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All Textbook Solutions for University Physics Volume 2
Two moles of a monatomic ideal gas at (5 MPa, 5 L) is expanded isothermally until the volume is doubled (step 1). Then it is cooled isochorically until the pressure is 1 MPa (step 2). The temperature drops in this process. The gas is now compressed isothermally until its volume is back to 5 L, but its pressure is now 2 MPa (step 3). Finally, the gas is heated isochorically to return to the initial state (step 4). (a) Draw the four pi-cresses in the pV plane. (b) Find the total work done by the gas.Consider a transformation from point A to B in a two-step process. First, the pressure is lowered from 3 MPa at point A to a pressure of 1 MPa, while keeping the volume at 2 L by cooling the system. The state reached is labeled C. Then the system is heated at a constant pressure to reach a volume of 6 L in the state B. (a) Find the amount of work on the ACB path. (b) Find the amount of heat exchanged by the system when it goes from A to B on the ACB path. (c) the change in the internal energy when the AB process occurs adiabatically with the AB change though the two-step process on the ACB path.Consider a cylinder with a movable piston containing n moles of an ideal gas. The entire apparatus is immersed in a constant temperature bath of temperature T kelvin. The piston is then pushed slowly so that pressure of the gas changes quasi-statically from p1 to p2 at constant temperature T. Find the work done by the gas in terms of n, R T, p1 , and p2 .An ideal gas expands isothermally along AB and does 700 J of work (see below). (a) How much heat does the gas exchange along AB? (b) The gas then expands adiabatically along BC and does 400 J of work. When the gas returns to A along CA, it exhausts 100 J of heat to its surroundings. How much work is done on the gas along this path?Consider the processes shown below. In the processes AB and BC, 3600 J and 2400 J of heat are added to the system, respectively. (a) Find the work done in each of the processes AB, BC, AD, and DC. (b) Find the internal energy change in processes AB and BC. (c) Find the internal energy difference between states C and A. (d) Find the total heat added in the ADC process. (e) From the information give, can you find the heat added in process AD? Why or why not?Two moles of helium gas axe placed in a cylindrical container with a piston. The gas is at room temperature 25 and under a pressure of 3.0105 Pa. When the pressure from the outside is decreased while keeping the temperature the same as the room temperature, the volume of the gas doubles. (a) Find the work the external agent does on the gas in the process. (b) Find the heat exchanged by the gas and indicate whether the gas takes in or gives up heat. Assume ideal gas behavior.An amount of n moles of a monatomic ideal gas in a conducting container with a movable piston is placed in a large thermal heat bath at temperature T1 and the gas is allowed to come to equilibrium. After the equilibrium is leached, the pressure on the piston is lowered so that the gas expands at constant temperature. The process is continued quasi-statically until the final pressure is 4/3 of the initial pressure p1 . (a) Find the change in the internal energy of the gas. (b) Find the work done by the gas. (c) Find the heat exchanged by the gas, and indicate, whether the gas takes in or gives up heat.The temperature of an ideal monatomic gas rises by 8.0 K. What is the change in the internal energy of 1 mol of the gas at constant volume?For a temperature increase of 10 at constant volume, what is the heat absorbed by (a) 3.0 mol of a dilute monatomic gas; (b) 0.50 mol of a dilute diatomic gas; and (c) 15 mol of a dilute polyatomic gas?If the gases of the preceding problem are initially at 300 K, what are their internal energies after they absorb the heat?Consider 0.40 mol of dilute carbon dioxide at a pressure of 0.50 atm and a volume of 50 L. What is the internal energy of the gas?When 400 J of heat are slowly added to 10 mol of an ideal monatomic gas, its temperature rises by 10 . What is the work done on the gas?One of a dilute diatomic gas occupying a volume of 10.00 L expands against a constant pressure of 2.000 atm when it is slowly heated. If the temperature of the gas rises by 10.00 K and 400.0 J of heat are added in the process, what is its final volume?A monatomic ideal gas undergoes a quasi-static adiabatic expansion in which its volume is doubled. How is the pressure of the gas changed?An ideal gas has a pressure of 0.50 atm and a volume of 10 L. It is compressed adiabatically and quasi-statically until its pressure is 3.0 atm and its volume is 2.8 L. Is the monatomic, diatomic, or polyatomic?Pressure and volume measurements of a dilute gas undergoing a quasi-static adiabatic expansion are shown below. Plot In p vs. V and determine 7 for this gas from your graph.An ideal monatomic gas at 300 K expands adiabatically and reversibly to twice its volume. What is its final temperature?An ideal diatomic gas at 80 K is slowly compressed adiabatically and reversibly to twice its volume. What is its final temperature?An ideal diatomic gas at 80 K is slowly compressed adiabatically to one-third its original volume. What is its final temperature?Compare the charge in internal energy of an ideal gas for a quasi-static adiabatic expansion with that for a quasi-static isothermal expansion. What happens to the temperature of an ideal gas in an adiabatic expansion?The temperature of n moles of an ideal gas changes from T1 to T2 in a quasi-static adiabatic transition. Show that the work done by the gas is given by W=nR1(T1T2).A dilute gas expands quasi-statically to three times its initial volume. Is the final gas pressure greater for an isothermal or an adiabatic expansion? Does your answer depend on whether the gas is monatomic, diatomic, or polyatomic?(a) An ideal gas expands adiabatically from a volume of 2.0103 m3 to 2.5103 m3. If the initial pressure and temperature 5.0105 Pa and 300 K, respectively, what are the final pressure and temperature of the gas? Use =5/3 for the gas. (b) In an isothermal process, an ideal gas expands from a of 2.0103 m3 to 2.5103 m3. If the initial pressure and temperature were 5.0105 Pa and 300 K, respectively, what are the final pressure and temperature of the gas?On an adiabatic process of an ideal gas pressure, volume and temperature change such that pV is constant with =5/3 for monatomic gas such as helium and =7/5 for diatomic gas such as hydrogen at room temperature. Use numerical values to plot two isotherms of 1 mol of helium gas using ideal gas law and two adiabatic processes mediating between them. Use T1=500K,V1=1L, and T2=300K for your plot.Two moles of a monatomic ideal gas such as helium is compressed adiabatically and reversibly from a state (3 atm, 5 L) to a state with pressure 4 atm. (a) Find the volume and temperature of the final state. (b) Find the temperature of the initial state of the gas. (c) Find the work done by the gas in the process. (d) Find the change in internal energy of the gas in the process.Consider the process shown below. During steps AB and BC, 3600 J and 2400 J of heat, respectively, are added to the system. (a) Find the work done in each of the processes AB, BC, AD, and DC. (b) Find the internal energy change in processes AB and BC. (c) Find the internal energy difference between states C and A (d) Find the total heat added in the ADC process. (e) From the information given, can you find the heat added in process AD? Why or why not?A car tile contains 0.0380 m3 of air at a pressure of 2.20105 Pa (about 32 psi). How much more internal energy does this gas have than the same volume has at zero gauge pressure (which is equivalent to normal atmospheric pressure)?A helium-filled toy balloon has a gauge pressure of 0.200 atm and a volume of 10.0 L. How much greater is the internal energy of the helium in the balloon than it would be at zero gauge pressure?Steam to drive an old-fashioned steam locomotive is supplied at a constant gauge pressure of 1.75106 N/m2 (about 250 psi) to a piston with a 0.200-m radius. (a) By calculating pv , find the work done by the steam when the piston moves 0.800 m. Note that this is the net work output, since gauge pressure is