A series circuit has a capacitor of 10−5
farad, a resistor of 3 × 102 ohms, and an inductor of 0.2 henry.
The initial charge on the capacitor is 10−6
coulomb and there is no initial current.
A) Set up an initial value problem modeling this circuit. (2 points)
The initial value problem is
0.2Q
00 + 300Q
0 + 105Q = 0, , Q(0) = 10−6
, Q0
(0) = 0
where Q(t) is the charge.
B) Find the charge on the capacitor and the current through the resistance at any time t.
The characteristic equation has roots
r =
−300 ±
p
(300)2 − 4(0.2)105
0.4
= −1000 or − 500
so the solution is of the form
Q(t) = C1e
−1000t + C2e
−500t
.
To find the constants C1 and C2 we use the initial conditions:
10−6 = Q(0) = C1 + C2
and the current is
Q
0
(t) = −1000C1e
−1000t − 500C2e
−500t
so
0 = Q
0
(0) = −1000C1 − 500C2
giving us
C1 = −10−6
and C2 = 2 × 10−6
therefor the charge is
Q(t) = −10−6
e
−1000t + 2 × 10−6
e
−500t
.
and the current is
I(t) = Q
0
(t) = 10−3
e
−1000t − 10−3
e
−500t