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A long, inexpensive extension cord is connected from inside the house to a refrigerator outside. The refrigerator doesn’t run as it should. What might be the problem?
In [link] , does the graph indicate the time constant is shorter for discharging than for charging? Would you expect ionized gas to have low resistance? How would you adjust $R$ to get a longer time between flashes? Would adjusting $R$ affect the discharge time?
An electronic apparatus may have large capacitors at high voltage in the power supply section, presenting a shock hazard even when the apparatus is switched off. A “bleeder resistor” is therefore placed across such a capacitor, as shown schematically in [link] , to bleed the charge from it after the apparatus is off. Why must the bleeder resistance be much greater than the effective resistance of the rest of the circuit? How does this affect the time constant for discharging the capacitor?
The timing device in an automobile’s intermittent wiper system is based on an $\text{RC}$ time constant and utilizes a $0\text{.}\text{500-\mu F}$ capacitor and a variable resistor. Over what range must $R$ be made to vary to achieve time constants from 2.00 to 15.0 s?
$\text{range 4}\text{.}\text{00 to 30}\text{.}\text{0 M}\Omega $
A heart pacemaker fires 72 times a minute, each time a 25.0-nF capacitor is charged (by a battery in series with a resistor) to 0.632 of its full voltage. What is the value of the resistance?
The duration of a photographic flash is related to an $\text{RC}$ time constant, which is $0\text{.}\text{100 \mu s}$ for a certain camera. (a) If the resistance of the flash lamp is $0\text{.}\text{0400}\phantom{\rule{0.15em}{0ex}}\Omega $ during discharge, what is the size of the capacitor supplying its energy? (b) What is the time constant for charging the capacitor, if the charging resistance is $\text{800}\phantom{\rule{0.25em}{0ex}}\text{k\Omega}$ ?
(a) $2\text{.}\text{50 \mu F}$
(b) 2.00 s
A 2.00- and a $7\text{.}\text{50-\mu F}$ capacitor can be connected in series or parallel, as can a 25.0- and a $\text{100-k\Omega}$ resistor. Calculate the four $\text{RC}$ time constants possible from connecting the resulting capacitance and resistance in series.
After two time constants, what percentage of the final voltage, emf, is on an initially uncharged capacitor $C$ , charged through a resistance $R$ ?
86.5%
A $\text{500-\Omega}$ resistor, an uncharged $1\text{.}\text{50-\mu F}$ capacitor, and a 6.16-V emf are connected in series. (a) What is the initial current? (b) What is the $\text{RC}$ time constant? (c) What is the current after one time constant? (d) What is the voltage on the capacitor after one time constant?
A heart defibrillator being used on a patient has an $\text{RC}$ time constant of 10.0 ms due to the resistance of the patient and the capacitance of the defibrillator. (a) If the defibrillator has an $8\text{.}\text{00-\mu F}$ capacitance, what is the resistance of the path through the patient? (You may neglect the capacitance of the patient and the resistance of the defibrillator.) (b) If the initial voltage is 12.0 kV, how long does it take to decline to $6.00\times {10}^{2}\phantom{\rule{0.25em}{0ex}}\text{V}$ ?
(a) $1\text{.}\text{25 k}\Omega $
(b) 30.0 ms
An ECG monitor must have an $\text{RC}$ time constant less than $1.00\times {10}^{2}\phantom{\rule{0.25em}{0ex}}\text{\mu s}$ to be able to measure variations in voltage over small time intervals. (a) If the resistance of the circuit (due mostly to that of the patient’s chest) is $1\text{.}\mathrm{00\; k\Omega}$ , what is the maximum capacitance of the circuit? (b) Would it be difficult in practice to limit the capacitance to less than the value found in (a)?
[link] shows how a bleeder resistor is used to discharge a capacitor after an electronic device is shut off, allowing a person to work on the electronics with less risk of shock. (a) What is the time constant? (b) How long will it take to reduce the voltage on the capacitor to 0.250% (5% of 5%) of its full value once discharge begins? (c) If the capacitor is charged to a voltage ${V}_{0}$ through a $\text{100-\Omega}$ resistance, calculate the time it takes to rise to $0\text{.}\text{865}{V}_{0}$ (This is about two time constants.)
(a) 20.0 s
(b) 120 s
(c) 16.0 ms
Using the exact exponential treatment, find how much time is required to discharge a $\text{250-\mu F}$ capacitor through a $\text{500-\Omega}$ resistor down to 1.00% of its original voltage.
Using the exact exponential treatment, find how much time is required to charge an initially uncharged 100-pF capacitor through a $\text{75}\text{.}0\text{-M}\Omega $ resistor to 90.0% of its final voltage.
$1\text{.}\text{73}\times {\text{10}}^{-2}\phantom{\rule{0.25em}{0ex}}\text{s}$
Integrated Concepts
If you wish to take a picture of a bullet traveling at 500 m/s, then a very brief flash of light produced by an $\text{RC}$ discharge through a flash tube can limit blurring. Assuming 1.00 mm of motion during one $\text{RC}$ constant is acceptable, and given that the flash is driven by a $\text{600-\mu F}$ capacitor, what is the resistance in the flash tube?
$3\text{.}\text{33}\times {\text{10}}^{-3}\phantom{\rule{0.25em}{0ex}}\Omega $
Integrated Concepts
A flashing lamp in a Christmas earring is based on an $\text{RC}$ discharge of a capacitor through its resistance. The effective duration of the flash is 0.250 s, during which it produces an average 0.500 W from an average 3.00 V. (a) What energy does it dissipate? (b) How much charge moves through the lamp? (c) Find the capacitance. (d) What is the resistance of the lamp?
Integrated Concepts
A $\text{160-\mu F}$ capacitor charged to 450 V is discharged through a $31\text{.}\text{2-k}\Omega $ resistor. (a) Find the time constant. (b) Calculate the temperature increase of the resistor, given that its mass is 2.50 g and its specific heat is $1\text{.}\text{67}\frac{\text{kJ}}{\text{kg}\phantom{\rule{0.15em}{0ex}}\cdot \phantom{\rule{0.15em}{0ex}}\text{\xbaC}}$ , noting that most of the thermal energy is retained in the short time of the discharge. (c) Calculate the new resistance, assuming it is pure carbon. (d) Does this change in resistance seem significant?
(a) 4.99 s
(b) $3\text{.}\text{87\xbaC}$
(c) $\text{31}\text{.}\text{1 k}\Omega $
(d) No
Unreasonable Results
(a) Calculate the capacitance needed to get an $\text{RC}$ time constant of $1.00\times {10}^{3}\phantom{\rule{0.25em}{0ex}}\text{s}$ with a $0\text{.}\text{100-\Omega}$ resistor. (b) What is unreasonable about this result? (c) Which assumptions are responsible?
Construct Your Own Problem
Consider a camera’s flash unit. Construct a problem in which you calculate the size of the capacitor that stores energy for the flash lamp. Among the things to be considered are the voltage applied to the capacitor, the energy needed in the flash and the associated charge needed on the capacitor, the resistance of the flash lamp during discharge, and the desired $\text{RC}$ time constant.
Construct Your Own Problem
Consider a rechargeable lithium cell that is to be used to power a camcorder. Construct a problem in which you calculate the internal resistance of the cell during normal operation. Also, calculate the minimum voltage output of a battery charger to be used to recharge your lithium cell. Among the things to be considered are the emf and useful terminal voltage of a lithium cell and the current it should be able to supply to a camcorder.
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