CHAPTER 20: ELECTRIC CURRENT, RESISTANCE, AND OHM S LAW

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1 College Physics Student s Manual Chapter CHAPTE : ELECTIC CUENT, ESISTANCE, AND OHM S LAW. CUENT. What is the current in milliamperes produced by the solar cells of a pocket calculator through which 4. C of charge passes in 4. h? Using the equation I, we can calculate the current given in the charge and the time, remembering that A C/s :. C h I h 6 s 4 4 A.78 ma 7. (a) A defibrillator sends a 6.- A current through the chest of a patient by applying a,- potential as in Figure.8. What is the resistance of the path? (b) The defibrillator paddles make contact with the patient through a conducting gel that greatly reduces the path resistance. Discuss the difficulties that would ensue if a larger voltage were used to produce the same current through the patient, but with the path having perhaps times the resistance. (Hint: The current must be about the same, so a higher voltage would imply greater power. Use this equation for power: P I.) (a) Using the equation I, we can calculate the resistance of the path given the current and the potential: I, so that,.667 Ω.67 kω I 6. A (b) If a times larger resistance existed, keeping the current about the same, the 48

2 College Physics Student s Manual Chapter power would be increased by a factor of about, causing much more energy to be transferred to the skin, which could cause serious burns. The gel used reduces the resistance, and therefore reduces the power transferred to the skin.. A large cyclotron directs a beam of He nuclei onto a target with a beam current of. ma. (a) How many He nuclei per second is this? (b) How long does it take for. C to strike the target? (c) How long before. mol of He ++ nuclei strike the target? (a) Since we know that a He ion has a charge of twice the basic unit of charge, we can convert the current, which has units of C/s, into the number of He ions per second: (. He C/s) ( He C) nuclei/s (b) Using the equation I, we can determine the time it takes to transfer. C of charge, since we know the current: I, so that. C 4. s 4 I. A (c) Using the result from part (a), we can determine the time it takes to transfer. mol of He (. mol He ions by converting units: ) mol ions s He ions 8 s. OHM S LAW: ESISTANCE AND SIMPLE CICUITS 9. Calculate the effective resistance of a pocket calculator that has a.- battery and through which. ma flows. 49

3 College Physics Student s Manual Chapter Using the equation I, given the voltage and the current, we can determine the. resistance: I, so that 6.7 Ω 6.7 kω 4 I. A. ESISTANCE AND ESISTIITY. The diameter of - gauge copper wire is 8. mm. Find the resistance of a.- km length of such wire used for power transmission. We know we want to use the equation ρl, so we need to determine the radius A for the cross- sectional area of A πr. Since we know the diameter of the wire is 8. mm, we can determine the radius of the wire: 8. m r d 4.6 m. 8 We also know from Table. that the resistivity of copper is.7 Ωm. These values give a resistance of: 8 ρl (.7 Ω m)(. A π (4.6 m) m). Ω. Of what material is a resistor made if its resistance is 4.% greater at C than at. C? We can use the equation ( + α ΔT ) to determine the temperature coefficient of resistivity of the material. Then, by examining Table., we can determine the type of material used to make the resistor. Since + αδt ).4, for a ( temperature change of 8. C, we can determine α :.4.4 α ΔT.4 α. / C ΔT 8. C So, based on the values of in Table., the resistor is made of iron.

4 College Physics Student s Manual Chapter 7. (a) Digital medical thermometers determine temperature by measuring the resistance of a semiconductor device called a thermistor (which has α.6 / C ) when it is at the same temperature as the patient. What is a patient s temperature if the thermistor s resistance at that temperature is 8.% of its value at 7. C (normal body temperature)? (b) The negative value for α may not be maintained for very low temperatures. Discuss why and whether this is the case here. (Hint: esistance can t become negative.) (a) [ + α ( T 7. C) ].8, where α.6 / C. Dividing by,+ α ( T 7. C). 8, so that.8 α ( T 7. C),.8.8 giving ( T 7. C). C. α.6 / C Finally, T 7. C +. C 4. C (b) If α is negative at low temperatures, then the term [ + ( T 7. C) ] α can become negative, which implies that the resistance has the opposite sign of the initial resistance, or it has become negative. Since it is not possible to have a negative resistance, the temperature coefficient of resistivity cannot remain negative to low temperatures. In this example the magnitude is α 7. C T 9. Unreasonable esults (a) To what temperature must you raise a resistor made of constantan to double its resistance, assuming a constant temperature coefficient of resistivity? (b) To cut it in half? (c) What is unreasonable about these results? (d) Which assumptions are unreasonable, or which premises are inconsistent? (a) Using the equation (+ α ΔT ) and setting the resistance equal to twice the initial resistance, we can solve for the final temperature: ( C) (+ α ΔT ) αδt α( T T ). T So the final temperature will be:

