726 CHAPTER 7 Work and Energy
SUMMARY
Electromagnetic Induction and Faraday’s Law
(Sections 21.1–21.3) A changing magnetic flux through a circuit loop
induces an emf in the circuit. Faraday’s law states that the magnitude E of the induced emf in a circuit equals the absolute value of
the time rate of change of magnetic flux through the circuit:
E 5 0 DFB Dt 0 (Equation 21.4). This relation is valid whether the
change in flux is caused by a changing magnetic field, motion of
the conductor, or both.
S
The magnet’s motion causes a
changing magnetic field through the
coil, inducing a current in the coil.
0
N
/
Lenz’s Law
S
(Section 21.4) Lenz’s law
N
states that an induced current or emf
always acts to oppose the change that caused it. Lenz’s law can be
derived from Faraday’s law and is a convenient way to determine
the correct sign for any induced effect.
S
BInduced Opposed
I
S
B
Flux at time t1— F1
Flux at time t2 — F2
Motional Electromotive Force
(Section 21.5) When a conductor moves in a magnetic field, the
charges in the conductor are acted upon by magnetic forces that
create a current. When a conductor with length L moves with speed
v perpendicular to a uniform magnetic field with magnitude B, the
induced emf is E 5 vBL (Equation 21.7).
S
v
FE 5 qE
S
B
DF — Change
in flux
FB 5 qvB
a +
q
L
Magnetic and
electric forces on
– b charges in rod
moving perpendicular to magnetic
field
Eddy Currents
(Section 21.6) When a bulk piece of conducting material, such as a
metal, is in a changing magnetic field or moves through a nonuniform field, eddy currents are induced in it.
Mutual Inductance and Self-Inductance
(Sections 21.7 and 21.8) When changing magnetic flux created by a
changing current i1 in one circuit links a second circuit, an emf
with magnitude E2 is induced in the second circuit. A changing
current i2 in the second circuit induces an emf with magnitude E1 in
the first circuit. The two emfs are given by
E2 5 P M
Di1
Dt P
E1 5 P M
and
Di2
,
Dt P
(21.12)
where M is a constant called the mutual inductance. A changing
current i in any circuit induces an emf E in that same circuit, called
a self-induced emf, given by
Di
,
Dt P
where L is a constant called inductance or self-inductance.
(21.14)
+
E 5 PL
S
B
i
Self-inductance: If the current i
in the coil is changing, the
changing flux through the coil
induces an emf in the coil.
SUMMARY
Transformers
(Section 21.9) A transformer is used to transform the voltage and
current levels in an ac circuit. An alternating current in either
winding results in an alternating flux in the other winding, inducing an emf. In an ideal transformer with no energy losses, if the
primary winding has N1 turns and the secondary winding has N2
turns, the two voltages are related by
V2 N2
5 .
V1 N1
Iron core
ac source I1
V1
N1
For an ideal
transformer,
N
V2
5 2
N1
V1
N2
V2
Primary
winding
R
FB
(21.15)
Secondary winding
Magnetic Field Energy
(Section 21.10) An inductor with inductance L carrying current I has
energy U 5 12 LI 2 (Equation 21.19). This energy is stored in the
magnetic field of the inductor. The energy density u (energy per
unit volume) is given by u 5 B2 2m0 (Equation 21.21).
/
R–L and L–C Circuits
In a circuit containing a resistor R, an
inductor L, and a source of emf E, the growth and decay of current
are exponential, with a characteristic time t called the time constant, given by t 5 L R (Equation 21.26). The time constant t is
the time required for the increasing current to approach within a
fraction 1 2 1 1 e 2 , or about 63%, of its final value.
A circuit containing an inductance L and a capacitance C
undergoes electrical oscillations with angular frequency v, where
v 5 "1 LC (Equation 21.30).
E
/
/
/
i 5
i
Switch
+
(Sections 21.11 and 21.12)
E
I5
R
E
(1 2 e2(R/L)t2
R
t
(
I 12
a
b
R
vab 5 iR
i
c
O
L
vbc 5 L
Di
Dt
t5t5
L
R
1
e
)
t