Physics 21
Fall, 2004
Solution, Practice Exam #2
Problem 1.
For the following circuit:
R
L
C
V
rms
(
frequency
ω
)
φ
ω
L
I
rms
1/
ω
C
I
rms
R
I
rms
V
rms
(a) Write the loop equation for this circuit.
V
−
IR
−
L
dI
dt
−
Q
C
=0
(b) If
R
=20Ω,
L
=0
.
025 H, and
C
=50
µ
F (50
×
10
−
6
F),
±nd the frequency
ω
for which the current lags the volt
age across the inductor by 30
◦
.
Hint:
Make sure
ω>
0
.
OOPS, the question SHOULD HAVE SAID, ±nd the
frequency
ω
for which the applied voltage lags the volt
age across the inductor by 30
◦
.
From the diagram, the phase angle
φ
can be written
tan
φ
=
ωL
−
1
ωC
R
.
If the applied voltage
V
∼
sin(
ωt
+
φ
) lags the voltage
across the inductor by 30
◦
, then
φ
=60
◦
and
tan
φ
=
√
3=
ωL
−
1
ωC
R
We transform this into a quadratic equation for
ω
Lω
2
−
√
3
Rω
−
1
/C
=0
,
whose solution is
ω
=
√
3
R
±
p
3
R
2
+4
L/C
2
L
.
We must take the plus sign so that
ω>
0; the result is
ω
=1
.
82 kH. A negative value of
ω
would reverse the
directions of the vertical vectors on the phasor diagram,
and change the phase relations.
(c) If
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 Fall '07
 Hickman
 Physics, Magnetic Field, VRMs, a1 a2 cos

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