Math 333 Logarithms Worksheet With Answers Page 7
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136
B2. Consider a series circuit with a 10 microfarad capacitor (a microfarad is 10
farad),
a 0.25 henry inductor, and a 10 ohm resistor.
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(a) Assume the impressed voltage is 0, the initial charge on the capacitor is 10
coulomb, and the initial current is 0.
Find Q(t), the charge on the capacitor, as a function of time.
Solution: The initial value problem here is
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5
5
Q + 10Q + 10
Q = 0;
Q(0) = 10
,
Q (0) = 0.
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1
2
5
The characteristic equation is
r
+ 10r + 10
= 0. The roots are
20
60i 111.
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The general solution to this differential equation is
20
Q(t) = e
(c
cos ωt + c
sin ωt)
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where ω = 60 111. From this we get
20
20
Q (t) =
20e
(c
cos ωt + c
sin ωt) + e
( ωc
sin ωt + ωc
cos ωt).
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2
1
2
The initial conditions say
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10
= Q(0) = c
1
and
0 = Q (0) =
20c
+ ωc
1
2
5
20c
10
1
5
Thus c
= 10
and c
=
=
. The complete solution is
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2
ω
3 111
1
5
20
Q(t) = 10
e
cos ωt +
sin ωt .
3 111
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