Inductors

Type of physical science: Classical physics

Field of study: Electromagnetism

Just as a capacitor is a conservative electrical device that can store electric field energy in the space between its plates, an inductor is a conservative electrical device that can store magnetic field energy in the space surrounding it. Inductors play important roles in distributing electrical energy through electrical circuits.

Overview

According to Michael Faraday's law of induction, a time-varying magnetic flux through a loop of wire causes an induced electromotive force (emf) to exist in the wire. Consider a pair of coils, one with a current I through it and one without current. The first coil sets up a magnetic field in the space surrounding it, a field that is intercepted by the second coil. When the current in the first coil is changing, the time-dependent magnetic flux φ intercepted by the second coil causes an induced emf in that coil and an induced current flows while the intercepted flux φ is varying.

Two coils are not necessary to demonstrate such an inductive effect. An induced emf can appear in a single coil if the current flowing in that coil is changing. This process is referred to as self-induction, and the emf is referred to as a self-induced emf. This self-induction process is described by Faraday's law of induction precisely as was the mutual induction process.

Consider a loop of wire that carries a current I. This generates a magnetic field B. The magnitude of this field can be determined analytically using either the Biot-Savart law or Ampere's law. In either case, the magnetic field strength B is proportional to the current I. Since the magnetic flux φ is proportional to the magnetic field strength B, which itself is proportional to the current I, the magnetic flux φ is proportional to the current I. The constant of proportionality between the magnetic flux φ and the current I is called the self-inductance L. Thus, the self-inductance L is the ratio of the magnetic flux φ to the current I that established it. The self-inductance L is independent of the current flowing through the coil and depends purely upon geometric factors such as the cross sectional area of the coil, the length of the coil, and the number of turns of wire in the coil. The unit of self-inductance is given as the henry in honor of the American physicist Joseph Henry. (A henry is given as a weber, the unit of magnetic flux, per unit ampere, the unit of current.)

In order to calculate the inductance of a given coil geometry, first imagine that a current I flows through the wire composing the coil. Next, the magnetic field B created by this current I must be calculated either directly from the Biot-Savart law or through judicious use of Ampere's law when a high degree of symmetry is present. Then, the magnitude of the magnetic flux φ must be evaluated. This will always be proportional to the current I if the calculation is correct. Finally, from the definition of self-inductance, calculate L from φ divided by I. Self-inductance L is always a positive constant.

The voltage across an inductor is self-induced, and by Faraday's law of induction, that is LdI/dt in magnitude. The power delivered to the inductor is given by the multiplication of the current and this voltage or LIdI/dt. Power is the time rate of change of energy; thus, in a time dt, an amount of energy dU = LIdI is delivered to the inductor. Integration yields U = 1/2LI2 as the energy stored by the inductor. Just as the energy stored in a capacitor is associated with the electric field between the plates rather than the plates themselves, the energy stored by an inductor is associated with the magnetic field surrounding the inductor rather than in the wire composing the inductor.

As a first example of an inductor, consider a solenoid. This is a long straight wire that is wrapped continuously in tightly consecutive turns around an object usually of circular or rectangular cross section. If the solenoid has N total turns, cross sectional area A, and length l, Ampere's law proves that the magnetic field that results from a current I flowing through the solenoid is spatially uniform inside the solenoid and given by B = m^oNI/l. The magnetic flux intercepted by the entire solenoid is the product of the magnetic field and total cross-sectional area or φ = &#03BC;^oN²IA/l. By the definition of self-inductance, L is given as φ divided by I or L = &#03BC;^oN²A/l for the case of the solenoid.

Another common example of an inductor is a toroid. This is constructed by winding a long wire continuously in tight turns around the circular cross section of a torus, a geometric shape resembling a doughnut or bagel. If the toroid has N total turns of wire, an inner diameter a, and an outer diameter b, it can be shown that its self-inductance L is given by &#03BC;^oN²(b-a).

In electrical circuits, inductors are often connected together in networks with other inductors or other circuit elements. What is the net effect of connecting several inductors together in series? Because circuit elements connected in series share the same current flowing through them, it can be shown that inductors in series add together linearly. What is the net effect of connecting several inductors together in parallel? Because circuit elements connected in parallel share the same potential difference across them, it can be shown that the effective inductance of such a combination is the inverse of the sum of inverses of the individual inductances.

