will grade life saver please help All RLC circuit consists of a 25 mH inductor,

Question

will grade life saver please help

All RLC circuit consists of a 25 mH inductor, a 0.001 inicroF capacitor and a variable resistor connected in series. The initial charge on the capacitor at t = 0 when the switch is closed is 1 mC. What is the critical resistance Rc for this circuit (the resistance that makes the oscillation of charge critically damped)? Consult section 31.3.3 in the text or your lecture notes for the relevant equations. Use Excel or other software to generate three plots of the charge on the capacitor vs. time: Q(t) from / = 0 to t = 1 ms for R = 0.01 Re Q(t) from t = 0 to t = 1 ms for R = 0.1 = Re Q(t) from t = 0 to t = 1 ms for R = Re from t = 0

Explanation / Answer

Given that L = 25 mH C = 0.001 mF Qo = 1 mC ------------------------------------------------------------------------------------------------- Critical resistance Rc = 2 [ L/C]^1/2 = 2 * [ 25 x 10^-3 / 0.001 x 10^-3]^1/2 = 316.22 ohms ------------------------------------------------------------------------------------------------- We know that Q(t) = Qo ( 1 + at) e^-wt Where a = R / 2L w = 1 / [LC]^1/2 = 1 / [ 25 x 10^-3 * 0.001 x 10^-3]^1/2 = 6.324 x 10^3 rad/s -------------------------------------------------------------------------------------------------- 1. Q(t) from t = 0 to t = 1 ms for R = 0.01 Rc -------------------------------------------------------------------------- Therefore Q(t = 0) = (Qo) [ 1 + a (0)] e^-w(0) = Qo [1] * [1] = Qo = 1 mC --------------------------------------------------------------------------- Q(t = 1 ms) = Qo [ 1 + a (1 ms)] e^-w(1ms) = (1 x 10^-3) [ 1 + (R / 2 L) * 1 x 10^-3] e^- [ 1 x 10^-3 *w ] = (1 x 10^-3) [ 1 + (0.01*Rc / 2 L) * 1 x 10^-3] e^- [ 1 x 10^-3 * 6.324 x 10^3] = (1 x 10^-3) [ 1 + (0.01*316.22 / 2 *25 x 10^-3) * 1 x 10^-3] e^- [1 x 10^-3 *6.324 x 10^3 = ........ C ( solve it) You will get Q (t = 1 ms) = ............. C ----------------------------------------------------------------------------------------------------- Now draw the graph between Q and t y - axis x - axis Q (t = 0) t = 0 Q(t = 1ms) t = 1 ms ------------------------------------------------- Repeat the same procedure to 2 and 3.

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