Algunos ejemplos de ejercicios: Inductancia William Oquendo
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RESUMEN
Inductors An inductor is a device that can be used to produce a known magnetic field in in a specified region. If a current i is established through each of the N wind windings ings of an indu inductor ctor,, a magne magnetic tic flux B links those windings.The inductance L of the inductor is L
N B i
(inductance defined).
(30-28)
The SI unit of inductance is the henry (H), where 1 henry 1 H 1 T m2/A. The inductance per unit unit length near the middle of a long solenoid of cross-sectional area A and n turns per unit length is L l
2 m 0 n A
(solenoid).
L
Series RL Circuits If a constant emf is introduced introduced into a single-loop circuit containing a resistance R and an inductance L, th the e current rises to an equilibrium value of /R: i
R
(1 e
t /t L
)
(rise of current).
(30-41)
Here t L ( L/R) is the inductive time constant. Wh When en the the source source of of consta con stant nt emf emf is remov removed, ed, the curr current ent decay decayss from from a valu value e i0 according to
i
i 0 e t /t L
(decay of current).
(30-45)
1
2 Li
2
(magnetic energy).
(30-49)
If B is the magnitude of a magnetic field at any point (in an inductor or anywhere else), else), the density of stored magnetic energy at that point is uB
(30-35)
The direction of L is found from Lenz’s law: The self-induced emf acts to oppose the change that produces it.
U B
(30-31)
Self-Induction If a current i in a coil changes with time, an emf is induced in the coil.This coil. This self-induced emf is di L . dt
Magnetic Energy If an inductor L carries a current i, the inductor’s magnetic field stores an energy given by
B2
2 m 0
(magnetic energy density).
(30-55)
Mutual Induction If coils 1 and and 2 are near near each other, other, a changing current in either coil can induce an emf in the other. This mutual induction is described by
and
2 M
di 1 dt
(30-64)
1 M
di 2 , dt
(30-65)
where M (measured in henries) is the mutual inductance.
Mutual inductance: When a changing current i 1 in one circuit causes a changing magnetic flux in a second circuit, an emf E 2 is induced in the second circuit. Likewise, a changing current i 2 in the second circuit induces an emf E 1 in the first circuit. If the circuits are coils of wire with N 1 and N 2 turns, the mutual inductance M can be expressed in terms of the average flux £ B2 through each turn of coil 2 caused by the current i 1 in coil 1, or in terms of the average flux £ B1 through each turn of coil 1 caused by the current i 2 in coil 2. The SI unit of mutual inductance is the henry, abbreviated H. (See Examples 30.1 and 30.2.)
Self-inductance: A changing current i in any circuit causes a self-induced emf E . The inductance (or self-inductance) L depends on the geometry of the circuit and the material surrounding it. The inductance of a coil of N turns is related to the average flux £ B through each turn caused by the current i in the coil. An inductor is a circuit device, usually including a coil of wire, intended to have a substantial inductance. (See Examples 30.3 and 30.4.)
Magnetic-field energy: An inductor with inductance L carrying current I has energy U associated with the inductor’s magnetic field. The magnetic energy density u (energy per unit volume) is proportional to the square of the magnetic field magnitude. (See Example 30.5.)
di1
E 2 = - M
Coil 1
and
dt
Coil 2
N 1 turns
N 2 turns i1
di 2
E 1 = - M
(30.4)
dt
i1
M
=
N 2 £ B2
L
U u
u
=
=
N 1 £ B1
S
N £ B
B2
2m
+
(30.6)
i
2m0
B
(30.7)
dt
B2
B
(30.5)
i2
di
1 2 = 2 LI
=
=
i1
E = - L
F B 2
S
i
B
(30.9)
I
(in vacuum) (in a material with magnetic permeability m)
I
(30.10)
(30.11)
Stored energy 1 U 5 2 L I 2
Energy density u 5 B 2 / 2m0
R-L circuits:
In a circuit containing a resistor R, an inductor L, and a source of emf, the growth and decay of current are exponential. The time constant t is the time required for the current to approach within a fraction 1> e of its final value. (See Examples 30.6 and 30.7.)
