Solved: Show that only half the total energy drawn from a battery in charging an RC

Are household electrical outlets connected in series or parallel? How do you know?

All the resistors in Fig. 25.24 have the same resistance. In which circuits does the battery supply the same current?

Can the voltage across a batterys terminals differ from the batterys rated voltage? Explain.

Can the voltage across a batterys terminals be higher than the batterys rated voltage? Explain.

In some cities, streetlights are wired in such a way that when one burns out, they all go out. Are the lights in series or parallel?

When the switch in Fig. 25.25 is open, whats the voltage across the resistor? Across the switch?

Two identical resistors in series dissipate equal power. How can this be, when electric charge loses energy in flowing through the first resistor?

When a large electric load such as a washing machine or oven comes on, lights throughout a house often dim. Why?

How would you connect a pair of equal resistors across an ideal battery in order to get the greatest power dissipation?

You have a battery whose voltage and internal resistance are unknown. Using an ideal voltmeter and an ideal ammeter, how would you determine each of these characteristics?

A student whos confused about voltage and current hooks a nearly ideal ammeter across a car battery. What happens?

A student whos confused about voltage and current tries to measure the voltage across a lighted lightbulb by inserting a voltmeter in series with the bulb. What happens to the bulb? Explain.

Sketch a circuit diagram for a circuit that includes a resistor R1 connected to the positive terminal of a battery, a pair of parallel resistors R2 and R3 connected to the lower-voltage end of R1 and then returned to the batterys negative terminal, and a capacitor across R2.

Sketch a diagram for a circuit consisting of two batteries, a resistor, and a capacitor, all in series. Does the circuit description allow you any flexibility?

Resistors R1 and R2 are in series, and the series combination is in parallel with R3. This parallel combination is connected across a battery. Draw a diagram of this circuit.

Whats the emf of a battery that delivers 27 J of energy as it moves 3.0 C between its terminals?

A 1.5-V battery stores 4.5 kJ of energy. How long can it light a flashlight bulb that draws 0.60 A?

If you accidentally leave your car headlights (current 5 A) on for an hour, how much energy drains from the cars 12-V battery?

A 47@k resistor and a 39@k resistor are in parallel, and the pair is in series with a 22@k resistor. Whats the resistance of the combination?

What resistance should you place in parallel with a 56@k resistor to make an equivalent resistance of 45 k?

A defective starter motor draws 300 A from a cars 12-V battery, dropping the battery terminal voltage to 6 V. A good starter should draw only 100 A. What will the battery terminal voltage be with a good starter?

Find the internal resistance of the battery in Exercise 21.

When a 9-V battery is temporarily short-circuited, a 200-mA current flows. Whats the batterys internal resistance?

You have a 1.0@, a 2.0@, and a 3.0@ resistor. What equivalent resistances can you form using all three?

Find all three currents in the circuit of Fig. 25.13, but now with E2 = 1.0 V

Whats the current through the 3@ resistor in Fig. 25.26? (Hint: This is trivial. Can you see why?)

Find I2 in Example 25.4 for the case E2 = 2.0 V.

A voltmeter with 200@k resistance is used to measure the voltage across the 10@k resistor in Fig. 25.27. By what percentage is the measurement in error because of the finite meter resistance?

An ammeter with 100@ resistance is inserted in the circuit of Fig. 25.27. By what percentage is the measured current in error because of the nonzero meter resistance?

A new mechanic foolishly connects an ammeter with 0.1@ resistance directly across a 12-V car battery with internal resistance 0.01 . Whats the power dissipation in the meter? (No wonder it gets destroyed!)

Show that the quantity RC has the units of time (seconds).

If capacitance is in F, what will be the units of the time constant RC when resistance is in (a) , (b) k, and (c) M? (Your answers eliminate the need for tedious power-of-10 conversions.)

Show that a capacitor is charged to approximately 99% of the applied voltage in five time constants 15RC2.

An uncharged 10@F capacitor and a 470@k resistor are in series, and 250 V is applied across the combination. How longdoes it take the capacitor voltage to reach 200 V?

