Chapter 30 Inductance


 Josephine Edwards
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1 Chapter 30 Inductance In this chapter we investigate the properties of an inductor in a circuit. There are two kinds of inductance mutual inductance and selfinductance. An inductor is formed by taken a length of copper wire and wrapping it around a cylinder to form a coil. If a changing current is applied to the coil it induces an emf in adjacent coils (mutual inductance) or itself (selfinductance). A second property of inductors is that it stores energy in its solenoidal magnetic field. Similar to a fullycharged capacitor where the energy is stored in the electric field, an inductor, supplied with a steady current, also stores energy in the form of a magnetic field. 1 Mutual inductance Consider two neighboring coils of wire as shown in the figure. A current flowing in coil 1 produces a magnetic field B and hence a magnetic flux through coil 2. If the current in coil 1 changes, the flux through coil 2 changes as well; and according to Faraday s law, this induces an emf in coil 2. As a result, a change in the current in one circuit can induce a current in a second circuit. Figure 1: The current i 1 in coil 1 gives rise to a magnetic flux through coil 2. 1
2 E 2 = N 2 dφ B2 dt We would like to write an equation that expresses the relationship between the flux in the 2nd coil in terms of the current i 1 in the first coil. N 2 Φ B2 = M 21 i 1 where Φ B2 is the flux for a single turn of coil 2, and M 21 is the mutual inductance of the two coils. Using this equation, we have a working definition for the mutual inductance Unit of Inductance M 21 = N 2 Φ B2 i 1 (Mutual Inductance) (1) 1 H = 1 W b/a = 1 V s/a The emf produced in the 2nd coil E 2 is N 2 dφ B2 /dt, so, we can write the following: E 2 = M 21 di 1 dt (2) 1.1 Calculating Mutual Inductance B 1 = µ o n 1 i 1 = µ o N 1 i 1 l The flux through a cross section of the solenoid equals B 1 A. This also equals the flux Φ B2 through each turn of the outer coil, independent of its crosssection area. M = N 2 Φ B2 i 1 In the example in the book, M = 25 µh. = N 2 B 1 A = N 2 µ o N 1 i 1 A = µ o A N 1 N 2 i 1 i 1 l l 2
3 Figure 2: A long solenoid with crosssectional area A and N 1 turns is surrounded at its center by a coil with N 2 turns. 2 SelfInductance and Inductors Figure 3: The current i in the circuit causes a magnetic field B in the coil and hence a flux through the coil. Selfinduced emfs can occur in any circuit, since there is always some magnetic flux through the closed loop of a currentcarrying circuit. However, the effect is enhanced if the circuit includes a coil with N turns of wire. As a result of the current i, there is an average magnetic flux Φ B through each turn of the coil. Similar to the mutual inductance defined earlier, we can define the selfinductance as: L = NΦ B i (SelfInductance) (3) 3
4 From Faraday s law for a coil with N turns, the selfinduced emf is E = N dφ B /dt, so it follows that: E = L di dt (4) 2.1 Inductors as Circuit Elements According to Faraday s Law E = E d l = dφb /dt E n d l = L di dt Figure 4: A circuit containing an emf source and an inductor. The emf source is variable, so the current i and its rate of change di/dt can be varied. 4
5 Figure 5: The potential difference across a resistor depends on the current, whereas the potential difference across an inductor (b), (c), (d) depends on the rate of change of the current. 5
6 2.2 Calculating SelfInductance Figure 6: Determining the selfinductance of a closely wound toroidal solenoid. Only a few turns of the winding are shown. Part of the toroid is cut away to show the crosssectional area A and radius r. 3 MagneticField Energy Figure 7: A resistor is a device in which energy is irrecoverably dissipated. By contrast, energy stored in a currentcarrying inductor can be recovered when the current decreases to zero. 3.1 Magnetic Energy Density 4 The RL Circuit 6
7 Current Growth in an RL Circuit Figure 8: An RL circuit. 7
8 Figure 9: Graph of i versus t for growth of current in an RL circuit with an emf in series. The final current is I = E/R; after one time constant τ, the current is 1 1/e of this value. 8
9 Current Decay in an RL Circuit Figure 10: Graph of i versus t for decay of current in an RL circuit. After one time constant τ, the current is 1/e of its initial value. 9
10 5 The LC Circuit Figure 11: In an oscillating LC circuit, the charge on the capacitor and the current through the inductor both vary sinusoidally with time. Energy is transferred between magnetic energy in the inductor (U B ) and electrical energy in the capacitor (U E ). As in simple harmonic motion, the total energy E remains constant. 5.1 Electrical Oscillations in an LC Circuit 5.2 Energy in an LC Circuit 6 the LRC Series Circuit 10
11 Figure 12: Graphs showing the capacitor charge as a function of time in an LRC series circuit with initial charge Q. 11
12 Figure 13: An LRC series circuit. 12
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