Capacitors and are connected to a resistor of resistance

In our study of electrostatics, we concluded that no electric field exists within the material of a conductor in electrostatic equilibrium. Why can we discuss electric fields within the material of conductors in this chapter?

Figure 25-12 shows a mechanical analog of a simple electric circuit. Devise another mechanical analog in which the current is represented by a flow of water instead of marbles. In the water circuit, what would be analogous to the battery? What would be analogous to the wire? What would be analogous to the resistor?

Wires A and B are both made of copper. The wires are connected in series, so we know they carry the same current. However, the diameter of wire A is twice the diameter of wire B. Which wire has the higher number density of charge carriers (number per unit charge)? (a) (b) (c) They have the same number density of charge carriers.

The diameters of copper wires A and B are equal. The current carried by wire A is twice the current carried by wire B. In which wire do the charge carriers have the higher drift speed? (a) (b) (c) They have the same drift speed.

Wire Aand wire B are identical copper wires. The current carried by wire Ais twice the current carried by wire B. Which wire has the higher current density? (a) (b) (c) They have the same current density. (d) None of the above

Consider a metal wire that has each end connected to a different terminal of the same battery. Your friend argues that no matter how long the wire is, the drift speed of the charge carriers in the wire is the same. Evaluate your friends claim.

In a resistor, the direction of the current must always be in the downhill direction, that is, in the direction of decreasing electric potential. Is it also the case that in a battery, the direction of the current must always be the downhill? Explain your answer. A, B, B, A, A, B,

Discuss the distinction between an emf and a potential difference.

Wire A and wire B are made of the same material and have the same length. The diameter of wire A is twice the diameter of wire B. If the resistance of wire B is then what is the resistance of wire A? (Neglet any effects that temperature may have on resistance.) (a) (b) (c) (d) (e)

Two cylindrical copper wires have the same mass. Wire A is twice as long as wire B. (Neglet any effects that temperature may have on resistance.) Their resistances are related by (a) (b) (c) (d)

If the current in a resistor is the power delivered to the resistor is If the current in the resistor is increased to what is the power then delivered to the resistor? (Assume the resistance of the resistor does not change.) (a) (b) (c) (d) (e)

If the potential drop across a resistor is the power delivered to the resistor is If the potential drop is increased to what is the power delivered to the resistor then equal to? (a) (b) (c) (d) (e)

A heater consists of a variable resistor (a resistor whose resistance can be varied) connected across an ideal voltage supply. (An ideal voltage supply is one that has a constant emf and a negligible internal resistance.) To increase the heat output, should you decrease the resistance or increase the resistance? Explain your answer.

One resistor has a resistance and another resistor has a resistance The resistors are connected in parallel. If the equivalent resistance of the combination is approximately (a) (b) (c) 0, (d) infinity.

One resistor has a resistance and another resistor has a resistance The resistors are connected in series. If the equivalent resistance of the combination is approximately (a) (b) R (c) 0, (d) infinity.

A parallel combination consisting of resistors A and B is connected across the terminals of a battery. The resistor Ahas twice the resistance of resistor B. If the current carried by resistor A is then what is the current carried by resistor B? (a) (b) (c) (d) (e)

A series combination consisting of resistors A and B is connected across the terminals of a battery. The resistor Ahas twice the resistance of resistor B. If the current carried by resistor A is then what is the current carried by resistor B? (a) (b) (c) (d) (e)

Kirchhoffs junction rule is considered to be a consequence of (a) conservation of charge, (b) conservation of energy, (c) Newtons laws, (d) Coulombs law, (e) quantization of charge.

True or false: (a) An ideal voltmeter has a zero internal resistance. (b) An ideal ammeter has a zero internal resistance. (c) An ideal voltage source has a zero internal resistance.