Find an expression for the voltage across the capacitor in Example 25.6 when its fully charged.

In Fig. 25.28, all resistors have the same value, R. What will be the resistance measured (a) between A and B or (b) between A and C?

In Fig. 25.28, take all resistors to be 1 k. Find the current in the vertical resistor when a 6.0-V battery is connected between A and B.

Three 1.5-V batteries, with internal resistances 0.01 , 0.1 , and 1 , each have 1@ resistors connected across their terminals. Whats the voltage between each batterys terminals, to three significant figures?

A partially discharged car battery can be modeled as a 9-V emf in series with a 0.08@ internal resistance. Jumper cables connect this battery to a fully charged battery, modeled as a 12-V emf in series with a 0.02@ internal resistance. The cables con nect + to + and - to -. What current flows through the discharged battery?

Your company is overstocked on 50@, 1 2-W resistors. Your project requires 50@ resistors that can be safely connected across a 12-V power source. How many of the available resistors will you need, and how will you connect them?

A 6.0-V battery has internal resistance 2.5 . If the battery is short-circuited, whats the rate of energy dissipation in its internal resistance?

How many 100-W, 120-V lightbulbs can be connected in parallel before they trip a 20-A circuit breaker?

You company is designing a battery-based backup power source, and your job is to assess its safety. You know that under damp or sweaty conditions, the resistance between two points of unbroken skin on the human body can be as low as 500 . Your product uses a 72-V battery whose internal resistance is 100 . Is it capable of passing a fatal 100 mA (Table 24.3) through a damp human body?

Take E = 12 V and R1 = 270 in Fig. 25.4. (a) Whats the resistance R2 if theres 4.5 V across it? (b) What will be the power dissipation in R2?

In Fig. 25.29, R1 is a variable resistor and the other two resistors have equal resistances R. (a) Find an expression for the voltage across R1, and (b) sketch a graph of this voltage as R1 varies from 0 to 10R.

In the circuit of Fig. 25.30, find (a) the current supplied by the battery and (b) the current through the 6@ resistor.

In Fig. 25.30, how much power is dissipated in the 4@ resistor?

Whats the ammeter reading in Fig. 25.31?

In Fig. 25.32, find the equivalent resistance measured between A and B.

Find all three currents in the circuit of Fig. 25.13 with the values given, but with battery E2 reversed.

The voltage across the 30@k resistor in Fig. 25.33 is measured with (a) a 50@k voltmeter, (b) a 250@k voltmeter, and (c) a 10@M digital meter. What does each read, to two significant figures?

In Fig. 25.34, what are the meter readings when an ideal (a) voltmeter or (b) ammeter is connected between A and B?

A resistor draws 1.00 A from an ideal 12.0-V battery. (a) If an ammeter with 0.10@ resistance is inserted in the circuit, what will it read? (b) If this current is used to calculate the resistance, by what percent will the result be in error?

The voltage across a charging capacitor in an RC circuit rises to 1 - 1/e of the battery voltage in 5.0 ms. (a) How long will it take to reach 1 - 1/e3 of the battery voltage? (b) If the capacitor is charging through a 22@k resistor, whats the capacitance?

Youre designing an external defibrillator that discharges a ca- BIO pacitor through the patients body, providing a pulse that stops ventricular fibrillation. Specifications call for a capacitor storing 250 J of energy; when discharged through a body with 40@ transthoracic resistance, the capacitor voltage is to drop to half its initial value in 10 ms. Determine the capacitance (to the nearest 10 F) and initial capacitor voltage (to the nearest 100 V) that meet these specs

A capacitor used to provide steady voltages in the power supply of a stereo amplifier charges rapidly to 35 V every 1/60 second. It must then hold that voltage to within 1.0 V for the next 1/60 s while it discharges through the amplifier. If the amplifier draws 1.2 A from the 35-V supply, (a) whats its effective resistance, and (b) what capacitance is needed?

