Physics Circuits: charging time, voltage, current flow, discharge
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Consider the following transient RC-circuit and answer these questions in the given order.
1) Find charging the constant.
2) Find full-charged time.
3) Find charging equations.
4) Find voltage across and current flow for capacitor at t = 4 second/charging.
5) Find discharging time constant.
6) Find full-discharged time.
7) Find discharging equations.
8) Find voltage across and current flow for capacitor at t = 16 seconds.
9) Plot voltage across and current flow for capacitor during time period o =< t =< 16
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Solution Summary
Using diagrams and step by step equations, this solution provides answers for charging the constant, full-charged time, charging equations, voltage, discharging time constant, full-discharged time, discharging equations, and voltage.
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1) Charging time constant of the R1C circuit = τC = R1C = 50x103x40x10-6 = 2 sec
2) Full charge time: Charging of the capacitor takes place as per the equation: Q = Q0(1- e-t/τ) where Q0 is the final charge on the capacitor. Strictly, the capacitor takes infinite time to charge fully. However, in practice after a period of 5 times the time constant, the capacitor is over 99% charged and 5τ is taken as the time to charge the capacitor fully. Hence, time to charge the capacitor fully = 5 x 2 = 10 sec.
3) Voltage across the capacitor VC = Q/C
By Kirchhoff's loop law: R1I + Q/C = 20
Current I = dQ/dt. Hence, R1(dQ/dt) + Q/C = 20
dQ/dt + Q/R1C = 20/R1
dQ/dt = (20/R1 - Q/R1C)
dQ/(20/R1 - Q/R1C) = dt
Integrating: ∫dQ/(20/R1 - Q/R1C) = ∫dt
Substituting (20/R1 - Q/R1C) = x, - (1/R1C) dQ = dx or dQ = - R1C dx
- R1C∫dx/x = t
- R1C logex = t + D
- R1C loge(20/R1 - Q/R1C) = t + D where D is the constant of integration
Assuming the capacitor is fully uncharged initially, substituting t = 0, Q = 0 we get:
D = - R1C loge(20/R1)
Substituting for D: - R1C loge(20/R1 - Q/R1C) = t - R1C loge(20/R1)
R1C [loge(20/R1 - Q/R1C) - loge(20/R1)] = - t
loge(20/R1 - Q/R1C) - loge(20/R1) = - t/R1C
loge[(20/R1 - Q/R1C)/(20/R1)] = - t/R1C
(20/R1 - Q/R1C)/(20/R1) = e-t/R1C
(20/R1 - Q/R1C) = (20/R1)e-t/R1C
Q/R1C = 20/R1[1 - e-t/R1C]
Q = 20C[1 - e-t/R1C] ............(1)
In the above ...
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