Before you and your classmates run an experiment, your professor lectures about safety. She reminds you that to measure the voltage across a resistor you connect a voltmeter in parallel with the resistor, and to measure the current in a resistor you connect an ammeter in series with the resistor. She also states that connecting a voltmeter in series with a resistor will not measure the voltage across the resistor, but also cannot do any damage to the circuit or the instrument. In addition, connecting an ammeter in parallel with a resistor will not measure the current in the resistor, but could cause significant damage to the circuit and the instrument. Explain why connecting a voltmeter in series with a resistor causes no damage while connecting an ammeter in parallel with a resistor can cause significant damage.

The capacitor in Figure 25-49 is initially uncharged. Just after the switch is closed, (a) the voltage across equals (b) the voltage across equals (c) the current in the circuit is zero, (d) both (a) and (c) are correct.

A capacitor is discharging through a resistor. If it takes a time for the charge on a capacitor to drop to half its initial value, how long (in terms of ) does it take for the stored energy to drop to half its initial value?

In Figure 25-50, the values of the resistances are related as follows: If power is delivered to what is the power delivered to and R

The capacitor in Figure 25-49 is initially uncharged. The switch is closed and remains closed for a very long time. During this time, (a) the energy supplied by the battery is (b) the energy dissipated in the resistor is (c) the energy in the resistor is dissipated at a constant rate, (d) the total charge passing through the resistor is

It is not a good idea to stick the ends of a metal paper clip into the two rectangular slots of a household electrical wall outlet in the United States. Explain why by estimating the current that a paper clip would carry until either the fuse blows or the breaker trips.

(a) Estimate the resistance of an automobile jumper cable. (b) Look up the current required to start a typical car. At that current, what is the potential drop that occurs across the jumper cable? (c) How much power is dissipated in the jumper cable when it carries that current?

ENGINEERING APPLICATION, CONTEXT-RICH Your manager wants you to design a new superinsulated water heater for the residential market. Acoil of Nichrome wire is to be used as the heating element. Estimate the length of wire required. Hint: You will need to determine the size of a typical water heater and a reasonable time period for creating hot water.

A compact fluorescent lightbulb costs about $6.00 and has a typical lifetime of The bulb uses of power but produces illumination equivalent to that of a incandescent bulb. An incandescent bulb costs about $1.50 and has a typical lifetime of Your family wonders whether it should buy fluorescent lightbulbs. Estimate the amount of money your household would save each year by using compact fluorescent lightbulbs instead of the incandescent bulbs.

CONTEXT-RICH The wires in a house must be large enough in diameter so that they do not get hot enough to start a fire. While working for a building contractor during the summer, you are involved in remodeling a house. The local building code states that the Joule heating of the wire used in houses should not exceed Estimate the maximum gauge of the copper wire that you can use during the rewiring of the house with circuits.

A laser diode used in making a laser pointer is a highly nonlinear circuit element. Its behavior is as follows. For any voltage drop across it that is less than about it behaves as if it has an infinite internal resistance, but for voltages higher than it has a very low internal resistanceeffectively zero. (a) A laser pointer is made by putting two watch batteries in series across the laser diode. If the batteries each have an internal resistance between and estimate the current in the laser diode. (b) About half of the power delivered to the laser diode goes into radiant energy. Using this fact, estimate the power of the laser beam, and compare this value to typical quoted values of about (c) If the batteries each have a capacity of (i.e., they can deliver a constant current of for approximately 1 hour before discharging), estimate how long one can continuously operate the laser pointer before replacing the batteries.

A copper wire carries a current equal to Assuming copper has one free electron per atom, calculate the drift speed of the free electrons in the wire. SSM

A thin nonconducting ring that has a radius and a linear charge density rotates with angular speed about an axis through its center and perpendicular to the plane of the ring. Find the current of the ring.

A length of copper wire and a length of copper wire are welded together end to end. The wires carry a current of (a) If there is one free electron for each copper atom in each wire, find the drift speed of the electrons in each wire. (b) What is the ratio of the magnitude of the current density in the length of wire to the magnitude of the current density in the length of wire?