A capacitor is charged until it holds 5.0 J of energy, then connected across a 10@k resistor. In 8.6 ms, the resistor dissipates 2.0 J. Find the capacitance.

In Fig. 25.35 the 2.0@F capacitor is charged to 150 V, while the 1.0@F capacitor is initially uncharged. Switch S is then closed. Find the total energy dissipated in the resistor as the circuit comes to equilibrium. (Hint: Think about charge conservation.)

For the circuit of Example 25.6, take E = 100 V, R1 = 4.0 k, and R2 = 6.0 k, and assume the capacitor is initially uncharged. Find the capacitor voltage and the currents in both resistors (a) just after the switch is closed, and (b) a long time after the switch is closed. Long after the switch is closed its re-opened. What are VC, I1, and I2 (c) just after this switch opening, and (d) a long time later?

In Fig. 25.36, the switch is initially open and both capacitors are initially uncharged. All resistors have the same value R. Find expressions for the current in R2 (a) just after the switch is closed, and (b) a long time after the switch is closed.

A batterys voltage is measured using a voltmeter with resistance 10.00 k; the result is 4.982 V. A 15.00@k meter gives 4.993 V. Find (a) the batterys voltage and (b) its internal resistance

An ammeter with resistance 1.42 is connected momentarily across a battery, and the meter reads 9.78 A. When the measurement is repeated with a 2.11@ meter, the reading is 7.46 A. Find (a) the battery voltage and (b) its internal resistance.

In Fig. 25.37, take E1 = 12.0 V, E2 = 6.00 V, E3 = 3.00 V, R1 = 1.00 , R2 = 2.00 , and R3 = 4.00 . Find the current in R2 and give its direction.

With all values except E2 as given in the preceding problem, (a) find E2 such that there is no current in this battery. (b) What are the currents in R1 and R3 under these conditions?

The voltage on a charged capacitor is monitored with a voltmeter whose resistance is 1.00 M. The table below gives the meter reading as a function of time. Determine a function of the voltage which, when you plot it against time, should give a straight line. Make your plot, establish a best-fit line, and use it to determine the capacitance.

Find the resistance needed in an RC circuit to bring a 20@f capacitor from zero charge to 45% charge in 140 ms.

Suppose the currents into and out of a circuit node differ by 1 A. If the node consists of a small metal sphere with diameter 1 mm, how long would it take for the electric field around the node to reach the 3-MV/m breakdown field in air?

Show that a battery delivers the most power when the load resistance across its terminals is equal to its internal resistance. (This is not the way to treat a battery, but its the basis for load matching in amplifiers; see Problem 69.)

Youre writing the instruction manual for a stereo amplifier with a maximum output of 100 W. The amplifier can be modeled as an emf in series with an 8@ resistance. What should you specify for the loudspeaker resistance to be used with the amplifier? How much power can the amplifier deliver to a speaker with half the optimum resistance?

Show that only half the total energy drawn from a battery in charging an RC circuit ends up stored in the capacitor. (Hint: What happens to the rest? Youll need to integrate.)

Find the equivalent resistance between A and B for the circuits in Fig. 25.38.

A 270@ resistor is connected across a battery and a 31-mA current flows. When the resistor is replaced with a 120@ resistor, the current increases to 63-mA. Find (a) the batterys voltage and (b) its internal resistance.

Obtain an expression for the rate of increase 1dV/dt2 of the voltage across a charging capacitor in an RC circuit. Evaluate your result at time t = 0, and show that if the capacitor continued charging steadily at this rate, it would reach full charge in exactly one time constant.

The circuit in Fig. 25.39 extends forever to the right, and all the resistors have the same value R. Show that the equivalent resistance measured across the two terminals at left is R11 + 152/2. (Hint: You dont need to sum an infinite series.)

Figure 25.40 on the next page shows the voltage across a capacitor thats charging through a 4700@ resistor in the circuit of Fig. 25.18. Use the graph to determine (a) the battery voltage, (b) the time constant, and (c) the capacitance.

