For the circuit of the figure below, let C = 13.0 nF, L = 19 mH, and R = 60.5 ?.
ID: 1261684 • Letter: F
Question
For the circuit of the figure below, let C = 13.0 nF, L = 19 mH, and R = 60.5 ?.
(a) Calculate the oscillation frequency of the circuit once the capacitor has been charged and the switch has been connected to point a.
kHz
(b) How long will it take for the amplitude of the oscillation to decay to 10.0% of its original value?
ms
(c) What value of R would result in a critically damped circuit?
?
I want the answer and the work please
When switch S is in this position. the emf charges the capacitor. When switch S is moved to this position, the capacitor discharges through the resistor and inductor.Explanation / Answer
(a) This is an LRC series circuit, and when we connect the switch at point a, the capacitor will discharge through the resistor and the inductor. The behavior of the charge on the capacitor depends on how large the resistance in the circuit is when compared to the capacitance and inductance. If R is relatively small, so that R2 < 4L/C, the circuit is called underdamped, and the charge on the capacitor as a function of time takes the form
q(t) = Ae?Rt/2L cos(?t),
where ? = sqrt(1/ LC ? R2 /4L2)
These two equations, which are on the equation sheet, show that the charge q(t) on the capacitor exhibits sinusoidal oscillation with an exponentially decaying amplitude. It can be easily checked that the given R, L, and C satisfy the condition of R2 < 4L/C for underdamped behavior. Since we are asked to find the frequency, and not the angular frequency, we note that ? = 2?f, and we just use the values given to evaluate f. (Don