An accelerator produces a beam of protons with a circular cross section that is in diameter and has a current of The current density is uniformly distributed throughout the beam. The kinetic energy of each proton is The beam strikes a metal target and is absorbed by the target. (a) What is the number density of the protons in the beam? (b) How many protons strike the target each minute? (c) What is the magnitude of the current density in this beam?

In one of the colliding beams of a planned proton supercollider, the protons are moving at nearly the speed of light and the beam current is The current density is uniformly distributed throughout the beam. (a) How many protons are there per meter of length of the beam? (b) If the cross-sectional area of the beam is what is the number density of protons? (c) What is the magnitude of the current density in this beam?

CONTEXT-RICH The solar wind consists of protons from the Sun moving toward Earth (the wind actually consists of about 95% protons). The number density of protons at a distance from the Sun equal to the orbital radius of Earth is about 7.0 protons per cubic centimeter. Your research team monitors a satellite that is in orbit around the Sun at a distance from the Sun equal to Earths orbital radius. You are in charge of the satellites mass spectrometer, an instrument used to measure the composition and intensity of the solar wind. The aperture of your spectrometer is a circle of radius The rate of collection of protons by the spectrometer is such that they constitute a measured current of What is the speed of the protons in the solar wind? (Assume the protons enter the aperture at normal incidence.)

A gold wire has a cross section. Opposite ends of this wire are connected to the terminals of a battery. If the length of the wire is how much time, on average, is required for electrons leaving the negative terminal of the battery to reach the positive terminal? Assume the resistivity of gold is

A wire has a resistance equal to and carries a current equal to (a) What is the potential difference across the entire length of the wire? (b) What is the electric-field strength in the wire?

A potential difference of across the terminals of a resistor produces a current of in the resistor. (a) What is the resistance of the resistor? (b) What is the current in the resistor when the potential difference is only (Assume the resistance of the resistor remains constant.) SSM

A block of carbon is long and has a square crosssection whose sides are long. Apotential difference of is maintained across its length. (a) What is the resistance of the block? (b) What is the current in this resistor?

An extension cord consists of a pair of 30-m-long 16-gauge copper wires. What is the potential difference that must be applied across one of the wires if it is to carry a current of

(a) How long is a copper wire that has a resistance of (b) How much current will it carry if a potential difference is applied across its length?

A cylinder of glass is long and has a resistivity of What length of copper wire that has the same crosssectional area will have the same resistance as the glass cylinder?

ENGINEERING APPLICATION While remodeling your garage, you need to temporarily splice, end to end, an copper wire that is in diameter with a aluminum wire that has the same diameter. The maximum current in the wires is (a) Find the potential drop across each wire of this system when the current is (b) Find the electric field in each wire when the current is

A wire has a resistance equal to A second wire made of identical material has a length of and a mass equal to the mass of the first wire. What is the resistance of the second wire?

A copper wire can safely carry currents up to (a) What is the resistance of a length of the wire? (b) What is the electric field in the wire when the current is (c) How long does it take for an electron to travel in the wire when the current is

A cube of copper has edges that are long. If copper in the cube is drawn to form a length of wire, what will the resistance of the length of wire be? Assume the density of the copper does not change.

Find an expression for the resistance between the ends of the half-ring shown in Figure 25-51. The resistivity of the material constituting the half-ring is Hint: Model the half-ring as a parallel combination of a large number of thin half-rings. Assume the current is uniformly distributed on a cross section of the half-ring.

Consider a wire of length in the shape of a truncated cone. The radius of the wire varies with distance from the narrow end according to where Derive an expression for the resistance of this wire in terms of its length radius radius and resistivity Hint: Model the wire as a series combination of a large number of thin disks. Assume the current is uniformly distributed on a cross section of the cone.

The space between two metallic concentric spherical shells is filled with a material that has a resistivity of If the inner metal shell has an outer radius of and the outer metal shell has an inner radius of what is the resistance between the conductors? Hint: Model the material as a series combination of a large number of thin spherical shells.

