 [T] In §3.4.2 we derived the EMF on a wire loop rotating in a magnetic field using the Lorentz force law to compute the forces on the mobile charges. Although Faraday’s law of induction (3.64) does not apply in a rotating reference frame, show that the same relation (3.62) follows from Faraday’s law.
3.21 [T] Explain why the integral that appears in eq. (3.65) is independent of the choice of surface S. [Hint: make use of eq. (3.46).]
3.22 [T] Consider a long, hollow solenoid of volume . Show that, ignoring end effects, its inductance is , and that the magnetic energy it stores can be written in the form of eq. (3.71).
3.23 [T] Prove that the mutual inductance is a symmetric relation, , by computing the energy stored when current is established in loop 1 and then current is established in loop 2. Then set up the same currents in the opposite order.
3.24 Design a transmission system to carry power from wind farms in North Dakota to the state of Illinois (about 1200 km). The system should handle Illinois’s summertime electricity demand of 42 GW. Land for transmission towers is at a premium, so the system uses very high voltage ( kV). Assume that the electricity is transmitted as ordinary alternating current, although in practice three-phase power (see §38.3.1) would be used. Assume that each tower can carry 36 aluminum cables (nine lines, each consisting of four conductors separated by non-conducting spacers). The conductors are 750 mm2 in cross section. How many separate strings of transmission towers are needed if the transmission losses are to be kept below 5%? Assume a purely resistive load.
3.25 [T] Consider the transformer in Figure 3.25. Suppose the load is a resistor R and that the transformer is ideal, with and all magnetic flux lines passing through both inductors. Show that the voltage drop across the resistor is and that the time-averaged power consumed in the resistor is .
3.26 [T] Take the divergence of both sides of eq. (3.74), use Coulomb’s law on the left and current conservation on right to show that the equation is consistent.
* * *
1 Energy density in this context is the volumetric energy density, meaning energy per unit volume. Elsewhere in this text, energy density generally refers to gravimetric energy density mass, also known as specific energy, though occasionally we deal with volumetric energy density. If not otherwise specified, the two types of energy density can be distinguished by units.
2 Note the difference between mechanical potential energy – for example of a massive object in a gravitational field, which has units of energy – and electrostatic potential, which is measured in units of energy per unit charge. The symbol V is widely used for both kinds of potential; we follow this standard terminology in the book but the reader is cautioned to be careful to keep track of which units are relevant in any given case.
3 This can be shown under the assumptions that the region containing charges is bounded and the electric field approaches zero at large distances outside the charge region.
4 Plasmas, which form at the extremely high temperatures found in the interiors of stars and in plasma fusion research devices here on Earth, are an exception where both positive ions and negative electrons can move about. Plasmas are discussed further in §19.
5 Unless the material is a superconductor, which has no electrical resistance and which can carry a current indefinitely without an external voltage to drive it. Superconductors are discussed further in §37 (Energy storage).
6 Other forms of oscillating voltages occur, in particular three-phase AC is quite common and is described further in §38. Here we limit ourselves to sinusoidally oscillating voltage in a single wire, and use the term AC voltage to refer to a voltage of the form eq. (3.34).
7 This, as noted above, provides the definition of the ampere in SI units.
8 For a further discussion of reference frames and Einstein’s special theory of relativity, see §21.
CHAPTER 4
Waves and Light
Waves have the capacity to propagate energy over great distances. Ocean waves can transmit energy halfway around the world, even though the molecules of water do not themselves experience any significant net motion. Sound waves in air and seismic waves propagating through Earth’s interior likewise transmit energy without net motion of material. Indeed, many of the energy systems described in this book depend at a fundamental level upon energy propagation by waves. Most notably, all of the energy that is transmitted from the Sun to Earth comes in the form of electromagnetic waves, composed of propagating excitations of the electric and magnetic fields (§22). This incident electromagnetic solar energy is the source of almost all energy used by humanity. Other examples of energy systems based on waves include technologies that harvest the energy in ocean surface waves (§31.2), the use of seismic waves to probe the structure of Earth’s interior in search of geothermal (§32) or fossil fuel (§33) resources, and the vibrational waves in crystals called phonons that facilitate the conversion of light into electricity in photovoltaic cells (§25). In the atomic and subatomic world, particles themselves are in many ways best characterized in terms of a quantum mechanical wave function that replaces the classical position of the particle with a diffuse probability distribution(§7).