Applications

As an example of the behavior of an inductor in an electrical circuit, consider the following simple circuit. Suppose a source of electromotive force (emf) (Vo) is connected to a switch, a resistor (R), and an inductor (L). Initially, the switch is open, meaning the circuit is not completed. At time t = 0, the switch is closed. Until that time, no current will be drawn out of the battery. At t = 0, the current will increase from zero to its final value. Gustav Robert Kirchhoff's circuital law, which states that the sum of net emfs equals the sum of potential drops in a circuital loop, applied to this simple circuit provides a differential equation that describes the time evolution of the current build-up from an initial zero value to a final equilibrium value. At a time t greater than zero, the current I flowing through the circuit will provide a potential drop across the resistor given by IR and a potential drop across the inductor given by LdI/dt. Thus, the circuit equation for the current is Vo = LdI/dt + IR. This is a first-order linear differential equation. Its solution is easily obtained. That solution involves an exponential behavior in time. The current I builds up from an initial zero value toward a final equilibrium value (given by Vo/R), with an exponential behavior whose decay from unity to zero (as time goes from zero to infinity) is regulated by the ratio of time to the inductive time constant tau, where &#03C4; = L/R.

In such a circuit, the inductance delays the ability of the current to achieve the value it would have if there was only resistance in the circuit. At t = 0, because the current is initially zero, there is no voltage across the resistor. Thus, at t = 0, the full strength of the emf is across the inductor. As time progresses and current builds up, the voltage across the resistor increases, while the voltage across the inductor decreases until after a very long time (as time goes toward infinity) the voltage across the inductor is zero and all the strength of the emf is across the resistor. (This explains why the equilibrium current is given as Io = Vo/R by Ohm's law.) The inductive time constant determines the rate at which that exchange of voltage between the inductor and resistor takes place. In a time equal to one inductive time constant the current builds up to 63 percent of its final equilibrium value Io.

During a time of a second inductive time constant, the current achieves 63 percent of the missing 37 percent. During each successive time constant period, 63 percent of the previously missing current is achieved.

Suppose now that a sufficiently long period of time has transpired since the switch was closed and equilibrium current Io has been achieved in the circuit. At this point, suppose the emf is disconnected and the measurement of time is reset to zero as the emf is removed from the circuit. Under these circumstances, Kirchhoff's circuital law yields the following simple ordinary linear differential equation for the time dependence of the circuit current: LdI/dt + IR = 0. The solution to this equation has the following characteristics: If the emf was removed from the circuit when time was reset to zero, the initial current following in the circuit is Io = Vo/R. Now, as time progresses, the current exponentially decays to zero, losing 63 percent of Io in a time equal to one inductive time constant. At the new time t = 0, the voltage across the resistor is Vo and the voltage across the inductor is -Vo. As time progresses, the voltage across the resistor exponentially decays to zero as the current decays to zero. As time progresses, the voltage across the inductor exponentially builds up from -Vo to zero. After a long time, there is no longer any current flow in the circuit.

Inductors are conservative circuit elements storing energy in the magnetic field surrounding them. Yet, a resistor is a dissipative circuit element. As a current I passes through a resistor R, joule heating results in an energy loss at a rate given by I2R. It is this power loss that causes the current to decay away to zero in the previous example, after the emf is removed from the circuit.

Consider the behavior of a circuit consisting of an initially charged capacitor C connected to a switch (initially open) and an inductor L. Assuming no resistance in the leads, this circuit is a completely conservative example of electromagnetic energy transformation. If the switch is closed at time t = 0, charge will flow off the capacitor plates and constitute a current I, whose time dependence is governed by the following equation obtained from Kirchhoff's circuital law: LdI/dt + q/c = 0 or Ld2q/dt2 + q/C = 0, since current I is the time derivative of charge q. Solutions to differential equations of this variety are oscillatory in nature rather than exponential. If the initial charge is qo, the charge on the plates after t = 0 will oscillate cosinusoidally with an angular frequency given by the square root of the inverse product of L and C. This circuit, often referred to as an LC circuit, is quite illustrative of electromagnetic energy oscillations. Initially, there is no current flow, hence, no magnetic field around the inductor. All the energy in the circuit is stored in the electric field between the plates of the capacitor. As charge bleeds off the capacitor plates, current builds up. Energy is diminished in the electric field of the capacitor and equally increased in the magnetic field of the inductor. This continues until one quarter of the way through an oscillation, when all charge has been removed from the capacitor and the circuit current is at maximum. Now, all the circuit energy is stored in the magnetic field surrounding the inductor. As time progresses, charge is returned to the capacitor plates but in the opposite direction, meaning that current flow has been reversed. Energy stored in the electric field of the capacitor increases at the expense of energy stored in the magnetic field around the inductor until the current has totally diminished at a time equal to one-half of a complete oscillation. At this point, all the charge is once again on the capacitor plates, but the electric field lines are reversed relative to the initial condition because the plate originally positive is now negative. Now, no energy is stored by the inductor. As time continues to progress, charge bleeds off the capacitor again, making current run oppositely directed to that in the previous quarter cycle. At a time given by three-quarters of the period of oscillation, the magnetic field is again at its maximum value but its direction is reversed relative to the orientation it had when it was previously at maximum. At this time, no energy is stored in the capacitor. During the final quarter cycle, charge returns to the capacitor plates in such a way as to restore the conditions of the circuit when the oscillation began. This cycle continues to transform electric energy repetitively into magnetic energy without loss.