The space between two metallic coaxial cylinders that have the same length L is completely filled with a nonmetallic material having a resistivity The inner metal shell has an outer radius and the outer metal shell has an inner radius (a) What is the resistance between the two cylinders? Hint: Model the material as a series combination of a large number of thin cylindrical shells. (b) Find the current between the two metallic cylinders if and a potential difference of is maintained between the two cylinders.

A tungsten rod is long and has a square cross section that has (a) What is its resistance at (b) What is its resistance at

At what temperature will the resistance of a copper wire be 10 percent greater than its resistance at

ENGINEERING APPLICATION You have a toaster that uses a Nichrome wire as a heating element. You need to determine the temperature of the Nichrome wire under operating conditions. First, you measure the resistance of the heating element at and find it to be Then you measure the current immediately after you plug the toaster into a wall outlet before the temperature of the Nichrome wire increases significantly. You find this startup current to be When the heating element reaches its operating temperature, you measure the current to be Use your data to determine the maximum operating temperature of the heating element.

ENGINEERING APPLICATION Your electric space heater has a Nichrome heating element that has a resistance of When is applied, the electric current heats the Nichrome wire to (a) What is the initial current in the heating element at (b) What is the resistance of the heating element at (c) What is the operating power of this heater?

A Nichrome resistor is wired into an electronic circuit using copper leads (wires) that have diameters equal to The copper leads have a total length of (a) What additional resistance is due to the copper leads? (b) What percentage error in the total resistance is produced by neglecting the resistance of the copper leads? (c) What change in temperature would produce a change in resistance of the Nichrome wire equal to the resistance of the copper leads? Assume that the Nichrome section is the only one whose temperature is changed.

A wire that has a cross-sectional area a length a resistivity and a temperature coefficient is connected end to end to a second wire that has the same cross-sectional area, a length a resistivity and a temperature coefficient so that the wires carry the same current. (a) Show that if then the total resistance is independent of temperature for small temperature changes. (b) If one wire is made of carbon and the other wire is made of copper, find the ratio of their lengths for which the total resistance is approximately independent of temperature.

The resistivity of tungsten increases approximately linearly with temperature from at to at A lightbulb is powered by a dc power supply. Under those operating conditions the temperature of the tungsten filament is the length of the filament is equal to and the power delivered to the filament is Estimate (a) the resistance of the filament and (b) the diameter of the filament.

A 5.00-V lightbulb used in an electronics class has a carbon filament that has a length of 3.00 cm and a diameter of 40.0 \(\mu\)m. At temperatures between 500 K and 700 K, the resistivity of the carbon used in making small lightbulb filaments is about \(3.00 \times 10^{-5}~ \Omega \cdot m\). (a) Assuming that the bulb is a perfect blackbody radiator, calculate the temperature of the filament under operating conditions. (b) One concern about carbon filament bulbs, unlike tungsten filament bulbs, is that the resistivity of carbon decreases with increasing temperature. Explain why this decrease in resistivity is a concern.

A heater is designed to operate at (a) What is the heaters resistance and what is the current in the wires that supply power to the heater? (b) What is the power delivered to the heater if it operates at Assume that its resistance remains the same.

Abattery has an emf of How much work does it do in if it delivers a current of

An automotive battery has an emf of When supplying power to the starter motor, the current in the battery is and the terminal voltage of the battery is What is the internal resistance of the battery?

(a) How much power is delivered by the battery in Problem 62 due to the chemical reactions within the battery when the current in the battery is (b) How much of this power is delivered to the starter when the current in the battery is (c) By how much does the chemical energy of the battery decrease if the current in the starter is for (d) How much energy is dissipated in the battery during those

A battery that has an emf of and an internal resistance of is connected to a variable resistor with resistance Find the current and power delivered by the battery when is (a) 0, (b) (c) and (d) infinite.