L-C circuits:
A circuit that contains inductance L and capacitance C undergoes electrical oscillations with an angular frequency v that depends on L and C . This is analogous to a mechanical harmonic oscillator, with inductance L analogous to mass m, the reciprocal of capacitance 1> C to force constant k , charge q to displacement x , and current i to velocity v x . (See Examples 30.8 and 30.9.)
t =
L R
1012
I
E
(
I 1 2
R
1 e
t
)
t
t5
L R t
v =
A
1
(30.22)
LC
+Qm
– Qm
+ + + +
E m
+
I m
L-R-C series circuits:
A circuit that contains inductance, resistance, and capacitance undergoes damped oscillations for sufficiently small resistance. The frequency v ¿ of damped oscillations depends on the values of L, R, and C . As R increases, the damping increases; if R is greater than a certain value, the behavior becomes overdamped and no longer oscillates. (See Example 30.10.)
i
(30.16)
v¿ =
B
1
LC
-
R2
4 L2
Bm
I m
q (30.29)
Q
O
Underdamped circuit (small R) t
Circuitos RL
a
z Figure 30-27 shows a circuit with two identical resistors and an ideal inFigure 30-26 Question 6. ductor. Is the current through the central resistor more than, less than, or the same as that through the other resistor (a) just after the closing of switch S, (b) a long time after that, (c) just after S is reopened a long time later, and (d) a long time after that? 7
+
a more; (b) same; (c) same; (d) same (zero)
–
S
Figure 30-27 Question 7.
The switch in the circuit of Fig. 30-15 has been closed on a for a very long time when it is then thrown to b. The resulting current through the inductor is indicated in Fig. 30-28 for four sets of values for the resistance R and inductance L: (1) R 0 and L 0, (2) 2R0 and L 0, (3) R0 and 2L0, (4) 2R0 and 2L0. Which set goes with which curve? 8
a +
S
b
–
R L
Figure 30-15 An RL circuit.When switch S is closed on a, the current rises and approaches a limiting value /R.
a
c
d
i
b c a
t Figure 30-29 shows three circuits Figure 30-28 Question 8. with identical batteries, inductors, and resistors. Rank the circuits, greatest first, according to the current through the resistor labeled R (a) long after the switch is closed, (b) just after the switch is reopened a long time later, and (c) long after it is reopened. 9
a a t e zero ; (b) 1 and 2 tie, then 3; (c) all tie (zero)
+ –
R
+
+
–
–
R
R
(1)
(2)
(3)
i y
x R + –
L
L
z
Figure 30-16 The circuit of Fig. 30-15 with the switch closed on a.We apply the loop rule for the circuit clockwise, starting at x.
Figure 30-30 gives the variation with time of the potential difference V R across a resistor in three circuits wired as shown in Fig. 30-16. The circuits contain the same resistance R and emf but differ in the inductance L. Rank the circuits according to the value of L, greatest first. 10
a R
V
b c
t
Figure 30-30 Question 10. b
The current in an RL circuit builds up to one-third of its steady-state value in 5.00 s.Find the inductive time constant. •50
The current in an RL circuit drops from 1.0 A to 10 mA in the first second following removal of the battery from the circuit. If L is 10 H, find the resistance R in the circuit. •51
ILW
The switch in Fig. 30-15 is closed on a at time t 0. What is the ratio L/ of the inductor’s self-induced emf to the battery’s emf (a) just after t 0 and (b) at t 2.00t L? (c) At what multiple of t L will L/ 0.500? •52
A solenoid having an inductance of 6.30 m H is connected in series with a 1.20 k resistor. (a) If a 14.0 V battery is connected across the pair, how long will it take for the current through the resistor to reach 80.0% of its final value? (b) What is the current through the resistor at time t 1.0t L? •53
SS M
In Fig. 30-62, 100 V, R 1 i 1 10.0 , R2 20.0 , R3 30.0 ,and S L 2.00 H. Immediately after switch R 1 R 3 S is closed, what are (a) i1 and (b) i2? + i 2 (Let currents in the indicated R 2 L – directions have positive values and currents in the opposite directions have negative values.) A long time Figure 30-62 Problem 54. later, what are (c) i1 and (d) i2? The switch is then reopened. Just then, what are (e) i1 and (f) i2? A long time later, what are (g) i1 and (h) i2? •54
50 51 52
53 54
12.3 s 46 (a) 1.00; (b) 0.135; (c) 0.693 (a) 8.45 ns; (b) 7.37 mA (a) 3.33 A; (b) 3.33 A; (c) 4.55 A; (d) 2.73 A; (e) 0; (f) -1.82 A (reversed); (g) 0; (h) 0
In Fig. 30-63, the inductor has 25 turns and the ideal battery as an emf of 16 V. Figure 30-64 gives the magnetic flux through ach turn versus the current i through the inductor. The vertical
56
) 2 Φs m
S
•
R
T 4 –
0 1 (
L
Figure 30-63 Problems 56,
80, 83, and 93.
