ELECTROMAGNETIC INDUCTION
27
E
XERCISES
Sections 27.2 Faraday’s Law and 27.3 Induction and Energy
15.
I
NTERPRET
In this problem we are asked to verify that the SI unit of the rate of change of magnetic flux is volt.
D
EVELOP
We first note that the lefthandside of Equation 27.2,
/
,
B
d
dt
ε
= − Φ
represents the induced emf which
has units of volt. From the definition of magnetic flux given in Equation 27.1a, we see that it has SI units of
2
T m .
⋅
E
VALUATE
The reasoning above shows that the units of
/
B
d
dt
Φ
are
2
2
T m /s
(N/A m)(m /s)
(N m/A s)
J/C
V
⋅
=
⋅
=
⋅
⋅
=
=
A
SSESS
Faraday’s law relates the induced emf to the change in flux. It is the rate of change of flux, and not the
flux or the magnetic field that gives rise to an induced emf.
16.
For a stationary plane loop in a uniform magnetic field, the integral for the flux in Equation 27.1a is just
B
B A
φ
=
⋅
=
G
G
2
cos
(80 mT)
(2.5 cm)
cos30
1.36
10
Wb
4
BA
θ
π
−
=
° =
×
,
.
(The SI unit of flux,
T m
2
⋅
is also called a weber, Wb.)
17.
I
NTERPRET
This problem is about the rate of change of magnetic flux through a loop due to a changing
magnetic field.
D
EVEL
P
For a stationary plane loop in a uniform magnetic field, the magnetic flux is given by Equation 27.1b,
O
cos .
B
B A
BA
θ
Φ
=
⋅
=
G
G
Note that the SI unit of flux,
T m
2
,
⋅
is also called a weber, Wb. The rate of change of
magnetic flux is
/
/
B
B
d
dt
t
Φ
= ΔΦ
Δ
.
)
E
VALUATE
(a)
The magnetic field at the beginning
1
(
0
t
=
is
2
2
1
1
1
1
1
(40 cm)
(5 mT)
6.28
10
Wb
4
4
B A
d B
π
π
−
Φ =
=
=
=
×
4
=
(b)
The magnetic field at
t
is
2
25 ms
2
2
2
2
2
1
1
(40 cm)
(55 mT)
6.91
10
Wb
4
4
B A
d B
π
π
−
Φ
=
=
=
=
×
3
(c)
Since the field increases linearly, the rate of change of magnetic flux is
3
3
2
1
2
1
6.91
10
Wb
0.628
10
Wb
0.251 V
25 ms
B
B
d
dt
t
t
t
−
−
Φ
ΔΦ
Φ
− Φ
×
−
×
=
=
=
=
Δ
−
From Faraday’s law, this is equal to the magnitude of the induced emf, which causes a current


0.251 V
=
2.5
100
I
R
ε
=
=
Ω
1 mA
in the loop.
sol
coil
sol
sol
coil
sol
 
(
)
B
dB
d
d
N
B
A
N
A
dt
dt
dt
ε
Φ
=
=
=
(d)
The direction must oppose the increase of the external field downward, hence the induced field is upward and
I
is CCW when viewed from above the loop.
A
SSESS
Since
with the area of the loop kept fixed, the induced emf and hence the current
scale linearly with the value
Δ
Δ
/
(
/
)
B
t
B
t
ΔΦ
Δ = Δ
Δ
A
/
.
B
t
27.1
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