of conservation o ve will solve for th
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Transcribed Image Text:4. You have seen how Kirchhoff's laws were used in your lectures to obtain a 2nd
order differential equation where we solved for the current. This time we will
use an even simpler concept: principle of conservation of energy to derive the
2nd order differential equation where we will solve for the charge. Take a look
at the circuit below.
.3F
14H
In the circuit above, we have a capacitor with capacitance 3 F, an inductor of
inductance 14 H and a resistor of 42
(a) The total energy that is supplied to the resistor is
LI, Q
20
E =
2
where L is the inductance, I is the current, C is the capacitance and Q
is the charge.
Write down the total energy supplied E in terms of Q and t only.
dQ
Remember that I =
dt
(b) Now you know that the power dissipation through a resistor is -1R.
Use the conservation of energy (energy gain rate = energy loss rate) to
derive the differential equation in terms Q and t only.
(c) Solve the differential equation for initial charge to be Qo with a initial
current of
(d) Given that the coefficient of your cosine function is the time-dependent
amplitude (for example A(t) is the amplitude of the function A(t) cos t).
At what time T will the amplitude of the charge oscillations in the circuit
be 65% of its initial value?
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