ENGINEERING APPLICATION, CONTEXT-RICH A 120-V automobile battery that has a negligible internal resistance can deliver a total charge of \(160 \mathrm{~A} \cdot \mathrm{h}\) (a) What is the amount of energy stored in the battery? (b) After studying all night for a calculus test, you try to drive to class to take the test. However, you find that the car’s battery is “dead” because you had left the headlights on! Assuming the battery was able to produce current at a constant rate until it died, how long were your lights on? Assume the pair of headlights together operates at a power of 150 W.

ENGINEERING APPLICATION The measured current in a circuit in your uncles house is . In this circuit, the only appliance that is on is a space heater that is being used to heat the bathroom. A pair of copper wires carries the current from the supply panel in your basement to the wall outlet in the bathroom, a distance of You measure the voltage at the supply panel to be exactly What is the voltage at the wall outlet in the bathroom that the space heater is connected to?

ENGINEERING APPLICATION A lightweight electric car is powered by a series combination of ten batteries, each having negligible internal resistance. Each battery can deliver a charge of before needing to be recharged. At a speed of the average force due to air drag and rolling friction is (a) What must be the minimum power delivered by the electric motor if the car is to travel at a speed of 80.0 km>h? (b) What is the combination of ten batteries before recharging is required? (c) What is the total electrical energy delivered by the ten batteries before recharging? (d) How far can the car travel (at before the batteries must be recharged? (e) What is the cost per kilometer if the cost of recharging the batteries is per kilowatt-hour?

A heater is designed to operate with an applied voltage of (a) What is the heaters resistance, and what current does the heater carry? (b) Show that if the potential difference across the heater changes by a small amount the power changes by a small amount where Hint: Approximate the changes by modeling them as differentials, and assume the resistance is constant. (c) Using the Part (b) result, find the approximate power delivered to the heater if the potential difference is decreased to 115 V. Compare your result to the exact answer.

If the potential drop from point to point (Figure 25-52) is find the current in each resistor.

If the potential drop between point a and b point (Figure 25-53) is 12.0 V, find the current in each resistor.

(a) Show that the equivalent resistance between point a and point b in Figure 25-54 is R. (b) How would adding a fifth resistor that has resistance R between point c and point d affect the equivalent resistance between point a and point b ?

The battery in Figure 25-55 has negligible internal resistance. Find (a) the current in each resistor and (b) the power delivered by the battery.

A power supply has an internal resistance of What is the smallest resistor that can be put in series with the power supply so that the voltage drop across the resistor is larger than

ENGINEERING APPLICATION You have been handed an unknown battery. Using your multimeter, you determine that when a resistor is connected across the batterys terminals, the current in the battery is When this resistor is replaced by an resistor, the current drops to From those data, find (a) the emf and (b) internal resistance of the battery.

(a) Find the equivalent resistance between point and point in Figure 25-56. (b) If the potential drop between point and point b is 12.0 V, find the current in each resistor.

(a) Find the equivalent resistance between point and point in Figure 25-57. (b) If the potential drop between point and point b is 12.0 V, find the current in each resistor.

A length of wire has a resistance of The wire is cut into pieces that have the same length, and then the wires are connected in parallel. The resistance of the parallel arrangement is 1.88 . Find the number of pieces into which the wire was cut. SSM

A parallel combination of an 8.00-\(\Omega\) resistor and a resistor of unknown resistance is connected in series with a 16.0-\(\Omega\) resistor and an ideal battery. The circuit is disassembled and the three resistors are then connected in series with each other and the same battery. In both arrangements, the current through the 8.00-\(\Omega\) resistor is the same. What is the resistance of the unknown resistor?

For the network shown in Figure 25-58, let denote the equivalent resistance between terminals and Find (a) so that (b) so that and (c) R so that Rab _ R1 .

Check your results for Problem 79 using the following specific values: (a) (b) ; and (c)

In Figure 25-59, the batterys emf is and The rate of Joule heating in (a) What is the current in the circuit? (b) What is the potential difference across (c) What is the resistance

The batteries in the circuit in Figure 25-60 have negligible internal resistance. (a) Find the current using Kirchhoffs loop rule. (b) Find the power delivered to or supplied by each battery. (c) Find the rate of Joule heating in each resistor.