Context

Self-induction is a process that clearly links magnetism to electricity. Investigation of induction phenomena led to the realization that the subjects of electricity and magnetism were intimately related and could be unified into a single theory under the title of electromagnetism.

Work of early researchers was capped by the Scottish physicist James Clerk Maxwell when he elucidated the set of four equations of electromagnetism that now bear his name. One immediately obvious aspect of Maxwell's equations is the ability of a time-dependent magnetic flux (Faraday's law) to generate an electric field and a time-dependent electric flux (Ampere's law) to generate a magnetic field. This symmetry is fundamental to electromagnetism.

One early experiment that helped pave the way toward this unification expressed in Maxwell's equations was performed by Hans Christian rsted. He was the first to observe that electricity would affect a magnetic field. rsted wound a coil of conducting wire around a magnetic compass. When that coil was connected to a source of emf, the current flowing through the wire established a magnetic field that moved the compass needle away from the North-South direction.

Another simple experiment that shows how a magnetic field can give rise to an electric field is to consider a coil of wire that is connected to a voltmeter but no source of emf. Without an emf, the voltmeter naturally reads zero voltage across the coil because no current is flowing in the coil. If a bar magnet were moved through the coil, during the time that the magnet was moving, the voltmeter would register a voltage. If the magnet is stopped while inside the coil, the voltmeter does not register a voltage, as only while the magnetic flux intercepted by the coil is changing is an emf induced in the coil. If the magnet were to be withdrawn from the coil, then the voltmeter would again register a voltage, but one of an opposite sense as compared with that registered when the magnet was inserted into the coil. This observation ultimately led to a full appreciation of the induction process as a means of generating electricity. In a simple generator, a coil is rotated inside a magnetic field to change the magnetic flux intercepted by the coil and to generate an emf across the coil.

Principal terms

CURRENT: charges in motion

INDUCTANCE: the ratio of magnetic flux to current, which generates the magnetic flux

MAGNETIC FIELD: a mathematical vector construct used to describe the interaction of currents; electromagnetic energy can be stored in a magnetic field

SOLENOIDS: a coil of wire formed by tightly wrapping individual turns of identical cross-sectional area; generally, the length is large compared to the diameter

TOROIDS: a coil of wire formed by tightly wrapping individual turns around an object with a doughnut shape

Bibliography

Frankyl, Daniel R. ELECTROMAGNETIC THEORY. Englewood Cliffs, N.J.: Prentice-Hall, 1986. Although targeted for the undergraduate physics student, the textbook qualitatively describes electric and magnetic interactions rather well.

Halliday, David, and Robert Resnick. FUNDAMENTALS OF PHYSICS. 3d rev. ed. New York: John Wiley & Sons, 1988. This version of the classic undergraduate physics text contains descriptive discussions of current topics in physics research as well as thoroughly covering basic physical theory. Excellent discourse on electromagnetic theory. Highly illustrated.

Lorrain, Paul, and Dale Corson. ELECTROMAGNETIC FIELDS AND WAVES. New York: W. H. Freeman, 1990. A classic text used for undergraduate instruction in electromagnetism. Many sections are accessible to the layperson with a good background of basic knowledge.

Nayfeh, Munir H., and Morton K. Brussel. ELECTRICITY AND MAGNETISM. New York: John Wiley & Sons, 1985. Undergraduate level instruction in mathematical and experimental aspects of electromagnetism. Accessible to those with modest skills in calculus of vectors.

Ohanian, Hans C. PHYSICS. New York: W. W. Norton, 1985. Although calculus-based, the text is not mathematically rigorous. Accessible to those with modest mathematics skills. Excellent descriptions of difficult physical concepts. Well illustrated and provides practical examples.

Reitz, John R., and Frederick J. Milford. FOUNDATIONS OF ELECTROMAGNETIC THEORY. Reading, Mass.: Addison-Wesley, 1987. Another classic text for undergraduate instruction in electromagnetism. Excellent field concept descriptions. Accessible to interested readers with moderate mathematical skills.

Tomboulian, D. H. ELECTRIC AND MAGNETIC FIELDS. New York: Harcourt, Brace & World, 1965. Although somewhat dated, this text is ideal for the amateur interested in electromagnetism without requiring higher-level mathematics. Well illustrated; includes many diagrams and sample calculations.

Wangness, Roald K. ELECTROMAGNETIC FIELDS. New York: John Wiley & Sons, 1986. Highly mathematical, but descriptions of basic concepts are highly illuminating. For the seriously interested person with good mathematical skills.

Wilson, Jerry D., and John Kinard. COLLEGE PHYSICS. Boston: Allyn & Bacon, 1990. Basic text, excellent for the reader not familiar with calculus. Targeted more for the undergraduate physics major, this text contains qualitative sections on the workings of devices such as inductors and capacitors, as well as electromagnetic interactions with matter.

Charges and Currents

Forces on Charges and Currents