The timing device in an automobile’s intermittent wiper system is based on an RC time constant and utilizes a 0.500-microfarad capacitor and a variable resistor. Over what range must 𝑅 be made to vary to achieve time constants from 2.00 to 15.0 seconds?
We can call the 0.500-microfarad capacitor 𝐶. And we want to solve for the range of 𝑅, the resistance. We’ll call the lower end of that range 𝑅 sub one and the upper end 𝑅 sub two that must be made so that the total time constants of our circuit vary from 2.00 to 15.0 seconds.
We’ll call the lower end of that 2.00 seconds 𝑡 sub one and the upper end we’ll call 𝑡 sub two. And RC time constant 𝑡 is simply the product of the resistance times the capacitance in a circuit. So we can write that 𝑡 one equals 𝑅 one times 𝐶 and 𝑡 two equals 𝑅 two times 𝐶.
Rearranging these equations to solve for the resistances, 𝑅 one is 𝑡 one over 𝐶 and 𝑅 two is 𝑡 two over 𝐶. We’re giving the capacitance 𝐶 as well as the time range in our problem statement. So we’re ready to plug in and solve for the range of 𝑅.
When we do and calculate these two fractions, we find that to three significant figures 𝑅 one is 4.00 million ohms and 𝑅 two is 30.0 million ohms. This is the lower and upper end of the range over which our resistor must vary.