Aseries 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