Φ
0
i s i (A)
Figure 30-64 Problem 56.
axis scale is set by s 4.0 104 Tm2, and the horizontal axis scale is set by i s 2.00 A. If switch S is closed at time t 0, at what rate di/dt will the current be changing at t 1.5t L? 7.1 102 A/s
Inductores e Inductancia
The inductance of a closely packed coil of 400 turns is 8.0 mH. Calculate the magnetic flux through the coil when the current is 5.0 mA. •40
0.10 Wb
A circular coil has a 10.0 cm radius and consists of 30.0 closely wound turns of wire. An externally produced magnetic field of magnitude 2.60 mT is perpendicular to the coil. (a) If no current is in the coil, what magnetic flux links its turns? (b) When the current in the coil is 3.80 A in a certain direction, the net flux through the coil is found to vanish. What is the inductance of the coil? •41
(a) 2.45 mWb; (b) 0.645 mH
Auto-induccion
A 12 H inductor carries a current of 2.0 A. At what rate must the current be changed to produce a 60 V emf in the inductor? •44
5.0 A/s
At a given instant the current L and self-induced emf in an inductor i are directed as indicated in Fig. 30-59. (a) Is the current increasing or deFigure 30-59 Problem 45. creasing? (b) The induced emf is 17 V, and the rate of change of the current is 25 kA/s; find the inductance. •45
(a) decreasing; b 0.68 mH
The current i through a 4.6 H inductor varies with time t as shown by the graph of Fig. 30-60, where the vertical axis scale is set by i s 8.0 A and the horizontal axis scale is set by t s 6.0 ms. The inductor has a resistance of 12 . Find the magnitude of the induced emf during time intervals (a) 0 to 2 ms, (b) 2 ms to 5 ms, and (c) 5 ms to 6 ms. (Ignore the behavior at the ends of the intervals.) ••46
i s
) A ( i
0
t s t (ms)
Figure 30-60 Problem 46.
(a) 16 kV; (b) 3.1 kV; (c) 23 kV
••47
Inductors in series. Two inductors L1 and L2 are connected in
series and are separated by a large distance so that the magnetic field of one cannot affect the other. (a) Show that the equivalent inductance is given by Leq L1 L2.
(Hint: Review the derivations for resistors in series and capacitors in series. Which is similar here?) (b) What is the generalization of (a) for N inductors in series?
••48
Inductors in parallel. Two inductors L1 and L2 are connected
in parallel and separated by a large distance so that the magnetic field of one cannot affect the other. (a) Show that the equivalent inductance is given by 1 Leq
1 L1
1 L2
.
(Hint: Review the derivations for resistors in parallel and capacitors in parallel. Which is similar here?) (b) What is the generalization of (a) for N inductors in parallel?
The inductor arrangement of Fig. 30-61, with L1 30.0 mH, L2 50.0 mH, L3 20.0 mH, and L4 15.0 mH, is to be connected to a varying current source. What is the equivalent inductance of the arrangement? (First see Problems 47 and 48.) 49
••
L 1
L 2
59.3 mH
L 4
Figure 30-61 Problem 49.
30.11 The inductor in Fig. E30.11 has Figure E30.11 inductance 0.260 H and carries a current in i the direction shown that is decreasing at a uniform rate, di > dt = - 0.0180 A> s. a b L (a) Find the self-induced emf. (b) Which end of the inductor, a or b, is at a higher potential? 30.12 The inductor shown in Fig. E30.11 has inductance 0.260 H and carries a current in the direction shown. The current is changing at a constant rate. (a) The potential between points a and b is V ab = 1.04 V, with point a at higher potential. Is the current increasing or decreasing? (b) If the current at t = 0 is 12.0 A, what is the current at t = 2.00 s? :- ;: .
E.PJ ×.- −F M
L 3
.
