Unformatted text preview: 4/11/10 CHAPTER 30
INDUCTANCE
Vahé Peroomian INDUCTANCE
Is there a difference between a straight
wire and a a coil made of the same wire
placed in a circuit? MUTUAL INDUCTANCE
In a system with two coils, current flowing in first
coil produces a B at coil 2 and ΦB through coil 2.
If current through coil 1 is changing, flux through
coil 2 changes, inducing an emf in coil 2, creating a
current in the second coil. The
reverse of this is also true.
Mutual inductance M is defined by M= N 2 d Φ B 2 N1d Φ B1
=
i1
i2 1 4/11/10 MUTUAL INDUCTANCE: UNITS
SI unit of mutual inductance is the henry (1H).
1 H = 1 Wb/A = 1 V . s/A = 1 Ω . S = 1 J/A2 CHAPTER  Inductance In mutual induction, the emf induced in one coil is always proportional to the rate
at which the current in the other coil is changing. Although the proportionality
constants M12 and M 21 have been treated separately, it can be shown that they are
equal. Therefore, with M12 5 M 21 5 M, Equations 32.16 and 32.17 become
I
e2 5 2M ddt1 and e1 5 2M dI2
dt These two equations are similar in form to Equation 32.1 for the selfinduced emf
e 5 2L (dI/dt). The unit of mutual inductance is the henry. EXAMPLE 1.
Quick Quiz In Figure 32.8, coil 1 is moved closer to coil 2, with the
orientation of both coils remaining fixed. Because of this movement, the . b y Braun GmbH, Kronberg An electricutual inductionhasthebasecoils (a) increases, (b) the
m toothbrush of a two designed to hold decreases, or (c) is
unaffected.
toothbrush handle when not in use. When the handle is
placed on the base, a changing current in a solenoid inside
the base cylinder induces a current in a coil inside the
handle. This induced current charges the battery in the
E x a m pl e .
“Wireless” Battery the base
handle. We can modelCharger as a solenoid of length ,
with NB turns, carrying a current I, and having a crossAn electric toothbrush has a base designed to hold the
sectional area A. The
toothbrush handle when not in use. As shown in Figure
Coil 1 (base)
handle coil contains NH
32.9a, the handle has a cylindrical hole that fits loosely
NB
over a matching cylinder on urns andWhen the handle
t the base. completely
is placed on the base, a changing current in a solenoid
Coil 2
surrounds the abaseinside
coil.
(handle)
inside the base cylinder induces a current in coil
Find charges the battery t he handle. This induced current the mutual inducin
NH
t he handle.
tance of the system. We can model the base as a solenoid of length , w ith
 Inductance
CHAPTER
N B turns (Fig. 32.9b), carrying a current I, and having a
crosssectional area A. The handle coil contains N H turns
and completely surrounds the base coil. Find the mutual emf a
In mutual induction, the
induced in one coil is always proportional to the rate
b
inductance of the system.
at which the current in the other coil is changing. Although the proportionality
constants M12 and M 21 haveFigure treated separately,(a) This electric toothbrushthey are
been . (Example 32.5) it can be shown that
uses the mutual induction of solenoids as part of its batterySOLUTION
equal. Therefore, with M 5 M 5 M, Equations 32.16 and 32.17 become
Conceptualize Be sure you can identify the two coils in the
situation and understand that a changing current in one
coil induces a current in the second coil. 12 charging system. (b) A coil of N t urns wrapped around the
21
H center of a solenoid of N B t urns. I
e2 5 2M ddt1 and e1 5 2M dI2
dt Categorize We will determine the Theseusingequationsdiscussed in this section,Equation 32.1 for the selfinduced emf
result two concepts are similar in form to so we categorize this example as a substitution problem.
e 5 2L (dI/dt). The unit of mutual inductance is the henry.
NB
Use Equation 30.17 to express the magnetic field in the
I
B 5 m0
,
interior of the base solenoid:
Quick Quiz . In Figure 32.8, coil 1 is moved closer to coil 2, with the
orientation of both coils 5 NHFBH 5fixed. Because BNH A movement, the
remaining NH BA 5 m N of this
Find the mutual inductance, noting that the magnetic
M
I
S coil caused by current IB of the two 106 (a) i ) It, NH =04 , NB
the the magand
f lux F BH t hrough the handle’s uppose mutual induction= (3.00 × coils A/sncreases, (b),decreases, or (c) is
netic field of the base coil is BA: 0, l = unaffected. and A = 1.00 × 104 m2.
=2
0.025 m, EXAMPLE 2 Wireless charging is used in(a number of other .5 µs, what averagesignificant example is the inductive charging
a) At time t = 2 “cordless” devices. One magnetic flux through
used by some manufacturers of electric cars that avoids direct metaltometal contact between the car and the charging
each turn of the solenoid in the handle is caused by
apparatus. E x a m pl e . current in the base coil?
