Intermittent windshield wipers use a variable resistor in an RC circuit to set t

Question

Intermittent windshield wipers use a variable resistor in an RC circuit 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 V_b = V_a. (Another part of the circuit, not shown, discharges the capacitor at this time so that the cycle can start again.) 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. To decrease the time, should the variable resistance be increased or decreased?

Explanation / Answer

Here ,

as Va = 12 * 47/(47 + 100)

Va = 3.83 V

as Vo = 12 V

Now , for the capacitor

V = Vo * (1 - e^(-t/RC))

3.83 = 12 * ( 1 - e^(-12/(100 *10^-6 * R)))

solving for R

R = 3.12 *10^5 Ohm

the variable resistance is 3.12 *10^5 Ohm

part B)

for decreasing the time

as time constant should be decreased

as T = R * C

R should decrease

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