Capacitor's Voltage Drop And Discharge Current

**Strings (S _{i}P_{j}A_{jk}) = S_{7}P_{3}A_{32} Base Sequence = 12735 String Sequence = 12735 - 3 - 32 **

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Capacitor's Voltage Drop And Discharge Current

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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) S_{7}P_{3}A_{32} (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*.

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/V_{0})

So, V = V_{0}e^{-t/RC} -------(3)

V is the voltage at time t secs during discharge; V_{0} 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 = V_{0} - V = V_{0}[1 - V_{0}e^{-t/(RC)}]

Initial voltage V_{0} = q/C (from eqn (1))

So, V_{0} = 10^{-4}/10^{-6} = 100 V.

So, capacitor's voltage drop after t = 1 ms of discharge:

(a) Time constant = t/(RC) = 10^{-3}/(10^{6} 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}/(10^{5} 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}/(10^{4} 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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