# Physics Circuits: charging time, voltage, current flow, discharge

Please see the attached problem.

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 Preview

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 ...

#### 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.