Electro- magnetic Induction

In This Chapter:

✔ *Electromagnetic Induction*

✔ *Faraday’s Law*

✔ *Lenz’s Law*

✔ *The Transformer*

✔ *Self-Inductance*

✔ *Inductors in Combination*

✔ *Energy of a Current-Carrying Inductor*

__Electromagnetic Induction__

A current is produced in a conductor whenever the cur- rent cuts across magnetic ﬁeld lines, a phenomenon known as *electromagnetic induction*. If the motion is parallel to the ﬁeld lines of force, there is no effect. Electromagnetic induction originates in the force a magnetic ﬁeld exerts on a moving charge. When a wire moves across a magnetic ﬁeld, the electrons it contains

experience sideways forces that push them along the wire to cause a cur- rent. It is not even necessary for there to be relative motion of a wire and a source of magnetic ﬁeld, since a magnetic ﬁeld whose strength is changing has moving ﬁeld lines associated with it and a current will be induced in a conductor that is in the path of these moving ﬁeld lines.

When a straight conductor of length *l *is moving across a magnetic ﬁeld **B **with the velocity **v**, the emf induced in the conductor is given by

Induced emf = *V** _{e }*=

*Blv*

when **B**, **v**, and the conductor are all perpendicular to one another.

**Solved Problem 16**.**1 **The vertical component of the earth’s magnetic ﬁeld in a certain region is 3 × 10−5 T. What is the potential difference be- tween the rear wheels of a car, which are 1.5 m apart, when the car’s ve- locity is 20 m/s?

**Solution**. The real axle of the car may be considered as a rod of 1.5 m long-moving perpendicular to the magnetic ﬁeld’s vertical component. The potential difference between the wheels is therefore

__Faraday’s Law__

__Faraday’s Law__

Figure 16-1 shows a coil (called a *solenoid *) of *N *turns that encloses an area *A*. The axis of the coil is parallel to a magnetic ﬁeld **B**. According to *Faraday’s law of electromagnetic induction*, the emf induced in the coil when the product *BA *changes by D(*B**A*) in the time D*t *is given byThe quantity *BA *is called the *magnetic ﬂux *enclosed by the coil and is de- noted by the symbol F (Greek capital letter *phi*):

Figure 16-1

The unit of magnetic ﬂux is the *weber *(Wb), where 1 Wb = 1 T·m2. Thus, Faraday’s law can be written

__Lenz’s Law__

The minus sign in Faraday’s law is a consequence of *Lenz’s law*:

An induced current is always in such a direction that its own magnetic ﬁeld acts to oppose the effect that created it.

For example, if **B **is decreasing in magnitude in the situation of Fig- ure 16-1, the induced current in the coil will be counterclockwise in or- der that its own magnetic ﬁeld will add to **B **and so reduce the rate at which **B **is decreasing. Similarly, if **B **is increasing, the induced current in the coil will be clockwise so that its own magnetic ﬁeld will subtract from **B **and thus reduce the rate at which **B **is increasing.

__The Transformer__

__The Transformer__

A *transformer *consists of two coils of wire, usually wound on an iron core. When an alternating current is passed through one of the windings, the changing magnetic ﬁeld it gives rise to induces an alternating current in the other winding. The potential difference per turn is the same in both

primary and secondary windings, so the ratio of turns in the winding determines the ratio of voltages across them:

__Secondary turns__

Since the power *I*1*V*1 going into a transformer must equal the power *I*2*V*2 going out, where *I*1 and *I*2 are the primary and secondary currents, respectively, the ratio of currents is inversely proportional to the ratio of turns:

__Self-Inductance__

__Self-Inductance__

When the current in a circuit changes, the magnetic ﬁeld enclosed by the circuit also changes, and the resulting change in ﬂux leads to a *self-induced emf *ofHere D *I*/D*t *is the rate of change of the current, and *L *is a property of the circuit called its *self-inductance*, or, more commonly, its *inductance*. The minus sign indicates that the direction of *V**e *is such as to oppose the change in current D*I *that caused it.

The unit of inductance is the *henry *(H). A circuit or circuit element that has an inductance of 1 H will have a self-induced emf of 1 V when the current through it changes at the rate of 1 A/s. Because the henry is a rather large unit, the *millihenry *and *microhenry *are often used, where

1 millihenry = 1 mH = 10^{−3} H

1 microhenry = 1 mH = 10^{−6} H

A circuit element with inductance is called an *inductor*. A solenoid is an example of an inductor. The inductance of a solenoid iswhere m is the permeability of the core material, *N *is the number of turns,

*A *is the cross-sectional area, and *l *is the length of the solenoid.

###### Inductors in Combination

When two or more inductors are sufﬁciently far apart for them not to in- teract electromagnetically, their equivalent inductances when they are connected in series and in parallel are as follows:

Connecting coils in parallel reduces the total inductance to less than that of any of the individual coils. This is rarely done because coils are relatively large and expensive compared with other electronic components; a coil of the required smaller inductance would normally be used in the ﬁrst place.

Because the magnetic ﬁeld of a current-carrying coil extends beyond the inductor itself, the total inductance of two or more connected coils will be changed if they are close to one another. Depending

on how the coils are arranged, the total inductance may be larger or smaller than if the coils were farther apart. This effect is called *mutual inductance *and is not considered in the above formula.

**Solved Problem 16**.**2 **Find the equivalent inductances of a 5- and an 8mH inductor when they are connected in (*a*) series and (*b*) parallel.

###### Energy of a Current-Carrying Inductor

Because a self-induced emf opposes any change in an inductor, work has to be done against this emf to establish a current in the inductor. This work is stored as magnetic potential energy. If *L *is the inductance of an inductor, its potential energy when it carries the current *I *is

This energy powers the self-induced emf that opposes any decrease in the current through the inductor.