Current and Resistance

A certain brand of hot-dog cooker works by applying a potential difference of 120 V across opposite ends of a hot dog and allowing it to cook by means of the thermal energy produced. The current is 10.0 A, and the energy required to cook one hot dog is 60.0 kJ. If the rate at which energy is supplied is unchanged, how long will it take to cook three hot dogs simultaneously?

Thermal energy is produced in a resistor at a rate of 100W when the current is 3.00 A .What is the resistance?

A 120 V potential difference is applied to a space heater whose resistance is $\mathbf{14}\mathbf{ }\mathbf{\Omega }$ when hot.   (a) At what rate is electrical energy transferred to thermal energy?  (b) What is the cost for 5.0h at US $0.05/kW.h?

An unknown resistor is connected between the terminals of a 3.00 V battery. Energy is dissipated in the resistor at the rate of 0.540 W.The same resistor is then connected between the terminals of a 1.50 V battery. At what rate is energy now dissipated?

A student kept his 9.0 V,  7.0 W  radio turned on at full volume from 9:00 P.M until 2:00 A.M. . How much charge went through it?

A 1250 W radiant heater is constructed to operate at 115 V. (a) What is the current in the heater when the unit is operating?  (b) What is the resistance of the heating coil?  (c) How much thermal energy is produced in 1.0 h?

A copper wire of cross-sectional area $\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}{\mathbf{m}}^{\mathbf{2}}$and length 4.00m has a current of 2.00A uniformly distributed across that area. (a) What is the magnitude of the electric field along the wire?  (b) How much electrical energy is transferred to thermal energy in 30min?

Exploding shoes. The rain-soaked shoes of a person may explode if ground current from nearby lightning vaporizes the water. The sudden conversion of water to water vapor causes a dramatic expansion that can rip apart shoes. Water has density $\mathbf{1000}\mathbf{kg}\mathbf{/}{\mathbf{m}}^{\mathbf{3}}$ and requires $\mathbf{2256}\mathbf{ }\mathbf{KJ}\mathbf{/}\mathbf{Kg}$ to be vaporized. If horizontal current lasts 2.00 ms and encounters water with resistivity $\mathbf{150}\mathbf{\Omega }\mathbf{.}\mathbf{m}$, length 12.0 cm, and vertical cross-sectional area , what average current is required to vaporize the water?

A 100 W lightbulb is plugged into a standard 120V outlet. (a) How much does it cost per 31-day month to leave the light turned on continuously? Assume electrical energy costs US $ 0.06/kWh. (b) What is the resistance of the bulb? (c) What is the current in the bulb?

The current-density magnitude in a certain circular wire is $\mathbf{J}\mathbf{=}\mathbf{(}\mathbf{2}\mathbf{.}\mathbf{75}\mathbf{\times }{\mathbf{10}}^{\mathbf{10}}\mathbf{A}\mathbf{/}{\mathbf{m}}^{\mathbf{4}}\mathbf{)}{\mathbf{r}}^{\mathbf{2}}$, where r is the radial distance out to the wire’s radius of 3.00 mm. The potential applied to the wire (end to end) is 6.00 V. How much energy is converted to thermal energy in 1.00 h?

Figure-a shows a rod of resistive material. The resistance per unit length of the rod increases in the positive direction of the x axis. At any position x along the rod, the resistance dR of a narrow (differential) section of width dx is given by dR = 5.00 xdx , where is in ohms and x is in meters. Figure-b shows such a narrow section. You are to slice off a length of the rod between x = 0 and some position x = L and then connect that length to a battery with potential difference V = 5.0V (Figure-c). You want the current in the length to transfer energy to thermal energy at the rate of 200 W . At what position x = L should you cut the rod?

