What value of the variable resistor will give 12 seconds from the start of a cyc

Question

What value of the variable resistor will give 12 seconds from the start of a cycle to a pass of the wipers?

Express your answer to two significant figures and include the appropriate units.

Intermittent windshield wipers use a variable resistor in an RCcircuit to set the delay between successive passes of the wipers. A typical circuit is shown in the figure. When the switch closes, the capacitor (initially uncharged) begins to charge and the potential at point b begins to increase. A sensor measures the potential difference between points a and b, triggering a pass of the wipers when Vb = Va. (Another part of the circuit, not shown, discharges the capacitor at this time so that the cycle can start again.) (Figure 1) What value of the variable resistor will give 12 seconds from the start of a cycle to a pass of the wipers? Express your answer to two significant figures and include the appropriate units.

Explanation / Answer

First let's find a differential equation for the unknown voltage Vb (here I'll just call it v).
Use ohm's law to get the current moving through the resistor in terms of v:
(V_batt - v)/R = i
Substitute into the constitutive relation for the capacitor:
i = C dv/dt
(V_batt - v)/R = C dv/dt
RC v' + v = V_batt

Now that we have a differential equation, we can solve it by finding a homogeneous solution and a particular solution.
RC v' + v = 0
Try a solution of the form v = e^(st).
RCs e^(st) + e^(st) = 0
RCs + 1 = 0
s = -1/RC
Our homogeneous solution is then
v = Ae^(-t/RC)
For a particular solution, try v = V_batt.
Add the solutions together.
v(t) = V_batt + Ae^(t/RC)
Finally, solve for A with the initial condition.
v(0) = V_batt + A
A = v(0) - V_batt = V_batt (since v(0) = 0, the capacitor is initially uncharged)
Vb(t) = V_batt (1 - e^(-t/RC))

Let's find Va now. Notice that I just call that ratio of resistances a.
Va = [(47k)/(47k + 0.1k)] V_batt = a V_batt

Now, we want to set Va = Vb(t) and solve for R.
Va = a V_batt = V_batt (1 - e^(-t/RC))
a = 1 - e^(-t/RC)
-t/RC = ln(1-a)
RC = -t/ln(1-a)

R = - t / (C ln(1-a))
R = - (12 s) / ((100*10^-6 F) ln(1 - [(47k)/(47k + 0.1k)]))
R = 19496.79 ohms = 20k

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