ENGINEERING APPLICATION An old car battery that has an emf of and an internal resistance of is connected to a 2.00- resistor. In an attempt to recharge the battery, you connect a second battery that has an emf of and an internal resistance of in parallel with the first battery and the resistor with a pair of jumper cables. (a) Draw a diagram of the circuit. (b) Find the current in each branch of the circuit. (c) Find the power supplied by the second battery and discuss where that power is delivered. Assume that the emfs and internal resistances of both batteries remain constant.

In the circuit in Figure 25-61, the reading of the ammeter is the same when both switches are open and when both switches are closed. What is the unknown resistance R?

In the circuit shown in Figure 25-62, the batteries have negligible internal resistance. Find (a) the current in each branch of the circuit, (b) the potential difference between point and point and (c) the power supplied by each battery.

In the circuit shown in Figure 25-63, the batteries have negligible internal resistance. Find (a) the current in each branch of the circuit, (b) the potential difference between point and point b, and (c) the power supplied by each battery.

Two identical batteries, each having an emf and an internal resistance can be connected across a resistance with the batteries connected either in series or in parallel. In each situation, determine explicitly whether the power supplied to is greater when is less than or when is greater than

ENGINEERING APPLICATION The circuit fragment shown in Figure 25-64 is called a voltage divider. (a) If is not attached, show that (b) If what is the smallest value of that can be used so that drops by less than 10 percent from its unloaded value? is measured with respect to ground.)

For the circuit shown in Figure 25-65, find the potential difference between point a and point b. SSM

For the circuit shown in Figure 25-66, find (a) the current in each resistor, (b) the power supplied by each source of emf, and (c) the power delivered to each resistor

The voltmeter shown in Figure 25-67 can be modeled as an ideal voltmeter (a voltmeter that has an infinite internal resistance) in parallel with a resistor. Calculate the reading on the voltmeter when (a) (b) (c) (d) and (e) (f ) What is the largest value of possible if the measured voltage is to be within 10 percent of the true voltage (i.e., the voltage drop across without the voltmeter in place)? SSM

You are given a DArsonval galvanometer that will deflect full scale if a current of runs through the galvanometer. At that current, there is a voltage drop of across the meter. What is the internal resistance of the galvanometer?

You are given a DArsonval galvanometer that will deflect full scale if a current of runs through the galvanometer. At that current, there is a voltage drop of across the meter. You wish to use the galvanometer to construct an ammeter that can measure currents up to Show that this can be done by placing a resistor in parallel with the meter, and find the value of its resistance.

You are given a DArsonval galvanometer that will deflect full scale if a current of runs through the galvanometer. At that current, there is a voltage drop of across the meter. You wish to use the galvanometer to construct a voltmeter that can measure potential differences up to Show that this can be done by placing a large resistance in series with the meter movement, and find the resistance needed.

For the circuit shown in Figure 25-68, and After having been at contact for a long time, the switch throw is rotated to contact . (a) What is the charge on the upper plate of the capacitor just as the switch throw is moved to contact (b) What is the current just after the switch throw is rotated to contact (c) What is the time constant of this circuit? (d) How much charge is on the upper plate of the capacitor after the switch throw is rotated to contact

At t = 0 the switch throw in Figure 25-68 is rotated to contact b after having been at contact a for a long time. (a) How much energy is stored in the capacitor at t = 0? (b) For t > 0, find the energy stored in the capacitor as a function of time. (c) Sketch a plot of the energy stored in the capacitor versus time t.

In the circuit in Figure 25-69, the emf equals and the capacitance equals Switch is opened after having been closed for a long time, and later the voltage drop across the resistor is Find the resistance of the resistor.

For the circuit shown in Figure 25-68, and The switch throw is rotated to contact after having been at contact for a long time, and later the potential difference across the capacitor is equal to What is the value of R?