A Nichrome heater dissipates 500 W when the applied potential difference is 110 V and the wire temperature is $\mathbf{800}\mathbf{°}\mathit{C}$ . What would be the dissipation rate if the wire temperature were held at $\mathbf{200}\mathbf{°}\mathit{C}$ by immersing the wire in a bath of cooling oil? The applied potential difference remains the same, and  for Nichrome at $\mathbf{800}\mathbf{°}\mathit{C}$ is  $\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{4}}{\mathit{k}}^{\mathbf{-}\mathbf{1}}$ .

A potential difference of 1.20 V will be applied to a 33.0 m length of 18-gauge copper wire (diameter = 0.0400 in). Calculate (a) the current, (b) the magnitude of the current density, (c) the magnitude of the electric field within the wire, and (d) the rate at which thermal energy will appear in the wire.

A 18 .0 W device has 9.00 V across it. How much charge goes through the device in 4.00 h ?

An aluminum rod with a square cross section is 1.3 m long and 5.2 mm on edge. (a) What is the resistance between its ends? (b)What must be the diameter of a cylindrical copper rod of length 1.3 m if its resistance is to be the same as that of the aluminum rod?

A cylindrical metal rod is 1.60 m long and 5.50 m in diameter. The resistance between its two ends (at $\mathbf{20}\mathbf{°}$) is $\mathbf{1}\mathbf{.}\mathbf{09}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{3}}\mathit{\Omega }$ . (a) What is the material? (b) A round disk, 2.00 cm in diameter and 1.00 mm thick, is formed of the same material. What is the resistance between the round faces, assuming that each face is an equi-potential surface?

The chocolate crumb mystery. This story begins with Problem 60 in Chapter 23 and continues through Chapters 24 and 25.The chocolate crumb powder moved to the silo through a pipe of radius R with uniform speed v and uniform charge density r. (a) Find an expression for the current i (the rate at which charge on the powder moved) through a perpendicular cross section of the pipe. (b) Evaluate i for the conditions at the factory: pipe radius R = 5.0 cm , speed v = 2.0 m/s , and charge density p = $\mathbf{1}\mathbf{.}\mathbf{1}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{3}}\mathbf{C}\mathbf{/}{\mathbf{m}}^{\mathbf{3}}$. If the powder were to flow through a change V in electric potential, its energy could be transferred to a spark at the rate P = iV . (c) Could there be such a transfer within the pipe due to the radial potential difference discussed in Problem 70 of Chapter 24? As the powder flowed from the pipe into the silo, the electric potential of the powder changed. The magnitude of that change was at least equal to the radial potential difference within the pipe (as evaluated in Problem 70 of Chapter 24). (d) Assuming that value for the potential difference and using the current found in (b) above, find the rate at which energy could have been transferred from the powder to a spark as the powder exited the pipe. (e) If a spark did occur at the exit and lasted for 0.20 s (a reasonable expectation), how much energy would have been transferred to the spark? Recall from Problem 60 in Chapter 23 that a minimum energy transfers of  is needed to cause an explosion. (f) Where did the powder explosion most likely occur: in the powder cloud at the unloading bin (Problem 60 of Chapter 25), within the pipe, or at the exit of the pipe into the silo?

A steady beam of alpha particles (q2e) traveling with constant kinetic energy  20 MeV carries a current of 0.25 A . (a) If the beam is directed perpendicular to a flat surface, how many alpha particles strike the surface in 3.0 s ? (b) At any instant, how many alpha particles are there in a given 20 cm length of the beam? (c) Through what potential difference is it necessary to accelerate each alpha particle from rest to bring it to energy of 20 MeV ?  

A resistor with a potential difference of 200 V across it transfers electrical energy to thermal energy at the rate of 300 V. What is the resistance of the resistor?

A cylindrical resistor of radius 5.00mm  and length 2.0 cm is made of material that has a resistivity of $\mathbf{3}\mathbf{.}\mathbf{5}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathbf{\Omega }\mathbf{.}\mathbf{m}$ .What are (a) the magnitude of the current density and (b) the potential difference when the energy dissipation rate in the resistor is 1.0 W?