used. (b) Now find the amount of work by calculating the force exerted times the distance traveled. Is the answer the same as in part (a)?A hand-driven tire pump has a piston with a 2.50-cm diameter and a maximum stroke of 30.0 cm. (a) How much work do you do in one stroke if the average gauge pressure is 2.1105 N/m2 (about 35 psi)? (b) What average force do you exert on the piston, neglecting friction and gravitational force?Calculate the net work output of a heat engine following path ABCDA as shown below.What is the net work output of a heat engine that follows path ABDA in the preceding problem with a straight line from B to D? Why is the work output less than for path ABCDA?Five moles of a monatomic ideal gas in a cylinder at 27 is expanded isothermally from a volume of 5 L to 10 L. (a) What is the change in internal energy? (b) How much work was done on the gas in the process? (c) How much heat was transferred to the gas?Four moles of a monatomic ideal gas in a cylinder at 27 is expanded at constant pressure equal to 1 atm until its volume doubles. (a) What is the change in internal energy? (b) How much work was done by the gas in the process? (c) How much heat was transferred to the gas?Helium gas is cooled from 20 to 10 by expanding from 40 atm to 1 atm. If there is 1.4 mol of helium. (a) What is the final volume of helium? (b) What is the change in internal energy?In an adiabatic process, oxygen gas in a container is compressed along a path that can be described by the following pressure in atm as a function of volume V, with V0=1L:P=(3.0atm)(V/V)1.2 . The initial and final volumes during the process were 2 L and 1.5 L, respectively. Find the amount of work done on the gas.A cylinder containing three moles of a monatomic ideal gas is heated at a constant pressure of 2 atm. The temperature of the gas changes from 300 K to 350 K as a result of the expansion. Find work done (a) on the gas; and (b) by the gas.A cylinder containing three moles of nitrogen gas is heated at a constant pressure of 2 atm. The temperature of the gas changes from 300 K to 350 K as a result of the expansion. Find work done (a) on gas, and (b) by the gas by using van der Waals equation of state instead of ideal gas law.Two moles of a monatomic ideal gas such as oxygen is compressed adiabatically and reversibly from a state (3 atm, 5 L) to a state with a pressure of 4 atm. (a) Find the volume and temperature of the final state. (b) Find the temperature of the initial state. (c) Find work done by the gas in the process. (d) Find the change in internal energy in the process. Assume Cv=5R and Cp=Cv+R for the diatomic ideal gas in the conditions given.An insulated vessel contains 1.5 moles of argon at 2 atm. The gas initially occupies a volume of 5 L. As a result of the adiabatic expansion the pressure of the gas is reduced to 1 atm (a) Find the volume and temperature of the final state. (b) Find the temperature of the gas in the initial state. (c) Find the work done by the gas in the process. (d) Find the change in the internal energy of the gas in the process.One mole of an ideal monatomic gas occupies a volume of 1.0102 m3 at a pressure of 2.0105 N/m2. (a) What is the temperature of the gas? (b) The gas undergoes a quasi-static adiabatic compression until its volume is decreased to 5.0103 m3. is the new gas temperature? (c) How much work is done on the gas during the compression? (d) What is the change in the internal energy of the gas?One mole of an ideal gas is initially in a chamber of volume 1.0102 m3 and at a temperature of 27 . (a) How much heat is absorbed by the gas when it slowly expands isothermally to twice its initial volume? (b) Suppose the gas is slowly transformed to the same final state by first decreasing the pressure at constant volume and then expanding it isobarically. What is the heat transferred for this case? (c) Calculate the heat transferred when the gas is transformed quasi-statically to the same final state by expanding it isobarically, then decreasing its pressure at constant volume.A bullet of mass 10 g is traveling horizontally at 200 m/s when it strikes and embeds in a pendulum bob of mass 2.0 kg. (a) How much mechanical energy is dissipated in the collision? (b) Assuming that Cv for the bob plus bullet is 3R, calculate the temperature increase of the system due to the collision. Take the molecular mass of the system to be 200 g/mol.The insulated cylinder shown below is closed at both ends and contains an insulating piston that is flee to move on frictionless bearings. The piston divides the chamber into two compartments containing gases A and B. Originally, each compartment has a volume of 5.0102 m3 and contains a monatomic ideal gas at a temperature of and a pressure of 1.0 atm. (a) How many moles of gas are in each compartment? (b) Heat Q is slowly added to A so that it expands and B is compressed until the pressure of both gases is 3.0 atm. Use the fact that the compression of B is adiabatic to determine the final volume of both gases. (c) What are their final temperatures? (d) What is the value of Q?In a diesel engine, the fuel is ignited without a spark plug. Instead, air in a cylinder is compressed adiabatically to a temperature above the ignition temperature of the fuel; at the point of maximum compression, the fuel is injected into the cylinder. Suppose that air at 20 C is taken into the cylinder at a volume V1 and then compressed adiabatically and quasi-statically to a temperature of 600 C and a volume V2 . If =1.4 , what is the ratio V1/V2 ? (Note: static. In an operating diesel engine, the compression is not quasi-Check Your Understanding What is the efficiency of a perfect heat engine? What is the coefficient of performance of a perfect refrigerator?Check your Understanding Show that QhQh=QcQc for the hypothetical engine of Figure 4.10 The second property to be demonstrated is that all reversible engines operating between the same two reservoirs have the same efficiency. To this, stat with the two engines D and E of Figure 4.10 (a), which are operating between two common heat reservoirs at temperatures Th and Tc . First, assume that D is a reversible engine and that E is a hypothetical irreversible engine that has a higher efficiency than D. If both engines perform the same amount of work W per cycle, it follows from Equation 4.2 that QhQh . It then follows from the first law that QcQc . Figure 4.10 (a) Two uncoupled engines D and E working between the same reservoirs. (b) The engines, With D working reverse. Suppose the cycle of D is so that it operates as a refrigerator, and the two engines are coupled such that the work output of E is used to drive D, as shown in Figure 4.10(b). Since QhQh and QcQc , the net result of each cycle is equivalent to a spontaneous transfer of heat from the cold reservoir to the hot reservoir, a process second law does not allow. The original assumption must therefore be wrong, and it is impossible to construct an irreversible engine such that E is more efficient than the reversible engine D. Now it is quite easy to demonstrate that the efficiencies of all reversible engines operating between the same reservoirs are equal. Suppose that D and E are reversible engines. If they are as shown in Figure 4.10(b), the efficiency of E cannot be greater than the efficiency of D, or second law would violated. If both engines are then reversed, the same reasoning implies that the efficiency of D cannot be greater than the efficiency of E. Combining these results leads to the conclusion that all reversible engines working between same two reservoirs have the same efficiency.Check Your Understanding A Carnot engine operates between reservoirs at 400 and 30 . (a) What is the efficiency of the engine? (b) If the engine does 5.0 J of work per cycle, how much heat per cycle does it absorb from the high-temperature reservoir? (c) How much heat per cycle does it exhaust to the cold-temperature reservoir? (d) What temperatures at the cold reservoir would give the minimum and maximum efficiency?Check Your Understanding A Carnot refrigerator operates between two heat reservoirs whose temperatures 0 and 25 . (a) What is the coefficient of performance of the refrigerator? (b) If 200 J of work done on the working substance cycle, how heat per cycle is from cold reservoir? (c) How