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Chapter 18: Problem 92

(a) In a charging \(R C\) circuit, how many time constants have elapsed when the capacitor has \(99.0 \%\) of its final charge? (b) How many time constants have elapsed when the capacitor has \(99.90 \%\) of its final charge? (c) How many time constants have elapsed when the current has \(1.0 \%\) of its initial value?

Short Answer

Expert verified
Answer: (a) Approximately 4.605 time constants have elapsed when the capacitor has 99.0% of its final charge. (b) Approximately 6.908 time constants have elapsed when the capacitor has 99.90% of its final charge. (c) Approximately 4.605 time constants have elapsed when the current has 1.0% of its initial value.

Step by step solution

01

(Step 1: Charging an RC circuit formula)

Recall that the charge on a capacitor as it charges in an RC circuit is given by the formula: \(Q(t) = Q_{final}(1 - e^{-t/(\tau)})\) Where \(Q(t)\) is the charge at time \(t\), \(Q_{final}\) is the final charge, and \(\tau\) is the time constant, given by the product of the resistance R and capacitance C.
02

(Step 2: Solve for number of time constants for 99% charging)

In this case, we want to find how many time constants have elapsed when the capacitor has 99% of its final charge. First, let's represent the ratio of charges as a percentage: \(\frac{Q(t)}{Q_{final}} = 0.99\) Substitute the charge formula and solve for the ratio of elapsed time to the time constant, \(t/\tau\): \(0.99 = 1 - e^{-t/(\tau)}\) To find \(t/\tau\), we rearrange and take the natural logarithm: \(t/\tau = \ln(\frac{1}{1 - 0.99}) \approx 4.605\)
03

(Step 3: Solve for number of time constants for 99.90% charging)

We'll follow the same procedure as before, but now the ratio of charges is 99.90%: \(\frac{Q(t)}{Q_{final}} = 0.999\) Substitute the charge formula and solve for the ratio of elapsed time to the time constant: \(t/\tau = \ln(\frac{1}{1 - 0.999}) \approx 6.908\)
04

(Step 4: Solve for number of time constants for 1% of initial current)

In this case, we want to find how many time constants have elapsed when the current has 1% of its initial value. First, recall the formula for current in an RC circuit: \(I(t) = I_{initial}e^{-t/\tau}\) Now, let's represent the ratio of currents as a percentage: \(\frac{I(t)}{I_{initial}} = 0.01\) Substitute the current formula and solve for the ratio of elapsed time to the time constant: \(t/\tau = \ln(\frac{I_{initial}}{0.01I_{initial}}) \approx 4.605\)
05

(Answers)

(a) Approximately 4.605 time constants have elapsed when the capacitor has 99.0% of its final charge. (b) Approximately 6.908 time constants have elapsed when the capacitor has 99.90% of its final charge. (c) Approximately 4.605 time constants have elapsed when the current has 1.0% of its initial value.

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