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 self-inductance. 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 (self-inductance). A second property of inductors is
that it stores energy in its solenoidal magnetic field. Similar to a fully-charged
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
~ and hence a magnetic flux through coil 2. If the
coil 1 produces a magnetic field B
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 i1 in coil 1 gives rise to a magnetic flux through coil 2.
1
E2 = −N2
dΦB2
dt
We would like to write an equation that expresses the relationship between the flux
in 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 inductance
of the two coils. Using this equation, we have a working definition for the mutual
inductance
M21 =
N2 ΦB2
i1
(Mutual Inductance)
(1)
Unit of Inductance
1 H = 1 W b/A = 1 V · s/A
The emf produced in the 2nd coil E2 is −N2 dΦB2 /dt, so, we can write the following:
E2 = −M21
1.1
di1
dt
(2)
Calculating Mutual Inductance
B1 = µo n1 i1 =
µ o N1 i 1
`
The flux through a cross section of the solenoid equals B1 A. This also equals
the flux ΦB2 through each turn of the outer coil, independent of its cross-section
area.
M =
N2 ΦB2
N2 B1 A
N2 µo N1 i1
µo A N1 N2
=
=
A =
i1
i1
i1
`
`
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 a
coil with N2 turns.
2
Self-Inductance and Inductors
~ in the coil and hence a flux through
Figure 3: The current i in the circuit causes a magnetic field B
the coil.
Self-induced emfs can occur in any circuit, since there is always some magnetic
flux through the closed loop of a current-carrying 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 ΦB through each turn of the coil. Similar to
the mutual inductance defined earlier, we can define the self-inductance as:
L =
N ΦB
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
2.1
di
dt
(4)
Inductors as Circuit Elements
According to Faraday’s Law → E =
I
H
~ · d~` = −dΦB /dt
E
~ n · d~` = − L di
E
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
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
2.2
Calculating Self-Inductance
Figure 6: Determining the self-inductance 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 cross-sectional area A and
radius r.
3
Magnetic-Field Energy
Figure 7: A resistor is a device in which energy is irrecoverably dissipated. By contrast, energy
stored in a current-carrying inductor can be recovered when the current decreases to zero.
3.1
4
Magnetic Energy Density
The R-L Circuit
6
Figure 8: An R-L circuit.
Current Growth in an R-L Circuit
7
Figure 9: Graph of i versus t for growth of current in an R-L 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
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.
9
5
The L-C Circuit
Figure 11: In an oscillating L-C 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 (UB ) and electrical energy in the capacitor (UE ). As in simple harmonic motion, the total
energy 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
10
Figure 12: Graphs showing the capacitor charge as a function of time in an L-R-C series circuit
with initial charge Q.
11
Figure 13: An L-R-C series circuit.
12