much heat cycle is discarded to the hot reservoir?Check Your Understanding In Example 4.7, the spontaneous flow of heat from a hot object to a cold object results in a net increase in entropy of the universe. Discuss how this result can be related to an increase in disorder of the system.Check Your Understanding A quantity of heat Q is absorbed from a reservoir at a temperature Th by a cooler reservoir at a temperature Tc . What is the entropy change of the hot reservoir, the cold reservoir, and the universe?Check Your Understanding A 50-g copper piece at a temperature of 20 is placed into a large insulated vat of water in 100 . (a) What is the entropy change of the copper piece when it reaches thermal equilibrium with the water? (b) What is the entropy change of the water? (c) What is the entropy change of the universe?State an example of a process that occurs in nature that is as close to reversible as it can be.Explain in practical terms why efficiency is defined asIf the refrigerator door is left what happens to the temperature of the kitchen?Is it possible for the efficiency of a reversible engine to greater than 1.0? Is it possible for the coefficient of performance of a reversible refrigerator to be less than 1.0?In the text, we showed that if the Clausius statement is false, the Kelvin statement must also be false. Now show the reverse, such that if the Kelvin statement is false, it follows that Clausius statement is false.Why don't we operate ocean liners by extracting heat from the ocean or operate airplanes by extracting heat from the atmosphere?Discuss the practical advantages and disadvantages of heat pumps and electric heating.The energy output of a heat pump is greater than the energy used to operate the pump. Why doesn't this statement violate the first law of thermodynamics?Speculate as to why nuclear power plants are less efficient than fossil-fuel plants based on temperature arguments.An ideal gas goes from state (pi,vi,) to state (pf,vf,) when it is allowed to expand freely. Is it possible to represent the actual process on a pVdiagram? Explain.To increase the efficiency of a Carnot engine, should the temperature of the hot reservoir be raised or lowered? What about the cold reservoir?How could you design a Carnot engine with 100% efficiency?What type of processes occur in a Carnot cycle?Does the entropy increase for a Carnot engine for each cycle?Is it possible for a system to have an entropy change if it neither absorbs nor emits heat during a reversible? transition? What happens it the process is irreversible?Are the entropy changes of the system in the following processes positive or negative? (a) water vapor that condenses on a cold surface; (b) gas in a that leaks into the surrounding atmosphere; (c) an ice cube that melts in a glass of lukewarm water; (d)the lukewarm water of part (c); a real heat engine performing a cycle; (f) food cooled in a refrigerator.Discuss the entropy changes in the systems of Question 21.10 in terms of disorder.A tank contains 111.0 g chlorine gas l2), which is at temperature 82.0 and absolute pressure 5.70105 Pa. The temperature of the air outside the tank is 20.0 . The molar mass of Cl2 is 70.9 g/mol. (a) What is the volume of the tank? (b) What is the internal energy of the gas? (c) What is the work done by the gas if the temperature and pressure inside the tank drop to 31.0 and 3.80105 Pa, respectively, due to a leak?A mole of ideal monatomic gas at 0 and 1.00 atm is warmed up to expand isobarically to triple its volume. How much heat is transferred during the process?A mole of an ideal gas at pressure 4.00 atm and temperature 298 K expands isothermally to double its volume. What is the work done by the gas?After a free expansion to quadruple its volume, a mole of ideal diatomic gas is compressed back to its original volume isobarically and then cooled down to its original temperature. What is the minimum heat removed from the gas in the final step to restoring its state?An engine is found to have an efficiency of 0.40. If it does 200 J of work per cycle, what are the corresponding quantities of heat absorbed and discharged?In performing 100.0 J of work, an engine discharges 50.0 J of heat. What is the efficiency of the engine?An engine with an efficiency of 0.30 absorbs 500 J of heat per cycle. (a) How much work does it perform per cycle? (b) How much heat does it discharge per cycle?It is found that an engine discharges 100.0 J while absorbing 125.0 J each cycle of operation. (a) What is the efficiency of the engine? (b) How much work does it perform per cycle?The temperature of the cold reservoir of the engine is 300 K. It has an efficiency of 0.30 and absorbs 500 J of heat per cycle. (a) How much work does it perform per cycle? (b) How much heat does it discharge cycle?An engine absorbs three times as much heat as it discharges. The work done by the engine per cycle is 50 J. Calculate (a) the efficiency of the engine, (b) the heat absorbed per cycle, and (c) the heat discharged per cycle.A coal power plant consumes 100,000 kg of coal per hour and produces 500 MW of power. If the heat of combustion of coal is 30 MJ/kg, what is the efficiency of the power plant?A refrigerator has a coefficient of performance of 3.0. (a) If it requires 200 J of work per cycle, how much heat per cycle does it remove the cold reservoir? (b) How much cycle is discarded to the hot reservoir?During one cycle, a refrigerator removes 500 J from a cold reservoir and discharges 800 J to its hot reservoir. (a) What is its coefficient of performance? (b) How much work per cycle does it require to operate?If a refrigerator discards 80 J of heat per cycle and its coefficient of performance is 6.0, what are (a) the quantity off heat it removes per cycle from a cold reservoir and (b) the amount of work per cycle required for its operation?A refrigerator has a coefficient of performance of 3.0. (a) If it requires 200 J of work per cycle, how much heat per cycle does it remove the cold reservoir? (b) How much heat per cycle is discarded to the hot reservoir?The temperature of the cold and hot reservoirs between which a Carnot refrigerator operates are 73 and 270 , respectively. Which is its coefficient of performance?Suppose a Carnot refrigerator operates between Tc and Th . Calculate the amount of work required to extract 1.0 J of heat from the cold reservoir if (a) Tc=7C,Th=27C; (b) Tc=73C,Th=27C; (c) Tc=173C,Th=27C; and (d) Tc=273C,Th=27C;A Carnot engine operates between reservoirs at 600 and 300 K. If the engine absorbs 100 J per cycle at the hot reservoir, what is its work output per cycle?A 500-W motor operates a Carnot refrigerator between 5 and 30 . (a) What is the amount of heat per second extracted from the inside of the refrigerator? (b)How much heat is exhausted to the outside air per second?Sketch a Carnot cycle on a temperature-volume diagram.A Carnot heat pump operates between 0 and 20 . How much heat is exhausted into the interior of a house for every 1.0 J of work done by the pump?An engine between heat reservoirs at 20 and 200 extracts 1000 J per cycle from the hot reservoir. (a) What is the maximum possible work that engine can do per cycle? (b) For this maximum work, how much heat is exhausted to the cold reservoir per cycle?Suppose a Carnot engine can be operated between two reservoirs as either a heat engine or a refrigerator. How is the coefficient of performance of the refrigerator related to the efficiency of the heat engine?A Carnot engine is used to measure the temperature of a heat reservoir. The engine operates between the heat reservoir and a reservoir consisting of water at its triple point. (a) If 400 J per cycle are removed from the heat reservoir while 200 J per cycle are deposited in the triple-point reservoir, what is the temperature of the heat reservoir? (b) If 400 J per cycle are removed from the triple-point reservoir while 200 J per cycle are deposited in the heat reservoir, what is the temperature of the heat reservoir?What is the minimum work required of a refrigerator if it is to extract 50 J per cycle from the inside of a freezer at 10 and exhaust heat to the air at 25 ?Two hundred joules of heat are removed from a heat at a temperature of 200 K. What is the entropy change of the reservoir?In an isothermal reversible expansion at 27 , an ideal gas does 20 J of work. What is the entropy change of the gas?An ideal gas at 300 K is compressed isothermally to one-fifth its original volume. Determine the entropy change per mole of the gas.What is the entropy change of 10 g of steam at 100 when it condenses to water at the same temperature?A metal is used to conduct heat between two reservoirs at temperatures Th and Tc , respectively. When an amount of heat Q flows through the rod from the hot to the cold reservoir, what is the net entropy change of the rod, the hot reservoir, the cold reservoir, and the universe?For the Carnot cycle of Figure 4.12, what is the entropy change of the hot reservoir, the cold reservoir, and the universe? Figure 4.11 The four processes of the Carnot cycle. The working substance is assumed to be an ideal gas whose thermodynamic path MNOP is represented in Figure 4.12. Figure 4.12 The total work done by the gas in the Carnot cycle is shown and given by the area enclosed by the loop MNOPM.A 5.0-kg piece of lead at a temperature of 600 is placed in a lake whose temperature is 15 . Determine the entropy change of (a) the lead piece, (b) the lake, and (c) the universe.One mole of an ideal gas doubles its volume in a reversible isothermal expansion. (a) What is the change in entropy of the gas? (b) If 1500 J of heat are added in this process, what is the temperature of the gas?One mole of an ideal monatomic gas is confined to a rigid container. When heat is added reversibly to die gas, its temperature changes from T1 to T2 . (a) How much heat is added? (b) What is the change in entropy of the gas?(a) A 5.0-kg rock at a temperature of 20 is dropped into a shallow lake also at 20 from a height of 1.0103 m. What is the resulting change in entropy of the universe? (b) If the temperature of the lock is 100 when it is dropped, what is the change of entropy of the universe? Assume that air friction is negligible (not a good assumption) and that c=860 J/kg K is the specific heat of the rock.A copper rod of cross-sectional area 5.0 cm2 and length 5.0 m conducts heat from a heat reservoir at 373 K to one at 273 K. What is the time rate of change of the universe's entropy for this process?Fifty grams of water at 20 is heated until it becomes vapor at 100 . Calculate the change in entropy of the water in this process.Fifty grams of water at 0 are changed into vapor at 100 . What is the change in entropy of the water in this process?In an isochoric process, heat is added to 10 mol of monoatomic ideal gas whose temperature increases from 273 to 373 K. What is the entropy change of the gas?Two hundred grams of water at 0 is brought into contact with a heat reservoir at 80 . After thermal equilibrium is reached, what is the temperature of the water? Of the reservoir? How much heat has been transferred in the process? What is the entropy change of the water? Of the reservoir? What is the entropy change of the universe?Suppose that the temperature of the water in the previous problem is raised by fist bringing it to thermal equilibrium with a reservoir at a temperature of 40 and then with a reservoir at 80 . Calculate the entropy changes of (a) each reservoir, (b) of the water, and (c) of the universe.Two hundred grams of water at 0 is brought into contact into thermal equilibrium successively with reservoirs at 20 , 40 , 60 , and 80 . (a) What is the entropy change of the water? (b) Of the reservoir? (c) What is the entropy change of the universe?(a) Ten grams of H2O stats as ice at 0 . The ice absorbs heat from the air (just above 0 ) until all of it melts. Calculate the entropy change of the H2O, of the air, and of the universe. (b) Suppose that the air in part (a) is at 20 rather than 0 and that the ice absorbs heat until it becomes water at 20 . Calculate the entropy change of the H2O, of the air, and of the universe. (c) Is either of these processes reversible?The Carnot cycle is represented by the temperature-entropy diagram shown below. (a) How much heat is absorbed per cycle at the high-temperature reservoir? (b) How much heat is exhausted per cycle at the low-temperature reservoir? (c) How much work is done per cycle by the engine? (d) What is the efficiency of the engine?A Carnot engine operating between heat reservoirs at 500 and 300 K absorbs 1500 J per cycle at the high-temperature reservoir. (a) Represent the engine's cycle on a temperature-entropy diagram. (b) How much per cycle is done by the engine?A monoatomic ideal gas (n moles) goes through a cyclic process shown below. Find the change in entropy of the gas in each step and the total entropy change over the entire cycle.A Carnot engine has an efficiency of 0.60. When the temperature of its cold reservoir the efficiency drops to 0.55. If initially Tc=27, determine (a) the constant value of Th and (b) the final value of Tc.A Carnot engine performs 100 J of work while discharging 200 J of heat each cycle. After temperature of the hot reservoir only is adjusted, it is found that the engine now does 130 J of work while discarding the same quantity of heat. (a) What are the initial and final efficiencies of the engine? (b) What is the fractional change in the temperature of the hot reservoir?A Carnot refrigerator exhausts heat to the air, which is at a temperature of 25 . How much is used by the refrigerator if it freezes 1.5 g of water per second? Assume the water is at 0 .A 300-W heat pump operates between the ground, whose temperature is 0 , and the interior of a house at 22 . What is the maximum amount of the heat per hour that the heat pump can supply to the house?An engineer must design a refrigerator that does 300 J of work per cycle to extract 2100 J of heat per cycle from a freezer whose temperature is 10 . What is the maximum air temperature for which this condition can be met? Is this a reasonable condition to impose on the design?A Carnot engine employs 1.5 mol of nitrogen gas as a working substance, which is considered as an ideal diatomic gas with =7.5 at the working temperatures of the engine. The Carnot cycle goes in the cycle ABCDA with AB being an isothermal expansion. The volume at points A and C of the cycle are 5.0103 m3 and 0.15 L, respectively. The engine operates between two thermal baths of temperature 500 K 300 K. (a) Find the values of volume at B and D. (b) How much heat is absorbed by the gas in the AB isothermal expansion? (c) How much work is done by the gas in the AB isothermal expansion? (d) How much heat is given up by the gas in the CD isothermal expansion? (e) How much work is done by the gas in the CD isothermal compression? (f) How much work is done by the gas in the BC adiabatic expansion? (g) How much work is done by the gas in the DA adiabatic compression? (h) Find the value of efficiency of the engine based on the net and heat input. Compare this value to the efficiency of a Carnot engine based on the temperatures of the baths.A 5.0-kg wood block starts with an initial speed of 8.0 m/s and slides across the floor until friction stops it. Estimate the resulting change in entropy of the universe. Assume that everything stays at a room temperature of 20 .A system consisting of 20.0 mol of a monoatomic ideal gas is cooled at constant pressure from a volume of 50.0 L to 10.0 L. The initial temperature was 300 K. What is the change in entropy of the gas?A glass beaker of mass 400 g contains 500 g of water at 27 . The beaker is heated reversibly so that the temperature of the beaker and water rise gradually to 57 . Find the change in entropy of the beaker and water together.A Carnot engine operates between 550 and 20 baths and produces 300 kJ of energy in each cycle. Find the change in entropy of the (a) hot bath and (b) cold bath, in each Carnot cycle?An ideal gas at temperature T is stored in the left half of an insulating container of volume V using a partition of negligible volume (see below). What is the entropy change per mole of the gas in each of the following cases? (a) The partition is suddenly removed and the gas quickly fills the entire container. (b) A tiny hole is punctured in the partition and after a long period, the gas reaches an equilibrium state such that there is no net flow through the hole. (c) The partition is moved very slowly and adiabatically all the way to the light wall so that the gas finally fills the entire