Chapter 30 INDUCTANCE. Copyright 2012 Pearson Education Inc.

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1 Chapter 30 INDUCTANCE

2 Goals for Chapter 30 To learn how current in one coil can induce an emf in another unconnected coil To relate the induced emf to the rate of change of the current To calculate the energy in a magnetic field To analyze circuits containing resistors and inductors To describe electrical oscillations in circuits and why the oscillations decay

Introduction How does a coil induce a current in a neighboring coil. A sensor triggers the traffic light to change when a car arrives at an intersection.

3 Introduction How does a coil induce a current in a neighboring coil. A sensor triggers the traffic light to change when a car arrives at an intersection. How does it do this? Why does a coil of metal behave very differently from a straight wire of the same metal? We ll learn how circuits can be coupled without being connected together.

4 Mutual inductance Mutual inductance: A changing current in one coil induces a current in a neighboring coil. " 2 = N 2 d B2 N 2 B2 = M 21 i 1 d N B2 di 2 = M 1 21 " 2 = M 21 di 1 M 21 = N 2 B2 i 1 " 2 = M di1 and " 1 = M di 2 M = N 2 B2 i 1 = N 1 B1 i 2

Mutual inductance examples 30.1. and 30.2.

A coil with N 2 turns surrounds it at its center. Find the mutual inductance M. (N 1 = 1000, N 2 = 10 turns, l = 0.

5 Mutual inductance examples and In one form of Tesla coil, a long solenoid with length l and crosssectional area A is closely wound with N 1 turns of wire. A coil with N 2 turns surrounds it at its center. Find the mutual inductance M. (N 1 = 1000, N 2 = 10 turns, l = 0.50 m, A = 10 cm 2 ) Suppose the current i 2 in the outer coil is given by i 2 = 2e6 A/s t. At t = 3.0 µs find the average magnetic flux through each turn of the solenoid due to the current in the outer coil? What is the induced emf in the solenoid?

6 Self-inductance Self-inductance L: A varying current in a circuit induces an emf in that same circuit. L = N B i N d B = L di " = L di

7 Inductors as circuit elements Purpose: oppose any variations of current through the circuit despite fluctuations in the applied emf. Device with a particular value of inductance is called inductor or a choke. How to include inductor in the circuit analysis? H! E n d! l = d B H! E n d! l = L di R b a R b a! E n d! l = L di! E c d! l = L di V ab = V a V b = L di! E c +! E n = 0 Assume negligible resistance. Potential difference across inductor.

8 Potential across an inductor The potential across an inductor depends on the rate of change of the current through it. Figure at the right compares the behavior of the potential across a resistor and an inductor. The self-induced emf does not oppose current, but opposes a change in the current.

9 Q30.1 A small, circular ring of wire (shown in blue) is inside a larger loop of wire that carries a current I as shown. The small ring and the larger loop both lie in the same plane. If I increases, the current that flows in the small ring Large loop Small ring I I A. is clockwise and caused by self-inductance. B. is counterclockwise and caused by selfinductance. C. is clockwise and caused by mutual inductance. D. is counterclockwise and caused by mutual inductance.

10 A30.1 A small, circular ring of wire (shown in blue) is inside a larger loop of wire that carries a current I as shown. The small ring and the larger loop both lie in the same plane. If I increases, the current that flows in the small ring Large loop Small ring I I A. is clockwise and caused by self-inductance. B. is counterclockwise and caused by selfinductance. C. is clockwise and caused by mutual inductance. D. is counterclockwise and caused by mutual inductance.

11 Q30.2 A current i flows through an inductor L in the direction from point b toward point a. There is zero resistance in the wires of the inductor. If the current is decreasing, A. the potential is greater at point a than at point b. B. the potential is less at point a than at point b. C. The answer depends on the magnitude of di/ compared to the magnitude of i. D. The answer depends on the value of the inductance L. E. both C. and D. are correct.

12 A30.2 A current i flows through an inductor L in the direction from point b toward point a. There is zero resistance in the wires of the inductor. If the current is decreasing, A. the potential is greater at point a than at point b. B. the potential is less at point a than at point b. C. The answer depends on the magnitude of di/ compared to the magnitude of i. D. The answer depends on the value of the inductance L. E. both C. and D. are correct.

Calculating self-inductance and self-induced emf (30.3+30.4.

13 Calculating self-inductance and self-induced emf ( ) Determine the self-inductance of a toroidal solenoid with cross-sectional area A and mean radius r, closely wound with N turns of wire on a nonmagnetic core. Assume that B is uniform across a cross-section (that is, neglect the variation of B with distance from the toroid axis). If the current in the toroidal solenoid increases uniformly from 0 to 6 A in 3 µs, find the magnitude and direction of the selfinduced emf.