A potential difference V is applied to a wire of cross-sectional area A , length L, and resistivity p. You want to change the applied potential difference and stretch the wire so that the energy dissipation rate is multiplied by 30.0 and the current is multiplied by 4.00 . Assuming the wire’s density does not change, what are (a) the ratio of the new length to L and (b) the ratio of the new cross-sectional area to  A?

The headlights of a moving car require about 10 A from the 12 V alternator, which is driven by the engine. Assume the alternator is efficient (its output electrical power is 80% of its input mechanical power), and calculate the horsepower the engine must supply to run the lights.

A 500 W heating unit is designed to operate with an applied potential difference of 115 V . (a) By what percentage will its heat output drop if the applied potential difference drops to 110 V ? Assume no change in resistance. (b) If you took the variation of resistance with temperature into account, would the actual drop in heat output be larger or smaller than that calculated in (a)?

The copper windings of a motor have a resistance of $\mathbf{50}\mathit{\Omega }$ at $\mathbf{20}\mathbf{°}\mathbf{C}$ when the motor is idle. After the motor has run for several hours, the resistance rises to $\mathbf{58}\mathit{\Omega }$.What is the temperature of the windings now? Ignore changes in the dimensions of the windings. (Use Table 26-1.)

How much electrical energy is transferred to thermal energy in 2.00h by an electrical resistance of $\mathbf{400}\mathbf{\Omega }$ when the potential applied across it is 90.0V?

A caterpillar of length 4.0cm crawls in the direction of electron drift along a 5.2-mm-diameter bare copper wire that carries a uniform current of 12A. (a) What is the potential difference between the two ends of the caterpillar? (b) Is its tail positive or negative relative to its head? (c) How much time does the caterpillar take to crawl 1.0cm if it crawls at the drift speed of the electrons in the wire? (The number of charge carriers per unit volume is$\mathbf{8}\mathbf{.}\mathbf{49}\mathbf{\times }{\mathbf{10}}^{\mathbf{28}}\mathbf{/}{\mathit{m}}^{\mathbf{3}}$.)

(a)At what temperature would the resistance of a copper conductor be double its resistance at${\mathbf{\text{20.0}}}^{\mathbf{0}}\mathbf{\text{C}}$ ? (Use ${\mathbf{\text{20.0}}}^{\mathbf{0}}\mathbf{\text{C}}$ as the reference point in Eq. 26-17; compare your answer withFig. 26-10.) 

(b) does this same “doubling temperature” hold for all copper conductors, regardless of shape or size?

A steel trolley-car rail has a cross-sectional area of $\mathbf{56}\mathbf{.}\mathbf{0}{\mathbf{cm}}^{\mathbf{2}}$. What is the resistance of 10.0 km of rail? The resistivity of the steel is $\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{7}}\mathbf{\Omega m}$.

A coil of current-carrying Nichrome wire is immersed in a liquid. (Nichrome is a nickel–chromium–iron alloy commonly used in heating elements.) When the potential difference across the coil is 12 V and the current through the coil is 5.2 A, the liquid evaporates at the steady rate of 21 mg/s. Calculate the heat of vaporization of the liquid (see Module 18-4).

The current density in a wire is uniform and has magnitude $\mathbf{2}\mathbf{.}\mathbf{0}\mathit{x}{\mathbf{10}}^{\mathbf{6}}\mathit{A}\mathbf{/}{\mathit{m}}^{\mathbf{2}}$, the wire’s length is 5.0m, and the density of conduction electrons is  $\mathbf{8}\mathbf{.}\mathbf{49}\mathit{x}{\mathbf{10}}^{28}\mathbf{/}{\mathit{m}}^3$. How long does an electron take (on the average) to travel the length of the wire?

A certain x-ray tube operates at a current of 7.00mA and a potential difference of 80.0kV. What is its power in watts?

