- #1
zenterix
- 771
- 84
- Homework Statement
- Consider the RLC circuit shown below. At
the switch is closed.
- Relevant Equations
- a) For what values of
will the initial sign of be reversed at some later times?
b) Assuming the value of satisfies the condition in (a), find and in terms of , and .
c) If it takes 10 cycles for the energy in the circuit to decrease to times its initial value, find the value of the resistance in terms of and .
Using Faraday's law we have
where
After rearranging the expression we get
If the system is critically- or over-damped then
Therefore, for the sign of
must be satisfied and we have oscillations.
(4) implies
The solution to the differential equation is
where
If we now impose the initial conditions
then we find
My question is about item (c).
What I learned about the quality factor is the following.
The quality factor is
It is also
Amplitude is proportional to
and since the period is
then in a time period of
Thus
If I apply this last formula I get
which, after subbing in for
and we can solve for
The answer from MIT OCW is
which is slightly different.
So, the first thing I want to tackle is why I didn't get (18).
But what I really want to know is the following.
The potential energy of the system is
(19) by itself is constant.
The resistor dissipates energy at a rate of
Thus, if the system starts with energy
then it simply loses energy through the resistor over time.
If we want to know the time it takes for this energy to drop by
is this correct?
This latter calculation seems very complicated, but does its solution give us the time it takes for the energy to drop by a factor of
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