OpenStax-CNX module: m Inductance. OpenStax College. Abstract

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1 OpenStax-CNX modue: m Inductance OpenStax Coege This work is produced by OpenStax-CNX and icensed under the Creative Commons Attribution License 3.0 Cacuate the inductance of an inductor. Cacuate the energy stored in an inductor. Cacuate the emf generated in an inductor. Abstract 1 Inductors Induction is the process in which an emf is induced by changing magnetic ux. Many exampes have been discussed so far, some more eective than others. Transformers, for exampe, are designed to be particuary eective at inducing a desired votage and current with very itte oss of energy to other forms. Is there a usefu physica quantity reated to how eective a given device is? The answer is yes, and that physica quantity is caed inductance. Mutua inductance is the eect of Faraday's aw of induction for one device upon another, such as the primary coi in transmitting energy to the secondary in a transformer. See Figure 1, where simpe cois induce emfs in one another. Version 1.4: Sep 11, :11 am

2 OpenStax-CNX modue: m Figure 1: These cois can induce emfs in one another ike an inecient transformer. Their mutua inductance M indicates the eectiveness of the couping between them. Here a change in current in coi 1 is seen to induce an emf in coi 2. (Note that "E 2 induced" represents the induced emf in coi 2.) In the many cases where the geometry of the devices is xed, ux is changed by varying current. We therefore concentrate on the rate of change of current, I/ t, as the cause of induction. A change in the current I 1 in one device, coi 1 in the gure, induces an emf 2 in the other. We express this in equation form as emf 2 = M I 1 t, (1) where M is dened to be the mutua inductance between the two devices. The minus sign is an expression of Lenz's aw. The arger the mutua inductance M, the more eective the couping. For exampe, the cois in Figure 1 have a sma M compared with the transformer cois in. Units for M are (V s) /A = Ω s, which is named a henry (H), after Joseph Henry. That is, 1H = 1 Ω s. Nature is symmetric here. If we change the current I 2 in coi 2, we induce an emf 1 in coi 1, which is given by emf 1 = M I 2 t, (2) where M is the same as for the reverse process. Transformers run backward with the same eectiveness, or mutua inductance M. A arge mutua inductance M may or may not be desirabe. We want a transformer to have a arge mutua inductance. But an appiance, such as an eectric cothes dryer, can induce a dangerous emf on its case if the mutua inductance between its cois and the case is arge. One way to reduce mutua inductance M is to counterwind cois to cance the magnetic ed produced. (See Figure 2.)

3 OpenStax-CNX modue: m Figure 2: The heating cois of an eectric cothes dryer can be counter-wound so that their magnetic eds cance one another, greaty reducing the mutua inductance with the case of the dryer. Sef-inductance, the eect of Faraday's aw of induction of a device on itsef, aso exists. When, for exampe, current through a coi is increased, the magnetic ed and ux aso increase, inducing a counter emf, as required by Lenz's aw. Conversey, if the current is decreased, an emf is induced that opposes the decrease. Most devices have a xed geometry, and so the change in ux is due entirey to the change in current I through the device. The induced emf is reated to the physica geometry of the device and the rate of change of current. It is given by emf = L I t, (3) where L is the sef-inductance of the device. A device that exhibits signicant sef-inductance is caed an inductor, and given the symbo in Figure 3.

