Aseries circuit has a capacitor of 10โ5
farad, a resistor of 3 ร 102 ohms, and an inductor of...
Mathematics, 11.12.2019 06:31 stichgotrich849
Aseries circuit has a capacitor of 10โ5
farad, a resistor of 3 ร 102 ohms, and an inductor of 0.2 henry.
the initial charge on the capacitor is 10โ6
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) = 10โ6
, 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 =
โ300 ยฑ
p
(300)2 โ 4(0.2)105
0.4
= โ1000 or โ 500
so the solution is of the form
q(t) = c1e
โ1000t + c2e
โ500t
.
to find the constants c1 and c2 we use the initial conditions:
10โ6 = q(0) = c1 + c2
and the current is
q
0
(t) = โ1000c1e
โ1000t โ 500c2e
โ500t
so
0 = q
0
(0) = โ1000c1 โ 500c2
giving us
c1 = โ10โ6
and c2 = 2 ร 10โ6
therefor the charge is
q(t) = โ10โ6
e
โ1000t + 2 ร 10โ6
e
โ500t
.
and the current is
i(t) = q
0
(t) = 10โ3
e
โ1000t โ 10โ3
e
โ500t
Answers: 3
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