container.A 0.50-kg piece of aluminum at 250 is dropped into 1.0 kg of water at 20 . After equilibrium is reached, what is the net entropy change of the system?Suppose 20 g of ice at 0 is added to 300 g of water at 60 . What is the total change in entropy of the mixture after it reaches thermal equilibrium?A heat engine operates between two temperatures such that the working substance of the engine absorbs 5000 J of heat from the high-temperature bath and discharges 3000 J to the low-temperature bath. The rest of the energy is converted into mechanical energy of the turbine. Find (a) the amount of work produced by the engine and (b) the efficiency of the engine.A thermal engine produces 4 MJ of electrical energy while operating between two thermal baths of different temperatures. The working substance of the engine discharges 5 MJ of heat to the cold temperature bath. What is the efficiency of the engine?A coal power plant consumes 100,000 kg of coal per hour and produces 500 MW of power. If the heat of combustion of coal is 30 MJ/kg, what is the efficiency of the power plant?A Carnot engine operates in a Carnot cycle between a heat source at 550 and a heat sink at 20 . Find the efficiency of the Carnot engine.A Carnot engine working between two heat baths of temperatures 600 K and 273 K completes each cycle in 5 sec. In each cycle, the engine absorbs 10 kJ of heat. Find the power of the engine.A Carnot cycle working between 100 and 30 is used to drive a refrigerator between 10 and 30 . How much energy must the Carnot engine produce per second so that the refrigerator is able to discard 10 J of energy per second?(a) infinitesimal amount of heat is added reversibly to a system. By combining the first and second laws, show that dU=TdSdW. (b) When heat is added to an ideal gas, its temperature and volume change from T1 and V1 to T2 and V2 . Show that the entropy change of n moles of the gas is given by S=CnvlnT2T1nRlnV2V1 .Using the result of the preceding problem, show that for an ideal gas undergoing an adiabatic process, TV1 is constant.With the help of the two preceding problems, show that S between states 1 and 2 of nmoles an ideal gas is given by S=CnplnT2T1nRlnP2P1.A cylinder contains 500 g of helium at 120 atm and 20 . The valve is leaky, and all the gas slowly escapes isothermally into the atmosphere. Use the results of the preceding problem to determine the resulting change in entropy of the universe.A diatomic ideal gas is brought from an initial equilibrium state at p1=0.50 atm and T1=300K to a final stage with p2=0.20 atm and T1=500K. Use the results of the previous problem to determine the entropy change per mole of the gas.The gasoline internal combustion engine operates in a cycle consisting of six parts. Four of these parts involve, among other things, friction, heat exchange through finite temperature differences, and accelerations of the piston; it is irreversible. Nevertheless, it is represented by the ideal reversible Otto cycle, which is illustrated below. The working substance of the cycle is assumed to be air. The six steps of the Otto cycle ale as follows: i. Isobaric intake stroke (OA). A mixture of gasoline and air is drawn into the combustion chamber at atmospheric pressure P0 as the piston expands, increasing the volume of the cylinder from zero to VA . ii. Adiabatic compression stroke (AB). The temperature of the mixture rises as the piston compresses it adiabatically from a volume VA to VB . iii. Ignition at constant volume (BC). The mixture is ignited by a spark. The combustion happens so fast that there is essentially no motion of the piston. During this process, the added heat Q1 causes the pressure to increase from pB to pc at the constant volume VB(=Vc) . iv. Adiabatic expansion (CD). The heated mixture of gasoline and air expands against the piston, increasing the volume from VC to VD . This is called the power stroke, as it is the part of the cycle that delivers most of the power to the crankshaft. v. Constant-volume exhaust (DA). When the exhaust valve opens, some of the combustion products escape. There is almost no movement of the piston during this part of the cycle, so the volume remains constant at VA(=VD) . Most of the available energy is lost here, as represented by the heat exhaust Q2 . vi. Isobaric compression (AO). The exhaust valve remains open, and the compression from VA to zero drives out the remaining combustion products. (a). Using (i)e=W/Q1; (ii)w=Q1Q2; and (iii)Q1=nCv(TCTB),Q2=nCv(TDTA), Show that e=1TDTATCTB. (b). Use the fact that steps (ii) and (iv) are adiabatic to show that e=11r1 where r=VA/VB . The quantity r is called the compression ratio of the engine. (c) In practice, r is kept less than around 7. For larger values, the gasoline-air mixture is compressed to temperatures so high that it explodes before the finely timed spark is delivered. This preignition causes engine knock and loss of power. Show that for r=6 and =1.4 (the value for air), e=0.51 , or an efficiency of 51%. Because of the many irreversible processes, an actual internal combustion engine has an efficiency much less than this ideal value. A typical efficiency for a tuned engine is about 25% to 30%.An ideal diesel cycle is shown below. This cycle consists of five strokes. In this case, only air is drawn into the chamber during the intake stroke OA. The air is then compressed adiabatically from state A to state B, raising its temperature high enough so that when fuel is added during the power stroke BC, it ignites. After ignition ends at C, there is a further adiabatic power stroke CD. Finally, there is an exhaust at constant volume as the pressure drops from pD to pA , followed by a further exhaust when the piston compresses the chamber volume to zero. (a) Use W=Q1Q2, Q1=nCp(TCTB) , and Q2=nCv(TDTA) to show that e=WQ11TDTA(TCTB). (b) Use the fact that AB and CD are adiabatic to show that e=11( V C V D )( V B V A )( V C V D )( V B V A ). (c) Since there is no preignition (remember, the chamber does not contain any fuel during the compression), the compression ratio can be larger than that for a gasoline engine. Typically, VA/VB=15 and VD/VC=5 . For these values and =1.4 , show that =0.56 , or an efficiency of 56%. Diesel engines actually operate at an efficiency of about 30% to 35% compared with 25% to 30% for gasoline engines.Consider an ideal gas Joule cycle, also called the Brayton cycle, shown below. Find the formula for efficiency of the engine using this cycle in terms of P1 , P2 and .Derive a formula for the coefficient of performance of a refrigerator using an ideal gas as a working substance operating in the cycle shown below in terms of the properties of the three states labeled 1, 2, and 3.Two moles of nitrogen gas, with =7/5 for ideal diatomic gases, occupies a volume of 102 m3 in an insulated cylinder at temperature 300 K. The gas is adiabatically and reversibly compressed to a volume of 5 L. The piston of the cylinder is locked in its place, and the insulation around the cylinder is removed. The heat-conducting cylinder is then placed in a 300-K bath. Heat from the compressed gas leaves the gas, and the temperature of the gas becomes 300 K again. The gas is then slowly expanded at the fixed temperature 300 K until the volume of the gas becomes 102 m3, thus making a complete cycle for the gas. For the entire cycle, calculate (a) the work done by the gas, (b) the heat into or out of the gas, (c) the change in the internal energy of the gas, and (d) the change in entropy of the gas.A Carnot refrigerator, working between 0 and 30 is used to cool a bucket of water containing 102 m3 of water at 30 to 5 in 2 hours. Find the total amount of work needed.Check Your Understanding What would be different if the election also had a positive charge?Check Your Understanding What would be different if q1 were negative?Check Your Understanding What is the electric field due to a single point particle?Check Your Understanding How would the strategy change to calculate the at a point a distance z above of the finite line segment?Check Your Understanding How would the above limit change with a uniformly rectangle instead of a disk?Check Your Understanding the electric field 100k like in a system with two parallel positively charged planes with equal charge densities?There are very large numbers of charged particles in most objects. Why, then, don't most objects exhibit static electricity?Why do most objects tend to contain nearly equal numbers of positive and negative charges?A positively charged It'd attracts a small piece of cork. (a) Can we conclude that the cork is negatively charged? (b) The rod repels another small piece of cork. Can we conclude that this piece is positively charged?Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel another.How would you determine whether the charge on a particular rod is positive or negative?An eccentlic inventor attempts to levitate a cork ball by wrapping it isith foil and placing a large negative charge on the ball and then putting a large positive charge on the ceiling of his workshop. Instead, while attempting to place a large negative charge on the ball, the foil flies off.When a glass rod is lubbed with silk, it becomes positive and the silk becomes negative—yet both attract dust. Does the dust have a third type of charge that is attracted to both positive and negative? FxplainWhy does a car always attract dust right after it is polished? (Note that car wax and car tires are insulators.)Does the uncharged conductor shown below expelience a net electric force?While walking on a mg, a person frequently becomes charged because of the tubbing between his shoes and the lug. This charge then causes a spark and a slight shock when the person gets close to a metal object. Why are these shocks so much more common on a day?Compare charging by conduction to charging by induction.Small pieces of tissue are attracted to a charged comb. Soon after stirlrinp to the comb, the pieces of tissue are frorn it. Explain.Trucks that cany gasoline often have chains dangling from their undercaniages and blushing ground. Why?Why do electrostatic experiments work so poorly in hunid weather?Why do some clothes cling together after being removed from the clothes dryer? Does this happen if they're still damp?Can induction be used to produce charge on an insulator?Suppose someone tells you that rubbing quartz with cotton cloth produces a third kind of charge on the quartz. Describe what you might do to test this claim.A handheld copper rod does not acquire a charge when you rub it with a cloth. Explain why.Suppose you place a charge q near a large metal plate. (a) If q is attracted to to the plate, is the plate necessarily charged? (b) If q is repelled by the plate, is the plate necessarily charged?Would defining the charge on an electron to be positive have any effect on Coulomb's law?An atomic nucleus contains positively charged protons and uncharged neutrons. Since nuclei do stay together, what must we conclude about the forces these nuclear particles?Is the fore between two fixed charges influenced by the plesence of other charges?When measuring an electlic field, could we use a negative rather than a positive test charge?During fair weather, the electric field due to the net charge on Earth points downward. Is Earth charged positively or negatively?If the electric field at a point on the line between two charges is zero, what do you know about the charges?Two charges lie along the x-axis. Is it nue that the net electric field always vanishes at some point (other than infinity) along the x-axis?Give a plausible argument as to why the electric field outside an infinite charged sheet is constant.Compare the electric fields of an infinite sheet of charge, an infinite, charged conducting plate, and infinite, oppositely charged parallel plates.Describe the electric fields of an infinite charged plate and of two infinite, charged parallel plates in terms of the electric field of an infinite sheet of charge.A negative charge is placed at center of a ring of uniform positive charge. What is the motion (if any) of the charge? What if the charge placed at a point on the axis of the ring other than the center?If a point charge is released fmm rest in a unifonn electlic field, will it follow a field line? Will it do so if the electlic field is not uniform?Under what conditions, if any, will the trajectory of a charged particle not follow a field line?How would you experimentally distinguish an electlic field from a gravitational field?A representation of an electric field shows 10 field lines perpendicular to a square plate. How many field lines should pass perpendicularly through the plate to depict a field with twice the magnitude?What is the ratio of the number of electlic field lines leaving a charge 10q and a charge q?What are the stable orientation(s) for a dipole in an external electric field? What happens if the dipole is slightly peltulbed from these orientations?Common static electricity involves charges ranging from nanocoulombs to microcoulombs. (a) How many electrons are needed to folm a chalge of 2.00 nC? (b) How many elections must be removed from a neutral object to leave a net chalge of 0.500 C ?If 1.801020 electrons move through a pocket calculator during a full day's operation, how many coulombs of charge moved through it?To stat a car engine, the car battery moves 3.751021 electrons through the staffer motor. How many coulombs of charge were moved?A certain lightning bolt moves 40.0 C of charge. How many fundamental units of charge is this?A 2.5-g copper penny is given a charge of 2.0109C . (a) How many excess electrons are on the penny? (b) By what percent do the excess electrons change the mass of the penny?A 2.5-g copper penny is given a charge of 4.0109C . (a) How many electrons ale removed from the penny? (b) If no more than one electron is removed from an atom, what percent of the atoms are ionized by this charging process?Suppose a speck of dust in an electrostatic precipitator has 1.00001012 protons in it and has a net charge of 5.00 nC (a very large charge for a small speck). How many electrons does it have?An amoeba has 1.001016 protons and a net charge of 0.300pC. (a) How many fewer electrons are there than protons? (b) If you paired them up, what fraction of the protons would have no electrons?A 50.0-g ball of copper has a net charge of 2.00C. What fraction of the copper's electrons has removed? (Each copper atom has 29 protons, and copper has an atomic mass of 63.3.)What net charge would you place on a 100-g piece of sulfur if you put an extra electron on 1 in 1012 of its atoms? (Sulfur has an atomic mass of 32.1 u.)How many coulombs of positive charge are there in 4.00 kg of plutonium given its atomic mass is 244 and that each plutonium atom has 94 protons?Two point particles with charges +3C and +5C are held in place by 3-N forces on each charge in appropriate directions. (a) Draw a free-body diagram for each particle. (b) Find the distance between the charges.Two charges +3C and +12C are fixed 1 m apart, with the second one to the right. Find the magnitude and direction of the net force on a 2nC charge when placed at the following locations: (a) halfway between the two (b) half a meter to the left of the +3C charge (c) half a meter above the +12C charge in a direction perpendicular to the line joining the two fixed chargesIn a salt crystal, the distance between adjacent sodium and chloride ions is 2.821010 m. What is the force of attraction between the two singly charged ions?Protons in an atomic nucleus ale typically 1015 m apart. What is the electric fore of repulsion between nuclear protons?Suppose Earth and the Moon each carried a net negative charge Q . Approximate both bodies as point masses and point charges. (a) What value of Q is required to balance the gravitational attraction between Earth and the Moon? (b) Does the distance between Earth and the Moon affect your answer? Explain. (c) How many electrons would be needed to produce this charge?Point charges q1=50C and q2=25C are placed 1.0 m apart. What is the force on a third charge q3=20C placed midway between q1 and q2 ?Where must q3 of the preceding problem be placed so that the net force on it is zero?Two small balls, each of mass 5.0 g, are attached to silk threads 50 cm long, which are in turn tied to the same point on the ceiling, as shown below. When the balls are given the same charge Q, the threads hang at 5.0( to the vertical, as shown below. What is the magnitude of Q? What are the signs of the two charges?Point charges q1=2.0C and q3=4.0C arelocated at