Figure 25.41 shows a portion of a circuit used to model muscle cells and neurons. All resistors have the same value R = 1.5 M, and the emfs are E1 = 75 mV, E2 = 45 mV, and E3 = 20 mV. Find the current through E3, including its direction.

An electrochemical impulse traveling along the cell modeled in Fig. 25.41 changes the value of E3 so now it supplies a 40-nA upward current. Assuming the rest of the circuit remains as described in Problem 76, whats the new value of E3?

A parallel-plate capacitor has plates of area 10 cm2 separated by a 0.10-mm layer of glass insulation with resistivity r = 1.2*1013 #m and dielectric constant k = 5.6. Because of the finite resistivity, charge leaks through the insulation. (a) How can such a leaky capacitor be represented in a circuit diagram? (b) Find the time constant for this capacitor to discharge through its insulation, and show that it depends only on the properties of the insulating material and not on its dimensions.

Write the node and loop equations for the circuit in Fig. 25.23a (Example 25.6), and find the time constant.

In Problem 60, take C1 = C2 = C, and find the current through R2 as a function of time. (Hint: Use the node and loop laws to get a differential equation for the current, and use the initial conditions on current and its derivative to evaluate the constants of integration.)

Youre about to purchase a battery. Normally, batteries are rated in ampere-hoursthe total charge they can deliver. Your application calls for a 5@A # h battery. But the 6-V battery you see while shopping online is rated at 50 watt-hours. Will it work?

In the circuit of Fig. 25.42 the switch is initially open and the capacitor is uncharged. Find expressions for the current I supplied by the battery (a) just after the switch is closed and (b) a long time after the switch is closed.

Stray voltage is a serious problem on dairy farms, often resulting from corroded wiring or poor wiring practices. These conditions can produce several volts between the ground and metal watering bowls, feed troughs, or milking equipment. Cows feel shocks that make them nervous, reducing milk output and sometimes leading to mammary gland infections. As a result, farmers can face serious financial losses. Figure 25.43 shows a typical stray-voltage situation, with the source of stray voltage modeled as a 6-V emf in series with a 1@k resistance. The current through the 500@ cow will be a. 3 mA. b. 4 mA. c. 6 mA. d. 12 mA.

Stray voltage is a serious problem on dairy farms, often resulting from corroded wiring or poor wiring practices. These conditions can produce several volts between the ground and metal watering bowls, feed troughs, or milking equipment. Cows feel shocks that make them nervous, reducing milk output and sometimes leading to mammary gland infections. As a result, farmers can face serious financial losses. Figure 25.43 shows a typical stray-voltage situation, with the source of stray voltage modeled as a 6-V emf in series with a 1@k resistance. The voltage across the cow shown is a. 2 V. b. 4 V. c. 6 V. d. nearly 0 V.

Stray voltage is a serious problem on dairy farms, often resulting from corroded wiring or poor wiring practices. These conditions can produce several volts between the ground and metal watering bowls, feed troughs, or milking equipment. Cows feel shocks that make them nervous, reducing milk output and sometimes leading to mammary gland infections. As a result, farmers can face serious financial losses. Figure 25.43 shows a typical stray-voltage situation, with the source of stray voltage modeled as a 6-V emf in series with a 1@k resistance. In an effort to diagnose the problem, a farmer connects an ideal voltmeter between the water bowl and ground, with the cow absent. The voltmeter reading is a. 2 V. b. 4 V. c. 6 V. d. none of the above

Stray voltage is a serious problem on dairy farms, often resulting from corroded wiring or poor wiring practices. These conditions can produce several volts between the ground and metal watering bowls, feed troughs, or milking equipment. Cows feel shocks that make them nervous, reducing milk output and sometimes leading to mammary gland infections. As a result, farmers can face serious financial losses. Figure 25.43 shows a typical stray-voltage situation, with the source of stray voltage modeled as a 6-V emf in series with a 1@k resistance. To explore the problem further, a farmer connects an ideal ammeter between the water bowl and ground, with the cow absent. The ammeter reading is a. 4 mA. b. 6 mA. c. 12 mA. d. infinite.