4 OpenStax-CNX modue: m Figure 3 The minus sign is an expression of Lenz's aw, indicating that emf opposes the change in current. Units of sef-inductance are henries (H) just as for mutua inductance. The arger the sef-inductance L of a device, the greater its opposition to any change in current through it. For exampe, a arge coi with many turns and an iron core has a arge L and wi not aow current to change quicky. To avoid this eect, a sma L must be achieved, such as by counterwinding cois as in Figure 2. A 1 H inductor is a arge inductor. To iustrate this, consider a device with L = 1.0H that has a 10 A current owing through it. What happens if we try to shut o the current rapidy, perhaps in ony 1.0 ms? An emf, given by emf = L ( I/ t), wi oppose the change. Thus an emf wi be induced given by emf = L ( I/ t) = (1.0H) [(10 A) / (1.0ms)] = 10,000 V. The positive sign means this arge votage is in the same direction as the current, opposing its decrease. Such arge emfs can cause arcs, damaging switching equipment, and so it may be necessary to change current more sowy. There are uses for such a arge induced votage. Camera ashes use a battery, two inductors that function as a transformer, and a switching system or osciator to induce arge votages. (Remember that we need a changing magnetic ed, brought about by a changing current, to induce a votage in another coi.) The osciator system wi do this many times as the battery votage is boosted to over one thousand vots. (You may hear the high pitched whine from the transformer as the capacitor is being charged.) A capacitor stores the high votage for ater use in powering the ash. (See Figure 4.) Figure 4: Through rapid switching of an inductor, 1.5 V batteries can be used to induce emfs of severa thousand vots. This votage can be used to store charge in a capacitor for ater use, such as in a camera ash attachment. It is possibe to cacuate L for an inductor given its geometry (size and shape) and knowing the magnetic ed that it produces. This is dicut in most cases, because of the compexity of the ed created. So in this text the inductance L is usuay a given quantity. One exception is the soenoid, because it has a very

5 OpenStax-CNX modue: m uniform ed inside, a neary zero ed outside, and a simpe shape. It is instructive to derive an equation for its inductance. We start by noting that the induced emf is given by Faraday's aw of induction as emf = N ( Φ/ t) and, by the denition of sef-inductance, as emf = L ( I/ t). Equating these yieds Soving for L gives emf = N Φ t = L I t. (4) L = N Φ I. (5) This equation for the sef-inductance L of a device is aways vaid. It means that sef-inductance L depends on how eective the current is in creating ux; the more eective, the greater Φ/ I is. Let us use this ast equation to nd an expression for the inductance of a soenoid. Since the area A of a soenoid is xed, the change in ux is Φ = (BA) = A B. To nd B, we note that the magnetic ed of a soenoid is given by B = µ 0 ni = µ 0 NI. (Here n = N/, where N is the number of cois and is. Substituting Φ into the soenoid's ength.) Ony the current changes, so that Φ = A B = µ 0 NA I L = N Φ I gives This simpies to L = N Φ I = N µ 0NA I. (6) I L = µ 0N 2 A (soenoid). (7) This is the sef-inductance of a soenoid of cross-sectiona area A and ength. Note that the inductance depends ony on the physica characteristics of the soenoid, consistent with its denition. Exampe 1: Cacuating the Sef-inductance of a Moderate Size Soenoid Cacuate the sef-inductance of a 10.0 cm ong, 4.00 cm diameter soenoid that has 200 cois. Strategy This is a straightforward appication of L = µ0n 2 A, since a quantities in the equation except L are known. Soution Use the foowing expression for the sef-inductance of a soenoid: L = µ 0N 2 A. (8) The cross-sectiona area in this exampe is A = πr 2 = ( ) ( m) 2 = m 2, N is given to be 200, and the ength is m. We know the permeabiity of free space is µ 0 = 4π 10 7 T m/a. Substituting these into the expression for L gives Discussion This soenoid is moderate in size. moderate. L = (4π 10 7 T m/a)(200) 2 ( m 2 ) 0.100m = mh. Its inductance of neary a miihenry is aso considered One common appication of inductance is used in trac ights that can te when vehices are waiting at the intersection. An eectrica circuit with an inductor is paced in the road under the pace a waiting car wi stop over. The body of the car increases the inductance and the circuit changes sending a signa to the trac ights to change coors. Simiary, meta detectors used for airport security empoy the same technique. A coi (9)

