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Capacitor's Voltage Drop And Discharge Current


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Strings (SiPjAjk) = S7P3A32     Base Sequence = 12735     String Sequence = 12735 - 3 - 32



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A capacitor C is connected in series with a resistor, R (fig.2.25).
Capacitor's Capacitance = 1 μF; initial charge = 10-4 C (Coulomb)'
Assume constant discharge current during 0 < t <1 ms.
Determine capacitor's voltage drop at t = 1 ms, for:
(a) R = 1 MΩ
(b) R = 100 kΩ
(c) R = 10 kΩ
(d) Determine actual discharge current.

The string:
(a) S7P3A32 (Force - Push).
The math:
Pj Problem of Interest is of type force. Problems of voltage, energy, and work in an electric circuit are force problems.
Capacitor Discharge
The relationship between charge (q), capacitance (C) and voltage (V) is as follows:
q = CV -------(1)
So, dq/dt = C(dv/dt)
So, i = C(dv/dt) ------- (2). Where i = current.
Discharge current has negative polarity
So, discharge current, i = -C(dv/dt)
So, v = iR = -RC(dv/dt)
So, dt = -RC(1/v)dv

So, t = -RCln(V/V0)
So, V = V0e-t/RC -------(3)
V is the voltage at time t secs during discharge; V0 is initial voltage at tim t = 0; R is the resistance; C is the capacitance.
The product RC is called the time constant.
Capacitor's voltage drop after t secs of discharge = V0 - V = V0[1 - V0e-t/(RC)]
Initial voltage V0 = q/C (from eqn (1))
So, V0 = 10-4/10-6 = 100 V.
So, capacitor's voltage drop after t = 1 ms of discharge:
(a) Time constant = t/(RC) = 10-3/(106 x 10-6) = 10-3
So, voltage drop = 100[1 - e-t/(RC)] = 100[1 - e-10-3] = 0.1 V
(b) Time constant = t/(RC) = 10-3/(105 x 10-6) = 10-2
So, voltage drop = 100[1 - e-t/(RC)] = 100[1 - e-10-2] = 1 V
(c) Time constant = t/(RC) = 10-3/(104 x 10-6) = 10-1
So, voltage drop = 100[1 - e-t/(RC)] = 100[1 - e-10-1] = 9.5 V
(d) Actual discharge current, i = (100/R)e-t/RC.
It is evident from problem (a), (b) and (c) that the capacitor's voltage drop increases as the resistance of the resistor decreases.

The point . is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.
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