r1=(4.0i2.0j+5.0k) and r2=(8.0i+5.0j9.0k) .What is the force of Q2 on Q1 ?The net excess charge on two small spheres (small enough to be treated as point charges) is Q. Show that the force of repulsion between the spheres is greatest when each sphere has an excess charge Q/2. Assume that the distance between the spheres is so large compared with their radii that the spheres can be treated as point charges.Two small, identical conducting spheres repel each other with a force of 0.050 N when they are 0.25 m apart. After a conducting wire is connected between the spheres and then removed, they repel each other with a force of 0.060 N. What is the original charge on each sphere?A charge q=2.0C is placed at the point P shown below. What is the fore on q?What is the net electric fore on the charge located at the lower light-hand comer of the triangle shown here?Two fixed particles, each of charge 5.0106C , are 24 cm apart. What force do they exert on a third particle of charge 2.5106C that is 13 cm from each of them?The charges q1=2.0107C, q2=4.0107C, and q3=1.0107C are placed at the corners of the triangle shown below. What is the fore on q1 ?What is the force on the charge q at the lower-right-hand comer of the square shown here?Point charges q1=10C and q2=30C are fixed at r1=(3.0i4.0j) m and r2=(9.0i6.0j) m What is the force of q2 on q1 ?A particle of charge 2.0108C experiences an upward force of magnitude 4.0106N when it is placed in a particular point in an electric field. (a) What is the electric field at that point? (b) If a charge q=1.0108C is placed there, what is the force on it?On a typical clear day, the atmospheric electric field points downward and has a magnitude of approximately 100 N/C. Compare the gravitational and electric forces on a small dust particle of mass 2.01015 g that carries a single electron charge. What is the acceleration (both magnitude and direction) of the dust particle?Consider an electron that is 1010 m from an alpha particle (q=3.21019C) . (a) What is the electric field due to the alpha particle at the location of the electron? (b) What is the electric field due to the electron at the location of the alpha particle? (c) What is the electric force on the alpha particle? On the electron?Each the balls shown below carries a charge q and has a mass m. The length of each thread is l, and at equilibrium, the balls are separated by an angle 2 . How does vary with q and l ? Show that satisfiesWhat is the electric field at a point where the force on a 2.0106C chargeis (4.0i6.0j)106N ?A proton is suspended in the air by an electric field at the surface of Earth. What is the strength of this electric field?The electric field in a particular thundercloud is 2.0105N/C . What is the acceleration of an electron in this field?A small piece of cork whose mass is 2.0 g is given a charge of 5.0107C . What electric field is needed to place the cork in equilibrium under the combined electric and gravitational forces?If the electric field is 100 N/C at a distance of 50 cm from a point charge q, what is the value of q?What is the electric field of a proton at the first Bohr orbit for hydrogen (r=5.291011m) ? What is the force on the electron in that orbit?(a) What is the electric field of an oxygen nucleus at a point that is 1010 m from the nucleus? (b) What is the force this electric field exerts on a second oxygen nucleus placed at that point?Two point charges, q1=2.0107C and q2=6.0108C , are held 25.0 cm apart. (a) What is the electric field at a point 5.0 cm from the negative charge and along the line between the two charges? (b)What is the force on an electron placed at that point?Point charges q1=50C and q2=25C are placed 1.0 m apart. (a) What is the electric field at a point midway between them? (b) What is the force on a charge q3=20C situated there?Can you arrange the two point charges q1=2.0106C and q2=4.0106C along the x-axis so that E=0 at the origin?Point charges q1=q2=4.0106C are fixed on the x-axis at x=3.0 m and x=3.0 m. What charge q must be placed at the origin so that the electric field vanishes at x=0,y=30m ?A thin conducting plate 1.0 m on the side is given a charge of 2.0106C . An electron is placed 1.0 cm above the center of the plate. What is the acceleration of the electron?Calculate the magnitude and direction of the electric field 2.0 m from a long wire that is charged uniformly at =4.0106C/m.Two thin conducting plates, each 25.0 cm on a side, are situated parallel to one another and 5.0 mm apart. If electrons are moved from one plate to the other, what is the electric field between the plates?The charge per unit length on the thin rod shown below is . What is the electric field at the point P? (Hint: Solve this problem by first considering the electric field dE at P due to a small segment dx of the rod, which contains charge dq=dx . Then find the net field by integrating dE over the length of the rod.)The charge per unit length on thin semicircular wire shown below is . What is the electric field at the point P?Two thin parallel conducting plates are placed 2.0 cm apart. Each plate is 2.0 cm on a side; one plate carries a net charge of 8.0 C , and the other plate carries a net charge of 8.0C . What is the charge density on the inside surface of each plate? What is the electric field between the plates?A thin conducing plate 2.0 m on a side is given a total charge of 10.0C . (a) What is the electric field 1.0 cm above the plate? (b) What is the force on an electron at this point? (c) Repeat these calculations for a point 2.0 cm above the plate. (d) When the electron moves from 1.0 to 2.0 cm above the plate, how much work is done on it by the electric field?A total charge q is distributed uniformly along a thin, straight rod of length L (see below). What is the electric field at P1?AtP2?Charge is distributed along the entire x-axis uniform density . How much work does the electric field of this charge distribution do on an electron that moves along the y-axis from y=a to y=b ?Charge is distributed along the entire x-axis withuniform density x and along the entire y-axis withuniform density y . Calculate the resulting electric field at(a) r=ai+bj and r=ck .A rod bent into the arc of a circle subtends an angle 2 at the center P of the circle (see below). If the rod is charged uniformly with a total charge Q, what is the electric field at P ?A pluton moves in the electric field E=200iN/C . (a) What are the force on and the acceleration of the proton? (b) Do the same calculation foran electron moving in this field.An electron and a proton, each starting from rest, are accelerated by the same uniform electric field of 200 N/C. Determine the distance and time for each particle to acquire a kinetic energy of 3.21016 J.A spherical water droplet of radius 25 m carries an excess 250 electrons. What vertical electric field is needed to balance the gravitational force on the droplet at the surface of the earth?A proton enters the uniform electric field produced by the two charged plates shown below. The magnitude of the electric field is 4.0105 N/C, and the speed of the proton when it enters is 1.5107 m/s. What distance d has the proton been deflected downward when it leaves the plates ?Shown below is a small sphere of mass 0.25 g that carries a charge of 9.01010 C. The sphere is attached to one end of a very thin silk string 5.0 cm long. The other end of the suing is attached to a large vertical conducting plate that has a charge density of 30106 C/m2. What is the angle that the string makes with the vertical?Two infinite rods, each carrying a uniform charge density , are parallel to one another and perpendicular to the plane of the page. (See below.) What is the electrical field at p1 ? At p2 ?Positive charge is distributed with a uniform density along the positive x-axis from r to , along the positive y-axis from r to , and along a 90( arc of a circle of radius r, as shown below. What is the electric field at O?From a distance of 10 cm, a proton is projected with a speed of v=4.0106 m/s directly at a large, positively charged plate whose charge density is =2.0105 C/m2. (See below.) (a) Does the proton reach the plate? (b) If not, how far from the plate does it turn around?A particle of mass m and charge q moves along a straight line away from a fixed particle of charge Q. When the distance