Inductance Chapter 29 opener. One of the great laws of physics is Faraday’s law of induction, which says that a changing magnetic flux produces an induced emf. This photo shows a bar magnet moving inside a coil of wire, and the galvanometer registers an induced current. This phenomenon of electromagnetic induction is the basis for many practical devices, including generators, alternators, transformers, tape recording, and computer memory.

Inductance (NOTE: “Coil”  Solenoid) Often, the induced flux must be included in a circuit analysis. When the switch is closed, a sudden change in current occurs in the coil. This current produces a magnetic field So, an emf and current are induced in the coil

Inductor A coil is type of circuit element called an inductor. Many inductors are constructed as small solenoids. But, almost any coil or loop will act as an inductor Whenever the current through an inductor changes, a voltage is induced in the inductor that opposes this change This phenomenon is called self-inductance The current changing through a coil induces a current in the same coil The induced current opposes the original applied current, from Lenz’s Law

Inductance of a Solenoid Faraday’s Law can be used to find the inductance of a solenoid. L is the symbol for inductance The EMF across the solenoid can be expressed in terms of the inductance L:

1 H = 1 V . s / A These results apply to all wire coils or loops. The value of L depends on the physical size and shape of the circuit element The voltage drop across an inductor is The SI unit of inductance is the Henry 1 H = 1 V . s / A

Mutual Inductance It is possible for the magnetic field of one coil to produce an induced current in a second coil. The coils are connected indirectly through the magnetic flux The effect is called mutual inductance

RL Circuit DC circuits may contain resistors, inductors, and capacitors. The voltage source is a battery or some other source that provides a constant voltage across its output terminals. Behavior of DC circuits with inductors Immediately after any switch is closed or opened, the induced emfs keep the current through all inductors equal to the values they had the instant before the switch was thrown After a switch has been closed or opened for a very long time, the induced emfs are zero

RL Circuit Example

RL Circuit Example, Analysis The presence of resistors and an inductor make the circuit an RL circuit The current starts at zero since the switch has been open for a very long time At t = 0, the switch is closed, inducing a potential across the inductor Just after t = 0, the current in the second loop is zero After the switch has been closed for a long time, the voltage across the inductor is zero

Time Constant for RL Circuit The current at time t is found by τ is called the time constant of the circuit For a single resistor in series with a single inductor, τ = L / R The voltage is given by VL = V e-t/τ

Real Inductors Most practical inductors are constructed by wrapping a wire coil around a magnetic material Filling a coil with magnetic material greatly increases the magnetic flux through the coil and therefore increases the induced emf The presence of magnetic material increases the inductance Most inductors contain a magnetic material inside which produces a larger value of L in a smaller package

Energy in an Inductor Energy is stored in the magnetic field of an inductor The energy stored in an inductor is PEind = ½ L I2 Very similar in form to the energy stored in the electric field of a capacitor The expression for energy can also be stated as g In terms of the magnetic field,

Energy contained in the magnetic field actually exists anywhere there is a magnetic field, not just in a solenoid Can exist in “empty” space The potential energy can also be expressed in terms of the energy density in the magnetic field. This expression is similar to the energy density contained in an electric field

Bicycle Odometers An odometer control unit is shown A permanent magnet is attached to a wheel A pickup coil is mounted on the axle support When the magnet passes over the pickup coil, a pulse is generated A computer keep tracks of the number of pulses Section 21.7

Ground Fault Interrupters A ground fault interrupter (GFI) is a safety device used in many household circuits It uses Faraday’s Law along with an electromechanical relay The relay uses the current through a coil to exert a force on a magnetic metal bar in a switch Section 21.7

GFI, cont. During normal operation, there is zero magnetic field in the relay If the current in the return coil is smaller, a non-zero magnetic field opens the relay switch and the current turns off

Electric Guitars An electric guitar uses Faraday’s Law to sense the motion of the strings The metal string passes near a pickup coil wound around a permanent magnet As the string vibrates, it produces a changing magnetic flux The resulting emf is sent to an amplifier and the signal can be played through speakers Section 21.7

Generators, Motors and Cars Motors and generators provide examples of conservation of energy and the conversion of energy from one type to another A hybrid car contains two motors and a generator The hybrid car “recaptures” some of the energy normally converted to heat when braking and stores it in batteries A hybrid car is a practical example of the conversion between mechanical and electrical energy Section 21.7

Induction from a Distance Assume a very long solenoid is inserted at the center of a single loop of wire The field from the solenoid at the outer loop is essentially zero Section 21.8

Induction from a Distance, cont. The field inside the solenoid at the center of the loop still produces a magnetic flux through the inner portion of the loop Energy is transferred across the empty space between the two conductors The energy is carried from the solenoid to the outer loop by an electromagnetic wave Section 21.8