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Chapter 27: Q59P (page 800)

Question: What multiple of the time constant τgives the time taken by an initially uncharged capacitor in an RC series circuit to be charged to99,0%of its final charge?

Short Answer

Expert verified

Answer:

The multiple of the time constant τ is 4.61 .

Step by step solution

01

The given data

The final charge of the capacitor, q=0.99q0

02

Understanding the concept of time constant

Using the charging and discharging concept of the capacitor, we can get the required charge relation using the time constant of the RC circuit. Thus, the condition will give the multiplication value of the time constant.

Formulae:

The charge equation of an RC circuit, q=q0e-t/RC (1)

The time constant of the RC circuit, τ=RC (2)

03

Calculation of the multiple of the time constant

Now, the multiple of the time constant can be given using equation (2) in equation (1) as follows:

0.99q0=q0e-t/τ0.99=e-t/τtτ=ln10099=4.61t=4.61τ

Hence, the value of the multiple of the time is 4.61

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Most popular questions from this chapter

Question: Figure shows a battery connected across a uniform resistor R-0. A sliding contact can move across the resistor from x = 0 at the left to x = 10 at the right. Moving the contact changes how much resistance is to the left of the contact and how much is to the right. Find the rate at which energy is dissipated in resistor as a function of x. Plot the function for ε=50V,R=2000Ω,andR0=100Ω, .

In Figure, ε1=6.00V, ε2=12.0V ,R1=100Ω , R2=200Ω ,andR3=300Ω . One point of the circuit is grounded(V=0) .(a)What is the size of the current through resistance 1? (b) What is the direction (up or down) of the current through resistance 1? (c) What is the size of the current through resistance 2?(d) What is the direction (left or right) of the current through resistance 2? (e) What is the size of the current through resistance 3? (f) What is the direction of the current through resistance 3? (g) What is the electric potential at point A?

The following table gives the electric potential differenceVTacross the terminals of a battery as a function of currentbeing drawn from the battery.

(a) Write an equation that represents the relationship betweenVTandi. Enter the data into your graphing calculator and perform a linear regression fit ofVTversus.iFrom the parameters of the fit, find

(b) the battery’s emf and

(c) its internal resistance.

Suppose that, while you are sitting in a chair, charge separation between your clothing and the chair puts you at a potential of 200 V, with the capacitance between you and the chair at 150 pF. When you stand up, the increased separation between your body and the chair decreases the capacitance to 10 pF. (a) What then is the potential of your body? That potential is reduced over time, as the charge on you drains through your body and shoes (you are a capacitor discharging through a resistance). Assume that the resistance along that route is 300GΩ. If you touch an electrical component while your potential is greater than 100V, you could ruin the component. (b) How long must you wait until your potential reaches the safe level of 100V?

If you wear a conducting wrist strap that is connected to ground, your potential does not increase as much when you stand up; you also discharge more rapidly because the resistance through the grounding connection is much less than through your body and shoes. (c) Suppose that when you stand up, your potential is 1400 Vand the chair-to-you capacitance is 10pF. What resistance in that wrist-strap grounding connection will allow you to discharge to100V in 0.30 s, which is less time than you would need to reach for, say, your computer?

Two identical batteries of emf ε=12.0Vand internal resistance r=0.200Ωare to be connected to an external resistanceR , either in parallel (Figure a) or in series (Figure b). (a) If ,R=2.00r whatis the current in the external resistance in the parallel arrangement? (b) If R=2.00r,what is the current iin the external resistance in the series arrangements? (c) For which arrangement isigreater? (d) IfR=r/2.00 , what is in the external resistance in the parallel? (e) If R=r/2.00, what is i in the external resistance in the series arrangements? (f) For which arrangement is i greater now?
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