6 OpenStax-CNX modue: m or inductor in the meta detector frame acts as both a transmitter and a receiver. The pused signa in the transmitter coi induces a signa in the receiver. The sef-inductance of the circuit is aected by any meta object in the path. Such detectors can be adjusted for sensitivity and aso can indicate the approximate ocation of meta found on a person. (But they wi not be abe to detect any pastic exposive such as that found on the underwear bomber.) See Figure 5. Figure 5: The famiiar security gate at an airport can not ony detect metas but aso indicate their approximate height above the oor. (credit: Aexbuirds, Wikimedia Commons) 2 Energy Stored in an Inductor We know from Lenz's aw that inductances oppose changes in current. There is an aternative way to ook at this opposition that is based on energy. Energy is stored in a magnetic ed. It takes time to buid up energy, and it aso takes time to depete energy; hence, there is an opposition to rapid change. In an inductor, the magnetic ed is directy proportiona to current and to the inductance of the device. It can be shown that the energy stored in an inductor E ind is given by This expression is simiar to that for the energy stored in a capacitor. E ind = 1 2 LI2. (10) Exampe 2: Cacuating the Energy Stored in the Fied of a Soenoid How much energy is stored in the mh inductor of the preceding exampe when a 30.0 A current ows through it? Strategy The energy is given by the equation E ind = 1 2 LI2, and a quantities except E ind are known. Soution

7 OpenStax-CNX modue: m Substituting the vaue for L found in the previous exampe and the given current into E ind = 1 2 LI2 gives E ind = 1 2 LI2 = 0.5 ( H ) (30.0 A) 2 = J. Discussion This amount of energy is certainy enough to cause a spark if the current is suddeny switched o. It cannot be buit up instantaneousy uness the power input is innite. (11) 3 Section Summary Inductance is the property of a device that tes how eectivey it induces an emf in another device. Mutua inductance is the eect of two devices in inducing emfs in each other. A change in current I 1 / t in one induces an emf emf 2 in the second: emf 2 = M I 1 t, (12) where M is dened to be the mutua inductance between the two devices, and the minus sign is due to Lenz's aw. Symmetricay, a change in current I 2 / t through the second device induces an emf emf 1 in the rst: emf 1 = M I 2 t, (13) where M is the same mutua inductance as in the reverse process. Current changes in a device induce an emf in the device itsef. Sef-inductance is the eect of the device inducing emf in itsef. The device is caed an inductor, and the emf induced in it by a change in current through it is emf = L I t, (14) where L is the sef-inductance of the inductor, and I/ t is the rate of change of current through it. The minus sign indicates that emf opposes the change in current, as required by Lenz's aw. The unit of sef- and mutua inductance is the henry (H), where 1H = 1 Ω s. The sef-inductance L of an inductor is proportiona to how much ux changes with current. For an N-turn inductor, L = N Φ I. (15) The sef-inductance of a soenoid is L = µ 0N 2 A (soenoid), (16) where N is its number of turns in the soenoid, A is its cross-sectiona area, is its ength, and µ 0 = 4π 10 7 T m/a is the permeabiity of free space. The energy stored in an inductor E ind is E ind = 1 2 LI2. (17)