In the circuit in Figure 25-69, the emf equals and has negligible internal resistance. The capacitance equals and the resistance equals Switch has been closed for a long time. Switch S is opened. After a time interval equal to one time constant of the circuit has elapsed, find (a) the charge on the capacitor plate on the right, (b) the rate at which the charge is increasing, (c) the current, (d) the power supplied by the battery, (e) the power delivered to the resistor, and (f ) the rate at which the energy stored in the capacitor is increasing.

Aconstant charge of is on the positively charged plate of the capacitor shown in Figure 25-70. Find (a) the battery current and (b) the resistances and R3 R .

Show that Equation 25-39 can be rearranged and written as Integrate this equation to derive the solution given by Equation 25-40.

Switch shown in Figure 25-71, is closed after having been open for a long time. (a) What is the initial value of the battery current just after switch is closed? (b) What is the battery current a long time after switch is closed? (c) What are the charges on the plates of the capacitors a long time after switch is closed? (d) Switch is reopened. What are the charges on the plates of the capacitors a long time after switch S is reopened?

For the circuit shown in Figure 25-72, switch S has been open for a long time. At time the switch is then closed. (a) What is the battery current just after switch is closed? (b) What is the battery current a long time after switch is closed? (c) What is the current in the 600- resistor as a function of time?

For the circuit shown in Figure 25-73, switch S has been open for a long time. At time the switch is then closed. (a) What is the battery current just after switch is closed? (b) What is the battery current a long time after switch is closed? (c) The switch has been closed for a long time. At time the switch is then opened. Find the current through the resistor as a function of time.

In the circuit shown in Figure 25-74, the capacitor has a capacitance of and the resistor has a resistance of Before the switch is closed, the potential drop across the capacitor is as shown. Switch is closed at (a) What is the current immediately after switch is closed? (b) At what time is the voltage across the capacitor

Repeat Problem 105 if the initial polarity of the capacitor is opposite to that shown in Figure 25-74.

In Figure 25-75, and the battery emf is Denote the currents through the resistors as and respectively. (a) Decide which of the following inequalities holds for the circuit. Explain your answer conceptually. (1) (2) (3) (4) None of the above (b) To verify that

A lightbulb is connected in series with a lightbulb and a potential difference of is placed across the combination. Assume the bulbs have constant resistance. (a) Which bulb should be brighter under those conditions? Explain your answer conceptually. Hint: What does the phrase 25.0-W lightbulb mean? That is, under what conditions is 25 W of power delivered to the bulb? (b) Determine the power delivered to each bulb under those conditions. Do your results support your answer to Part (a)?

The circuit shown in Figure 25-76 is a Wheatstone bridge, and the variable resistor is being used as a slide-wire potentiometer. The resistance is known. This bridge is used to determine an unknown resistance The resistances and comprise a wire long. Point is a sliding contact that is moved along the wire to vary the resistances. Resistance is proportional to the distance from the left end of the wire (labeled to point and is proportional to the distance from point to the right end of the wire (labeled The sum of and remains constant. When points and are at the same potential, there is no current in the galvanometer and the bridge is said to be balanced. (Because the galvanometer is used to detect the absence of a current, it is called a null detector.) If the fixed resistance find the unknown resistance if (a) the bridge balances at the mark, (b) the bridge balances at the mark, and (c) the bridge balances at the 95.0-cm mark.

For the Wheatstone bridge in Problem 109, suppose the bridge balances at the mark. (a) What is the unknown resistance? (b) What is the percentage error in the measured value of if there is an error of in the location of the balance point? (c) To what value should be changed to so that the balance point for the unknown resistor will be nearer the mark? (d) If the balance point is at the mark, what is the percentage error in the measured value of if there is an error of in the location of the balance point?