 A current is established in a gas discharge tube when a sufficiently high potential difference is applied across the two electrodes in the tube. The gas ionizes; electrons move toward the positive terminal and singly charged positive ions toward the negative terminal.(a)What is the current in a hydrogen discharge tube in which $\mathbf{3}\mathbf{.}\mathbf{1}\mathbf{\times }{\mathbf{10}}^{\mathbf{18}}$ electrons and $\mathbf{1}\mathbf{.}\mathbf{1}\mathbf{\times }{\mathbf{10}}^{\mathbf{18}}$ protons move past a cross-sectional area of the tube each second? (b) Is the direction of the current density $\overrightarrow{\mathbf{J}}$ toward or away from the negative terminal?

In Fig. 26-37, a resistance coil, wired to an external battery, is placed inside a thermally insulated cylinder fitted with a frictionless piston and containing an ideal gas. A current i = 240mA flows through the coil, which has a resistance $\mathit{R}\mathbf{=}\mathbf{550}\mathbf{ }\mathit{\Omega }$. At what speed v must the piston, of mass m=12 kg, move upward in

Order that the temperature of the gas remains unchanged?

An insulating belt moves at speed 30 m/s and has a width of 50 cm. It carries charge into an experimental device at a rate corresponding to 100 mA. What is the surface charge density on the belt?

In Fig. 26-37, a resistance coil, wired to an external battery, is placed inside a thermally insulated cylinder fitted with a frictionless piston and containing an ideal gas. A current i = 240mA flows through the coil, which has a resistance. At what speed v must the piston, of mass m=12kg , move upward in

Order that the temperature of the gas remains unchanged?

When a metal rod is heated not only its resistance, but also its length and cross-sectional area is changed. The relation $\mathit{R}\mathbf{=}\mathit{\rho }\mathit{L}\mathbf{/}\mathit{A}$ suggests that all three factors should be taken into account in measuring r at various temperatures. If the temperature changes by $\mathbf{1}\mathbf{.}{\mathbf{0}}^{\mathbf{0}}\mathit{C}$, what percentage changes in (a) L, (b) A, and (c) R occur for a copper conductor? (d) What conclusion do you draw? The coefficient of linear expansion is $\mathbf{1}\mathbf{.}\mathbf{70}\mathit{x}{\mathbf{10}}^{\mathbf{-}\mathbf{5}}{\mathbf{K}}^{\mathbf{-}\mathbf{1}}$.

A beam of 16 MeV deuterons from a cyclotron strikes a copper block. The beam is equivalent to current of 15 mA . (a) At what rate do deuterons strike the block? (b) At what rate is thermal energy produced in the block?

A linear accelerator produces a pulsed beam of electrons. The pulse current is 0.50 A, and the pulse duration is 0.10 ms. (a) How many electrons are accelerated per pulse? (b) What is the average current for a machine operating at 500 pulses/s ? If the electrons are accelerated to energy of 50 MeV, what are the (c) average power and (d) peak power of the accelerator?

A  immersion heater is placed in a pot containing 2.00 L of water at $\mathbf{20}\mathbf{°}\mathit{C}$. (a) How long will the water take to rise to the boiling temperature, assuming that 80% of the available energy is absorbed by the water? (b) How much longer is required to evaporate half of the water?

A 400W immersion heater is placed in a pot containing 2.00 L of water at $\mathbf{20}\mathbf{°}\mathit{C}$. (a) How long will the water take to rise to the boiling temperature, assuming that 80% of the available energy is absorbed by the water? (b) How much longer is required to evaporate half of the water?

A 30 $\mathbf{\mu F}$ capacitor is connected across a programmed power supply. During the interval from t = 0 to t = 0.300s the output voltage of the supply is given by $\mathbf{V}\left(t\right)\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{00}\mathbf{+}\mathbf{4}\mathbf{.}\mathbf{00}\mathbf{t}\mathbf{-}\mathbf{2}\mathbf{.}\mathbf{00}{\mathbf{t}}^{\mathbf{2}}\mathbf{volts}$ volts. At $\mathbf{t}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{500}\mathbf{s}$  find (a) the charge on the capacitor, (b) the current into the capacitor, and (c) the power output from the power supply.