8 OpenStax-CNX modue: m Conceptua Questions Exercise 1 How woud you pace two identica at cois in contact so that they had the greatest mutua inductance? The east? Exercise 2 How woud you shape a given ength of wire to give it the greatest sef-inductance? The east? Exercise 3 Verify, as was concuded without proof in Exampe 1 (Cacuating the Sef-inductance of a Moderate Size Soenoid), that units of T m 2 /A = Ω s = H. 5 Probems & Exercises Exercise 4 (Soution on p. 10.) Two cois are paced cose together in a physics ab to demonstrate Faraday's aw of induction. A current of 5.00 A in one is switched o in 1.00 ms, inducing a 9.00 V emf in the other. What is their mutua inductance? Exercise 5 If two cois paced next to one another have a mutua inductance of 5.00 mh, what votage is induced in one when the 2.00 A current in the other is switched o in 30.0 ms? Exercise 6 (Soution on p. 10.) The 4.00 A current through a 7.50 mh inductor is switched o in 8.33 ms. What is the emf induced opposing this? Exercise 7 A device is turned on and 3.00 A ows through it ms ater. What is the sef-inductance of the device if an induced 150 V emf opposes this? Exercise 8 Starting with emf 2 = M I1 t, show that the units of inductance are (V s) /A = Ω s. Exercise 9 Camera ashes charge a capacitor to high votage by switching the current through an inductor on and o rapidy. In what time must the A current through a 2.00 mh inductor be switched on or o to induce a 500 V emf? Exercise 10 (Soution on p. 10.) A arge research soenoid has a sef-inductance of 25.0 H. (a) What induced emf opposes shutting it o when 100 A of current through it is switched o in 80.0 ms? (b) How much energy is stored in the inductor at fu current? (c) At what rate in watts must energy be dissipated to switch the current o in 80.0 ms? (d) In view of the answer to the ast part, is it surprising that shutting it down this quicky is dicut? Exercise 11 (a) Cacuate the sef-inductance of a 50.0 cm ong, 10.0 cm diameter soenoid having 1000 oops. (b) How much energy is stored in this inductor when 20.0 A of current ows through it? (c) How fast can it be turned o if the induced emf cannot exceed 3.00 V? Exercise 12 (Soution on p. 10.) A precision aboratory resistor is made of a coi of wire 1.50 cm in diameter and 4.00 cm ong, and it has 500 turns. (a) What is its sef-inductance? (b) What average emf is induced if the 12.0 A current through it is turned on in 5.00 ms (one-fourth of a cyce for 50 Hz AC)? (c) What is its inductance if it is shortened to haf its ength and counter-wound (two ayers of 250 turns in opposite directions)?

9 OpenStax-CNX modue: m Exercise 13 The heating cois in a hair dryer are cm in diameter, have a combined ength of 1.00 m, and a tota of 400 turns. (a) What is their tota sef-inductance assuming they act ike a singe soenoid? (b) How much energy is stored in them when 6.00 A ows? (c) What average emf opposes shutting them o if this is done in 5.00 ms (one-fourth of a cyce for 50 Hz AC)? Exercise 14 (Soution on p. 10.) When the 20.0 A current through an inductor is turned o in 1.50 ms, an 800 V emf is induced, opposing the change. What is the vaue of the sef-inductance? Exercise 15 How fast can the 150 A current through a H inductor be shut o if the induced emf cannot exceed 75.0 V? Exercise 16 (Soution on p. 10.) Integrated Concepts A very arge, superconducting soenoid such as one used in MRI scans, stores 1.00 MJ of energy in its magnetic ed when 100 A ows. (a) Find its sef-inductance. (b) If the cois go norma, they gain resistance and start to dissipate therma energy. What temperature increase is produced if a the stored energy goes into heating the 1000 kg magnet, given its average specic heat is 200 J/kg ºC? Exercise 17 Unreasonabe Resuts A 25.0 H inductor has 100 A of current turned o in 1.00 ms. (a) What votage is induced to oppose this? (b) What is unreasonabe about this resut? (c) Which assumption or premise is responsibe?

10 OpenStax-CNX modue: m Soutions to Exercises in this Modue Soution to Exercise (p. 8) 1.80 mh Soution to Exercise (p. 8) 3.60 V Soution to Exercise (p. 8) (a) 31.3 kv (b) 125 kj (c) 1.56 MW (d) No, it is not surprising since this power is very high. Soution to Exercise (p. 8) (a) 1.39 mh (b) 3.33 V (c) Zero Soution to Exercise (p. 9) 60.0 mh Soution to Exercise (p. 9) (a) 200 H (b) 5.00ºC Gossary Denition 1: inductance a property of a device describing how ecient it is at inducing emf in another device Denition 2: mutua inductance how eective a pair of devices are at inducing emfs in each other Denition 3: henry the unit of inductance; 1 H = 1 Ω s Denition 4: sef-inductance how eective a device is at inducing emf in itsef Denition 5: inductor a device that exhibits signicant sef-inductance Denition 6: energy stored in an inductor sef-expanatory; cacuated by E ind = 1 2 LI2