You are running an experiment that uses an accelerator that produces a proton beam. Each proton in the beam has of kinetic energy. The protons impinge on, and come to rest inside, a copper target within a vacuum chamber. You are concerned that the target will get too hot and melt the solder on some connecting wires that are crucial to the experiment. (a) Determine the number of protons that strike the target per second. (b) Find the amount of energy delivered to the target each second. (c) Determine how much time elapses before the target temperature increases to (Neglect any heat released by the target.)

The belt of a Van de Graaff generator carries a surface charge density of The belt is wide and moves at (a) What current does the belt carry? (b) If the potential of the dome of the generator is above ground, what is the minimum power of the motor needed to drive the belt?

ENGINEERING APPLICATION Large conventional electromagnets use water cooling to prevent excessive heating of the magnet coils. A large laboratory electromagnet has a current equal to when a voltage of is applied to the terminals of the energizing coils. To cool the coils, water at an initial temperature of is circulated around the coils. How many liters of water must circulate by the coils each second if the temperature of the coils is not to exceed

(a) Give support to the assertion that a leaky capacitor (one for which the resistance of the dielectric is finite) can be modeled as a capacitor that has an infinite resistance in parallel with a resistor. (b) Show that the time constant for discharging the capacitor is given by (For simplicity, assume the capacitor is a parallel- plate variety filled completely with a leaky dielectric.) (c) Mica has a dielectric constant equal to about and a resistivity equal to about Calculate the time it takes for the charge of a mica-filled capacitor to decrease to 10 percent of its initial value.

ENGINEERING APPLICATION Figure 25-77 shows the basis of the sweep circuit used in an oscilloscope. Switch is an electronic switch that closes whenever the potential across the switch increases to a value and opens when the potential across the switch decreases to The emf which is much greater than Vc charges the capacitor through a resistor The resistor represents the small but finite resistance of the electronic switch. In a typical circuit, and (a) What is the time constant for charging of the capacitor (b) Show that as the potential across switch increases from to the potential across the capacitor increases almost linearly with time. Hint: Use the approximation for (This approximation of can be derived using the differential approximation.) (c) What should the value of be changed to so that the capacitor charges from to in (d) How much time elapses during the discharge of the capacitor when switch closes? (e) At what average rate is energy delivered to the resistor during charging and to the switch resistance R during discharge?

In the circuit shown in Figure 25-78, and The capacitor is initially without charge on either plate. At switch is closed, and at switch is opened. (a) Sketch a graph of the voltage across and the current in between and (b) Find the voltage across the capacitor at t _ 2.00 s and at t _ 8.00 s.

Two batteries that have emfs and and internal resistances and are connected in parallel. Prove that if a resistor of resistance is connected in parallel with combination, the optimal load resistance (the value of at which maximum power is delivered) is given by

Capacitors and are connected to a resistor of resistance and an ideal battery that has as shown in Figure 25-79. The throw of switch is initially at contact and both capacitors are without charge. The throw is then rotated to contact and left there for a long time. Finally, at time the throw is returned to contact (a) Quantitatively compare the total energy stored in the two capacitors at t _ 0 and a long time later. (b) Find the current to the resistor as a function of . (d) Find the total energy dissipated in the resistor and compare it with the loss of stored energy found in Part (a).

(a) Calculate the equivalent resistance (in terms of the resistance of each individual resistor) between points and for the infinite ladder of resistors shown in Figure 25-80 assuming the resistors are identical. That is, assuming (b) Repeat Part (a) but do not assume that and express your answer in terms of and (c) Check your results by showing that your result from Part (b) agrees with your result from Part (a) if you substitute for both and R2 R .

A graph of current as a function of voltage for an Esaki diode is shown in Figure 25-81. (a) Make a graph of the differential resistance of the diode as a function of voltage. The differential resistance of a circuit element is defined as where is the voltage drop across the element and is the current in the element. (b) At what value of the voltage drop does the differential resistance become negative? (c) What is the maximum differential resistance for this diode in the range shown and at what voltage does it occur? (d) Are there any places in the voltage range shown where the diode exhibits a differential resistance equal to zero? If so, under what value(s) of the voltage does this (do these) occur?