“Wireless” Battery Charger
(b) What is Oscillations in in the handle’s coil?
. the induced emf an LC Circuit . b y Braun GmbH, Kronberg An electric toothbrush has a base designed to hold the
When a capacitor is connected to an inductor as illustrated in Figure 32.10, the
toothbrush handle when not in use. As shown in Figure
combination is an LC circuit. If the capacitor is initially charged and the switch is
Coil 1 (base)
32.9a, the handle has a cylindrical hole that fits loosely
NB
over a matching cylinder on the base. When the handle
is placed on the base, a changing current in a solenoid
Coil 2
(handle)
inside the base cylinder induces a current in a coil inside
t he handle. This induced current charges the battery in
NH
t he handle.
27819_32_c32_p927952.indd 936
27819_32_c32_p927952.indd 936
We can model the base as a solenoid of length , w ith
N B turns (Fig. 32.9b), carrying a current I, and having a
crosssectional area A. The handle coil contains N H turns
and completely surrounds the base coil. Find the mutual
a
b
inductance of the system.
Figure . (Example 32.5) (a) This electric toothbrush
uses the mutual induction of solenoids as part of its batterySOLUTION
Conceptualize Be sure you can identify the two coils in the
situation and understand that a changing current in one
coil induces a current in the second coil. 10/6/09 8:50:05 AM
10/6/09 8:50:05 AM charging system. (b) A coil of N H t urns wrapped around the
center of a solenoid of N B t urns. Categorize We will determine the result using concepts discussed in this section, so we categorize this example as a substitution problem.
NB
Use Equation 30.17 to express the magnetic field in the
I
B 5 m0
,
interior of the base solenoid:
NBNH
NHFBH
NH BA
Find the mutual inductance, noting that the magnetic
5
5 m0
A
M5
I
I
,
f lux F BH t hrough the handle’s coil caused by the magnetic field of the base coil is BA:
Wireless charging is used in a number of other “cordless” devices. One significant example is the inductive charging 2 4/11/10 SELFINDUCTANCE
Selfinduced emfs can occur in any circuit with a
changing current: selfinductance. SELFINDUCTANCE
Selfinductance is greatly enhanced if a coil is
present in the circuit.
Selfinductance or
inductance of a coil with
N turns is L= N ΦB
i INDUCTORS AS CIRCUIT ELEMENTS
Inductor is shown with Purpose of inductor in a circuit is to
oppose variations in the current through
the circuit. 3 4/11/10 A QUICK REVIEW OF
KIRCHHOFF’S LOOP RULES
Kirchhoff’s junction rule: the algebraic sum of the
currents into any junction is zero ∑I = 0 A QUICK REVIEW OF
KIRCHHOFF’S LOOP RULES
Kirchhoff’s loop rule: the algebraic sum of the
potential differences in any loop, including those
associated with emfs and those of resistive elements,
must equal zero. ∑V = 0 SIGN CONVENTION FOR
KIRCHHOFF’S LOOP RULE
Minus to plus across emf is positive
Through resistor in same direction as current is
negative. 4 4/11/10 EXAMPLE 26.4
In the circuit shown, a 12V power supply with unknown
internal resistance r is connected to a rundown
rechargeable battery with unknown emf ε and internal
resistance 1 Ω and to an indicator light bulb of resistance
3 Ω carrying a current of 2 A. The current through the rundown battery is 1 A in the direction shown. Find the
unknown current I, the internal resistance r, and the emf ε. INDUCTORS AS CIRCUIT ELEMENTS
Potential across inductor is
Vab = Va − Vb = L di
dt POTENTIAL
DIFFERENCE
ACROSS
INDUCTOR
Potential difference
across inductor
depends on the
rate of change
of the current. 5 4/11/10 EXAMPLE 3
Consider a uniformly wound solenoid having N turns and
length l. Assume l is much longer than the radius of the
windings and the core of the solenoid is air.
(a) Find the inductance of the solenoid.
(b) Calculate the inductance of the solenoid if it contains
300 turns, its length is 25.0 cm, and its crosssectional
area is 4.00 cm2.
(c) Calculate the selfinduced emf in the solenoid if the
current it carries decreases at the rate of 50.0 A/s. ENERGY STORED IN AN INDUCTOR
Energy input U needed to establish a final current I
(initial current zero) in and inductor of inductance L is
given by U=L 1 I ∫0 i di = 2 LI 2 MAGNETIC ENERGY DENSITY
For a toroidal solenoid, magnetic energy density is
given by u= B2
2 µ0 (vacuum) u= B2
2µ (material) 6 ...
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 Spring '10
 peroomian
 Physics, Current, Inductance, Magnetic Field, Solenoid, Inductor, Faraday's law of induction

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