Capacitor Resistor Inductor A charged capacitor connected to an inductor causes a current to flow through the inductor until the capacitor is fully discharged. The current in the inductor, in turn, charges up the capacitor until the capacitor is fully charged again. If Q(t) is the charge on the capacitor at time t, and I is the current, then dQ I = dt If the circuit resistance is zero, then the charge Q and the current I in the circuit satisfy the differential equation IP L Q = 0, dt where C is the capacitance and L is the inductance, so Q = 0. dt2 C Then, just as as a spring can have a damping force which affects its motion, so can a circuit; this is introduced by the resistor, so that if the resistance of the resistor is R, d²Q dQ + R 1 Q = 0. C L- dt? dt If L = 1 henry, R = 2 ohm, and C = 9 farads, find a formula for the charge when (a) Q(0) = 0 and Q'(0) = 8: Q(t) = (b) Q(0) = 8 and Q' (0) = 0: Q(t) =

Capacitor Resistor Inductor A charged capacitor connected to an inductor causes a current to flow through the inductor until the capacitor is fully discharged. The current in the inductor, in turn, charges up the capacitor until the capacitor is fully charged again. If Q(t) is the charge on the capacitor at time t, and I is the current, then dQ I = dt If the circuit resistance is zero, then the charge Q and the current I in the circuit satisfy the differential equation IP L Q = 0, dt where C is the capacitance and L is the inductance, so Q = 0. dt2 C Then, just as as a spring can have a damping force which affects its motion, so can a circuit; this is introduced by the resistor, so that if the resistance of the resistor is R, d²Q dQ + R 1 Q = 0. C L- dt? dt If L = 1 henry, R = 2 ohm, and C = 9 farads, find a formula for the charge when (a) Q(0) = 0 and Q'(0) = 8: Q(t) = (b) Q(0) = 8 and Q' (0) = 0: Q(t) =

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

Сараcitor
Resistor
Inductor
A charged capacitor connected to an inductor causes a current to flow through the inductor until the capacitor is fully discharged. The current in the inductor, in turn,
charges up the capacitor until the capacitor is fully charged again. If Q(t) is the charge on the capacitor at time t, and I is the current, then
dQ
I =
dt
If the circuit resistance is zero, then the charge Q and the current I in the circuit satisfy the differential equation
IP
L
dt
Q
= 0,
C
where C is the capacitance and L is the inductance, so
d?Q
L-
= 0.
dt?
Then, just as as a spring can have a damping force which affects its motion, so can a circuit; this is introduced by the resistor, so that if the resistance of the resistor
is R.
d?Q
dQ
+R
dt2
1
Q = 0.
L
dt
If L = 1 henry, R =
ohm, and C = 9 farads, find a formula for the charge when
(a) Q(0) = 0 and Q' (0) = 8:
Q(t) =
(b) Q(0) = 8 and Q' (0) = 0:
Q(t) =

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