Inductancia Mutua
Coil 1 has L1 25 mH and N 1 100 turns. Coil 2 has L2 40 mH and N 2 200 turns. The coils are fixed in place; their mutual inductance M is 3.0 mH.A 6.0 mA current in coil 1 is changing at the rate of 4.0 A/s. (a) What magnetic flux 12 links coil 1, and (b) what self-induced emf appears in that coil? (c) What magnetic flux 21 links coil 2, and (d) what mutually induced emf appears in that coil? •72
72
Two coils are at fixed locations. When coil 1 has no current and the current in coil 2 increases at the rate 15.0 A/s, the emf in coil 1 is 25.0 mV. (a) What is their mutual inductance? (b) When coil 2 has no current and coil 1 has a current of 3.60 A, what is the flux linkage in coil 2? •73
SS M
73 74
(a) 1.5 Wb; (b) 1.0 102 mV; (c) 90 nWb; (d) 12 mV (a) 1.67 mH; (b) 6.00 mWb 13 H
Two solenoids are part of the spark coil of an automobile. When the current in one solenoid falls from 6.0 A to zero in 2.5 ms, an emf of 30 kV is induced in the other solenoid. What is the mutual inductance M of the solenoids? •74
A rectangular loop of N closely packed turns is positioned near a long straight wire as shown in Fig. 30-68. What is the mutual inductance M for the loop– wire combination if N 100, a 1.0 cm, b 8.0 cm, and l 30 cm? ••75
ILW
a
i N turns
13 H
b l
30.1 Two coils have mutual inductance M = 3.25 * 10 -4 H. The current i 1 in the first coil increases at a uniform rate of 830 A> s. (a) What is the magnitude of the induced emf in the second coil? Is it constant? (b) Suppose that the current described is in the second coil rather than the first. What is the magnitude of the induced emf in the first coil? 30.2 Two coils are wound around the same cylindrical form, like the coils in Example 30.1. When the current in the first coil is decreasing at a rate of - 0.242 A> s, the induced emf in the second coil has magnitude 1.65 * 10 -3 V. (a) What is the mutual inductance of the pair of coils? (b) If the second coil has 25 turns, what is the flux through each turn when the current in the first coil equals 1.20 A? (c) If the current in the second coil increases at a rate of 0.360 A> s, what is the magnitude of the induced emf in the first coil? A 10.0-cm-long solenoid of diameter 0.400 cm is wound 30.3 uniformly with 800 turns. A second coil with 50 turns is wound around the solenoid at its center. What is the mutual inductance of the combination of the two coils? .
-.,N- O ε .
=
-.,N- O:
.
Q.L, ?K G.,N × .-−H RD
,.HQ ?O
.
Q:G, µ K:
1.00 V
.
.
781 pH .
.
.
Energía y densidad de energía en un campo magnetico
A coil is connected in series with a 10.0 k resistor. An ideal 50.0 V battery is applied across the two devices, and the current reaches a value of 2.00 mA after 5.00 ms. (a) Find the inductance of the coil. (b) How much energy is stored in the coil at this same moment? •61
SS M
A coil with an inductance of 2.0 H and a resistance of 10 is suddenly connected to an ideal battery with 100 V. At 0.10 s after the connection is made, what is the rate at which (a) energy is being stored in the magnetic field, (b) thermal energy is appearing in the resistance, and (c) energy is being delivered by the battery? •62
At t 0, a battery is connected to a series arrangement of a resistor and an inductor. If the inductive time constant is 37.0 ms, at what time is the rate at which energy is dissipated in the resistor equal to the rate at which energy is stored in the inductor’s magnetic field? •63
ILW
At t 0, a battery is connected to a series arrangement of a resistor and an inductor. At what multiple of the inductive time constant will the energy stored in the inductor’s magnetic field be 0.500 its steady-state value? •64
61 62
63 64
(a) 97.9 H; (b) 0.196 mJ (a) 2.4 102 W; (b) 1.5 102 W; (c) 3.9 102 W 25.6 ms 1.23
A circular loop of wire 50 mm in radius carries a current of 100 A. Find the (a) magnetic field strength and (b) energy density at the center of the loop. •66
A solenoid that is 85.0 cm long has a cross-sectional area of 17.0 cm2. There are 950 turns of wire carrying a current of 6.60 A. (a) Calculate the energy density of the magnetic field inside the solenoid. (b) Find the total energy stored in the magnetic field there (neglect end effects). •67
SS M
A toroidal inductor with an inductance of 90.0 mH encloses a volume of 0.0200 m3. If the average energy density in the toroid is 70.0 J/m3, what is the current through the inductor? •68
What must be the magnitude of a uniform electric field if it is to have the same energy density as that possessed by a 0.50 T magnetic field? •69
Figure 30-67a shows, in cross section, two wires that are straight, parallel, and very long. The ratio i1/i2 of the current carried by wire 1 to that carried by wire 2 is 1/3. Wire 1 is fixed in place. Wire 2 can be moved along the positive side of the x axis so as to change the magnetic energy density uB set up by the two currents at the origin. Figure 30-67b gives uB as a function of the position x of wire 2. The curve has an asymptote of uB 1.96 nJ/m3 as x , and the horizontal axis scale is set by x s 60.0 cm. What is the value of (a) i1 and (b) i2? ••70
:
66
ILW
67
y 1
2
x
(a ) 2 )
3
m / J 1 n ( B
u
0 (b )
x s x (cm)
Figure 30-67 Problem 70.