between the two particles is r0 , q is moving with a speed v0 . (a) Use the work-energy theorem to calculate the maximum separation of the charges. (b) What do you have to assume about v0 to make this calculation? (c) What is the minimum value of v0 such that q escapes from Q?Which of the following electric field lines are incorrect for point charges? Explain why.In this exercise, you practice electric field lines. Make sure you represent both the magnitude and direction of the electric field adequately. Note that the number of lines into or out of charges is proportional to the charges. (a) Draw the electric field lines map for two charges +20C and 20C situated 5 cm from each other. (b) Draw the electric field lines map for two charges +20C and +20C situated 5 cm from each other. (c) Draw the electric field lines map for two charges +20C and 30C situated 5 cm from each other.Draw the electric field for a system of three particles of charges +1C , +2C , and 3C fixed at the corners of an equilateral triangle of side 2 cm.Two charges of equal magnitude but opposite sign make up an electric dipole. A quadrupole consists of two electric dipoles are placed anti-parallel at two edges of a square as shown. Draw the electric field of the charge distribution.Suppose the electric field of an isolated point charge decreased with distance as 1/r2+ rather than as 1/r2 . Show that it is then impossible to draw continuous field lines so that their number per unit area is proportional to E.Consider the equal and opposite charges shown below. (a) Show that at all points on the x-axis for which |x|a,EQal20x3. (b) Show that at all points on they-axis for which |y|a,EQal0y3.(a) What is the dipole moment of the configuration shown above? If Q=4.0C , (b) what is the torque on this dipole with an electric field of 4.0105N/Ci ? (c) What is the torque on this dipole with an electric field of 4.0105N/Ci ? (d) is the torque on this dipole with an field of 4.0105N/Cj ?A water molecule consists of two hydrogen atoms bonded with one oxygen atom. The bond angle between the two hydrogen atoms is 104( (see below). Calculate the net dipole moment of a water molecule that is placed in a uniform, horizontal electric field of magnitude 2.3108N/C . (You are missing some information for solving this problem; you will need to determine what information you need, and look it up.)Point charges q1=2.0C and q1=4.0C are located at r1=(4.0i2.0j+2.0k)m and r2=(8.0i+5.0j9.0k)m. What is the force of q2 on q1 ?What is the force on the 5.0C charge shown below?What is the force on the charge placed at the 2.0C center of the square shown below?Four charged particles are positioned at the corners of parallelogram as shown below. If q=5.0C and Q=8.0C , What is the net force on q?A charge Q is fixed at the origin and a second charge q moves along the x-axis, as shown below. How much work is done on q by the electric force q moves from x1 to x2 ?A charge q=2.0C is released from rest when it is 2.0 m from a fixed charge q=6.0C . What is the kinetic energy of q when it is 1.0 m from Q?What is the electric field at the midpoint M of the hypotenuse of the triangle shown below?Find the electric field at P for the charge configurations shown below.(a) What is the electric field at the lower-light-hand corner of the square shown below? (b) What is the force on a charge q placed at that point?Point charges are placed at the four corner of a rectangle as shown below: q1=2.0106C , q1=2.0106C , q3=4.0106C and q4=1.0106C . What is the electric field at P ?Three charges are positioned at the cornets of a parallelogram as shown below. (a) If Q=8.0C what is the electric field at the unoccupied comer? (b) What is the force on a 5.0C charge placed at this corner?119APA particle of charge q and mass m is placed at the center of a uniformly charged ring of total charge Q and radius R. The particle is displaced a small distance along the axis perpendicular to the plane of the ring and released. Assuming that the particle is constrained to move along the axis, show that the particle oscillates in simple harmonic motion with a frequency f=12qQ40mR3.Charge is distributed uniformly along the entire y-axis with a density y and along the positive x-axis from x=a to x=b with a density x. What is force between the two distributions?The circular are shown below carries a charge per unit length =0 cos . Where is measured from the x-axis. What is the electric field at the origin?Calculate the electric field due to a uniformly charged rod of length L, aligned with the x-axis with one end at the origin; at a point P on the z-axis.The charge unit length on the thin shown below is .What is the force on the point charge q? Solve this problem by first considering the electric force dF on q due to a small segment dx of the rod, which contains charge dx . Then, find the net force by integrating dF over the length of the rod.The charge per unit length on the thin rod shown here is . What is the electric force on the point charge q? (See the preceding problem.)The charge per unit length on the thin semicircular wire shown below is . What is the electric force on die point charge q? (See the preceding problems.)Check Your Understanding What angle should there be between the electric field and the surface show in Figure 6.11 in the previous example so that no electric flux passes through the surface?Check Your Understanding If the electric field in Example 6.4 is E=mxk . what is the through the rectangular area?Check Your Understanding Calculate the electric flux through the closed cubical surface for each distribution shown in Figure 6.20. Figure 6.20 A cubical Gaussian surface with various charge distributions.Check Your Understanding Check that the electric fields for the sphere reduce to the correct values for a point charge.Check Your Understanding A thin straight wire has a uniform linear charge density 0 . Find the electric field at a distance d from the wire, where d is much less than the length of the wire.Check Your Understanding How will the System above change if there are charged objects external to the sphere?Discuss how to orient a planar surface of area A in a uniform electric field of magnitude E0 to obtain (a) the maximum flux and (b) the minimum flux through the area.What are the maximum and minimum values of the flux in the preceding question?The net electric flux crossing a closed surface is always zero. True or false?The net electric flux crossing an open surface is never zero. True or false?Two concentric spherical surfaces enclose a point charge q. The radius of the outer sphere is twice that of the inner one. Compare the electric fluxes crossing the two surfaces.Compare the electric flux through the surface of a cube of side length a that has a charge q at its center to the flux through a spherical surface of radius a with a charge q at its center.(a) If the electric flux through a closed surface is zero, is the electric field necessarily zero at all points on the surface? (b) What is the net charge inside the surface?Discuss how Gauss's law would be affected if the electric field of a point charge did not vary as 1/r2.Discuss the similarities and differences between the gravitational field of a point mass m and the electric field of a point charge q.Discuss whether Gauss's law can be applied to other forces, and if so, which ones.Is the term in Gauss's law the electric field produced by just the charge inside the Gaussian surface?Reformulate Gauss's law by choosing the unit normal of the Gaussian surface to be the one directed inward.Would Gauss's law be helpful for determining the electric field of two equal but opposite charges a fixed distance apart?Discuss the role that symmetry plays in the application of Gauss's law. Give examples of continuous charge distributions in which Gauss's law is useful and not useful in determining the electric field.Discuss the restrictions on the Gaussian surface used to discuss planar symmetry. For example, is its length important? Does the cross-section have to be square? Must the end faces be on opposite sides of the sheet?Is the electric field inside a metal always zero?Under electrostatic conditions, the excess charge on a conductor resides on its surface. Does this mean that all the conduction electrons in a conductor are on the surface?