Chapter 30

Inductance

In this chapter we investigate the properties of an inductor in a circuit. There aretwo kinds of inductance–mutual inductance and self-inductance. An inductor isformed by taken a length of copper wire and wrapping it around a cylinder to forma coil. If a changing current is applied to the coil it induces an emf in adjacent coils(mutual inductance) or itself (self-inductance). A second property of inductors isthat it stores energy in its solenoidal magnetic field. Similar to a fully-chargedcapacitor where the energy is stored in the electric field, an inductor, supplied witha 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 incoil 1 produces a magnetic field ~B and hence a magnetic flux through coil 2. If thecurrent in coil 1 changes, the flux through coil 2 changes as well; and according toFaraday’s law, this induces an emf in coil 2. As a result, a change in the currentin one circuit can induce a current in a second circuit.

Figure 1: The current i1 in coil 1 gives rise to a magnetic flux through coil 2.

1

E2 = −N2dΦB2

dt

We would like to write an equation that expresses the relationship between the fluxin the 2nd coil in terms of the current i1 in the first coil.

N2 ΦB2 = M21 i1

where ΦB2 is the flux for a single turn of coil 2, and M21 is the mutual inductanceof the two coils. Using this equation, we have a working definition for the mutualinductance

M21 =N2 ΦB2

i1(Mutual Inductance) (1)

Unit of Inductance

1 H = 1 Wb/A = 1 V · s/A

The emf produced in the 2nd coil E2 is −N2 dΦB2/dt, so, we can write the follow-ing:

E2 = −M21di1dt

(2)

1.1 Calculating Mutual Inductance

B1 = µo n1 i1 =µoN1 i1

`

The flux through a cross section of the solenoid equals B1A. This also equalsthe flux ΦB2 through each turn of the outer coil, independent of its cross-sectionarea.

M =N2 ΦB2

i1=

N2B1A

i1=

N2

i1

µoN1 i1`

A =µoAN1N2

`

In the example in the book, M = 25 µH.

2

Figure 2: A long solenoid with cross-sectional area A and N1 turns is surrounded at its center by acoil with N2 turns.

2 Self-Inductance and Inductors

Figure 3: The current i in the circuit causes a magnetic field ~B in the coil and hence a flux throughthe coil.

Self-induced emfs can occur in any circuit, since there is always some magneticflux through the closed loop of a current-carrying circuit. However, the effect isenhanced if the circuit includes a coil withN turns of wire. As a result of the currenti, there is an average magnetic flux ΦB through each turn of the coil. Similar tothe mutual inductance defined earlier, we can define the self-inductance as:

L =NΦB

i(Self-Inductance) (3)

3

From Faraday’s law for a coil with N turns, the self-induced emf is E = −N dΦB/dt,so it follows that:

E = −L didt

(4)

2.1 Inductors as Circuit Elements

According to Faraday’s Law → E =∮~E · d~̀= −dΦB/dt∮

~En · d~̀ = −Ldidt

Figure 4: A circuit containing an emf source and an inductor. The emf source is variable, so thecurrent i and its rate of change di/dt can be varied.

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Figure 5: The potential difference across a resistor depends on the current, whereas the potentialdifference across an inductor (b), (c), (d) depends on the rate of change of the current.

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2.2 Calculating Self-Inductance

Figure 6: Determining the self-inductance of a closely wound toroidal solenoid. Only a few turnsof the winding are shown. Part of the toroid is cut away to show the cross-sectional area A andradius r.

3 Magnetic-Field Energy

Figure 7: A resistor is a device in which energy is irrecoverably dissipated. By contrast, energystored in a current-carrying inductor can be recovered when the current decreases to zero.

3.1 Magnetic Energy Density

4 The R-L Circuit

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Figure 8: An R-L circuit.

Current Growth in an R-L Circuit

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Figure 9: Graph of i versus t for growth of current in an R-L circuit with an emf in series. Thefinal current is I = E/R; after one time constant τ , the current is 1− 1/e of this value.

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Current Decay in an R-L Circuit

Figure 10: Graph of i versus t for decay of current in an R-L circuit. After one time constant τ ,the current is 1/e of its initial value.

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5 The L-C Circuit

Figure 11: In an oscillating L-C circuit, the charge on the capacitor and the current through theinductor both vary sinusoidally with time. Energy is transferred between magnetic energy in theinductor (UB) and electrical energy in the capacitor (UE). As in simple harmonic motion, the totalenergy E remains constant.

5.1 Electrical Oscillations in an L-C Circuit

5.2 Energy in an L-C Circuit

6 the L-R-C Series Circuit

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Figure 12: Graphs showing the capacitor charge as a function of time in an L-R-C series circuitwith initial charge Q.

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Figure 13: An L-R-C series circuit.

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