14 Q30.3 A steady current flows through an inductor. If the current is doubled while the inductance remains constant, the amount of energy stored in the inductor A. increases by a factor of 2. B. increases by a factor of 2. C. increases by a factor of 4. D. increases by a factor that depends on the geometry of the inductor. E. none of the above

15 A30.3 A steady current flows through an inductor. If the current is doubled while the inductance remains constant, the amount of energy stored in the inductor A. increases by a factor of 2. B. increases by a factor of 2. C. increases by a factor of 4. D. increases by a factor that depends on the geometry of the inductor. E. none of the above

16 Magnetic field energy The energy stored in an inductor U. Lets assume that the current is increasing at di/ rate. V ab = L di The energy du supplied to inductor during is: du = Li di So energy stored in inductor is: P = V ab i = Li di U = L R I 0 idi = 1 2 LI2

17 Magnetic energy density Energy in the inductor is stored in the magnetic field within the coil. L = ¹ 0N 2 A 2¼r U = 1 2 LI2 = 1 2 u = ¹ 0 N 2 A 2¼r I 2 B = U 2¼rA = 1 2 ¹ 0 N 2 I 2 (2¼r) 2 ¹ 0 NI 2¼r N 2 I 2 (2¼r) 2 = B2 ¹ 2 0 u = B2 u = B2 2¹ 0 2¹

18 Magnetic field energy The energy stored in an inductor is U = 1/2 LI 2. The energy density in a magnetic field is u = B 2 /2µ 0 (in vacuum) and u = B 2 /2µ (in a material).

19 Example 30.5: Storing energy in an inductor The electric power industry would like to find efficient ways to store electric energy generated during low-demand hours to help meet customer requirements during high-demand hours. Could a large inductor be used? What inductance would be needed to store 1 kwh of energy in a coil carrying 200-A current?

20 The R-L circuit An R-L circuit contains a resistor and inductor and possibly an emf source. At t = 0, switch S 1 closes. v ab = ir If we apply Kirhoff s loop rule: At t = 0, i = 0 and v ab = 0: v bc = L di " ir L di = 0 = " L R L i di = " ir L ( di ) initial = " L Greater the inductance, the slower the current increase. As the current approaches final value I, rate of increase drops to zero: ( di ) final = 0 = " L R L I and I = " R I = " R

21 Current growth in an R-L circuit Lets derive current as a function of time in R-L circuit. di i ("=R) = R L So the current is: i = " R (1 e (R=L) t) At t = 0, I = 0 and di/ = ε/l. At t, i ε/r and di/ 0 Time constant measures how quickly current builds up to its final value: = L R "i = i 2 R + Li di

22 Current decay in an R-L circuit Current varies according to: i = I 0 e (R=L)t Energy analysis for the current decay: 0 = i 2 R + Li di It is all very similar to charging and discharging capacitor in RC circuit.

23 Example A sensitive electronic device of resistance R= 175 Ω is to be connected to a source of emf (of negligible internal resistance) by a switch. The device is designed to operate with a 36 ma current, but to avoid damage to the device, the current can rise to no more than 4.9 ma in the first 58 µs after the switch is closed. An inductor is therefore connected in series with the device: the switch in question is S 1. a) What is the required emf source ε 1? b) What is the required inductance L? c) What is the R-L time constant τ?

24 Example 30.7: Energy in R-L circuit When the current in an R-L circuit is decaying, what fraction of the original energy stored in the inductor has been dissipated after 2.3 time constants?

25 The L-C circuit An L-C circuit contains an inductor and a capacitor and is an oscillating circuit.

26 Electrical oscillations in an L-C circuit Kirhoff s loop rule: L di q C = 0 i = dq= and di= = d 2 q= 2 d 2 q LC q = 0 q = Qcos(!t + Á)! = q 1 LC i =!Qsin(!t + Á)

27 Electrical and mechanical oscillations Table on the right summarizes the analogies between SHM and L-C circuit oscillations.

28 Example 30.8: An oscillating circuit A 300-V dc power supply is used to charge a 25-µF capacitor. After the capacitor is fully charged, it is disconnected from the power supply and connected across a 10-mH inductor. The resistance in the circuit is negligible. a) Find the frequency and period of oscillation of the circuit. b) Find the capacitor charge and the circuit current 1.2-ms after the inductor and capacitor are connected.

29 The L-R-C series circuit An L-R-C circuit exhibits damped harmonic motion if the resistance is not too large. ir L di q C = 0 d 2 q 2 + R L q q = Ae (R=2L)t cos(! 0 = q dq + 1 LC q = 0 1 LC R2 4L 2 t + Á) 1 LC R2 4L 2

30 Example 30.10: An underdamped L-R-C circuit What resistance R is required (in terms of L and C) to give an L-R-C series circuit a frequency that is one-half the undamped frequency?

Handout 10: Inductance. Self-Inductance and inductors

Handout 10: Inductance. Self-Inductance and inductors 1 Handout 10: Inductance Self-Inductance and inductors In Fig. 1, electric current is present in an isolate circuit, setting up magnetic field that causes a magnetic flux through the circuit itself. This