68 69 70
(a) 1.3 mT; 3 (b) 0.63 J/m (a) 34.2 J/m ; (b) 49.4 mJ 5.58 A 8 1.5 10 V/m (a) 23 mA; (b) 70 mA
8.06 MJ
6.32 kJ
Circuito LC
Figure 31-19 shows three oscillating LC circuits with identical inductors and capacitors. At a particular time, the charges on the capacitor plates (and thus the electric fields between the plates) are all at their maximum values. Rank the circuits according to the time taken to fully discharge the capacitors during the oscillations, greatest first. 1
(a )
(b )
Figure 31-19 Question 1.
(c )
b, a, c
Figure 31-20 shows graphs of capacitor voltage vC for LC circuits 1 and 2, which contain identical capacitances and have the same maximum charge Q. Are (a) the inductance L and (b) the maximum current I in circuit 1 greater than, less than, or the same as those in circuit 2? 2
Figure 31-21 Question 5. Charges on the capacitors in three oscillating LC circuits vary as: (1) q 2 cos 4t , (2) q 4 cos t , (3) q 3 cos 4t (with q in coulombs and t in seconds). Rank the circuits according to (a) the current amplitude and (b) the period, greatest first. (a) 3, 1, 2; (b) 2, then 1 and 3 tie 6
v
C
t 2 1
(a) less; (b) greater
Figure 31-20 Question 2.
A charged capacitor and an inductor are connected at time t 0. In terms of the period T of the resulting oscillations, what is the first later time at which the following reach a maximum: (a) U B, (b) the magnetic flux through the inductor, (c) di/dt , and (d) the emf of the inductor? 3
(a) T /4;
In a certain oscillating LC circuit, the total energy is converted from electrical energy in the capacitor to magnetic energy in the inductor in 1.50 m s. What are (a) the period of oscillation and (b) the frequency of oscillation? (c) How long after the magnetic energy is a maximum will it be a maximum again? •3
What is the capacitance of an oscillating LC circuit if the maximum charge on the capacitor is 1.60 m C and the total energy is 140 m J?
3
•4
In an oscillating LC circuit, L 1.10 mH and C 4.00 m F. The maximum charge on the capacitor is 3.00 m C. Find the maximum current. •5
A 0.50 kg body oscillates in SHM on a spring that, when extended 2.0 mm from its equilibrium position, has an 8.0 N restoring force.What are (a) the angular frequency of oscillation, (b) the period of oscillation, and (c) the capacitance of an LC circuit with the same period if L is 5.0 H? •6
The energy in an oscillating LC circuit containing a 1.25 H inductor is 5.70 m J. The maximum charge on the capacitor is 175 m C. For a mechanical system with the same period, find the (a) mass, (b) spring constant, (c) maximum displacement, and (d) maximum speed. •7
SS M
4 5 6
7
(a) 6.00 s; (b) 167 kHz; (c) 3.00 s 9.14 nF 45.2 mA (a) 89 rad/s; (b) 70 ms; (c) 25 F (a) 1.25 kg; (b) 372 N/m; -4 (c) 1.75 10 m; (d) 3.02 mm/s
608 pF
20.0 V
Circuito RLC
A single-loop circuit consists of a 7.20 resistor, a 12.0 H inductor, and a 3.20 m F capacitor. Initially the capacitor has a charge of 6.20 m C and the current is zero. Calculate the charge on the capacitor N complete cycles later for (a) N 5, (b) N 10,and (c) N 100. ••24
What resistance R should be connected in series with an inductance L 220 mH and capacitance C 12.0 m F for the maximum charge on the capacitor to decay to 99.0% of its initial value in 50.0 cycles? (Assume v v .) ••25
ILW
24
25
(a) 5.85 C; (b) 5.52 C; (c) 1.93 C 8.66 m