5 College Physics Student s Manual Chapter T T C T C 6 α / C (b) Again, using the equation (+ α ΔT ), we can solve for the final temperature when the resistance is half the initial resistance: ( + αδt ) α( T T ), so the final temperature will be:. T T. C or T. C. 6 / C (c) In part (a), the temperature is above the melting point of any metal. In part (b) the temperature is far below K, which is impossible. (d) The assumption that the resistivity for constantan will remain constant over the derived temperature ranges in part (a) and (b) above is wrong. For large temperature changes, α may vary, or a non- linear equation may be needed to find ρ..4 ELECTIC POWE AND ENEGY 4. erify that the units of a volt- ampere are watts, as implied by the equation P I. Starting with the equation P I, we can get an expression for a watt in terms of P W, [ I ] A. (C/s)(J/C) J/s W, so that a watt is current and voltage: [ ] equal to an ampere - volt.. A cauterizer, used to stop bleeding in surgery, puts out. ma at. k. (a) What is its power output? (b) What is the resistance of the path? (a) Using the equation P I, we can determine the rms power given the current and the voltage: P I (. A)(. ). W

6 College Physics Student s Manual Chapter (b) Now, using the equation result from part (a): I I, we can solve for the resistance, without using the 4 ( 6 I.. ) 7. A Ω 7. MΩ Note, this assume the cauterizer obeys Ohm's law, which will be true for ohmic materials like good conductors. 9. Integrated Concepts Cold vaporizers pass a current through water, evaporating it with only a small increase in temperature. One such home device is rated at. A and utilizes AC with 9.% efficiency. (a) What is the vaporization rate in grams per minute? (b) How much water must you put into the vaporizer for 8. h of overnight operation? (See Figure.4.) (a) From the equation P I, we can determine the power generated by the vaporizer. P I (. A)( ) 4 J/s.4 kj/s and since the vaporizer has an efficiency of 9.%, the heat that is capable of vaporizing the water is Q (.9) Pt. This heat vaporizes the water according to the equation Q ml v, where L v 6 kj/kg, from Table 4., so that (.9) Pt ml, or v (.9) Pt m L v (.9)(.4 kj/s)(6. s) 6 kj/kg.6 kg.6 g/min (b) If the vaporizer is to run for 8. hours, we need to calculate the mass of the water by converting units: 6 min m required (.6 g/min)(8. h).9 g.9 kg h In other words, making use of Table. to get the density of water, it requires m L.9 kg.9 L of water to run overnight. kg m

7 College Physics Student s Manual Chapter 6. Integrated Concepts A light- rail commuter train draws 6 A of 6- DC electricity when accelerating. (a) What is its power consumption rate in kilowatts? (b) How long 4 does it take to reach. m/s starting from rest if its loaded mass is. kg, assuming 9.% efficiency and constant power? (c) Find its average acceleration. (d) Discuss how the acceleration you found for the light- rail train compares to what might be typical for an automobile. (a) Using the equation P I, we can determine the power generated: P I (6 A)(6 ) 4. W 4 kw (b) Since the efficiency is 9.%, the effective power is P (.9) P 89. kw. Then we can calculate the work done by the effective train: W P ) t. Setting that equal to the change in kinetic energy gives us an ( eff expression for the time it takes to reach. m/s from rest: W mv mv ( Peff ) t, so that (/ ) mv t (/ ) mv P eff 4.(. kg)(. m/s).89 W v. m/s (c) We recall that v v + at, so that a. 74 m/s t 7. s 7. s 7. s (d) A typical automobile can go from to 6 mph in seconds, so that its v 6 mi/hr hr 69 m acceleration is: a. 7 m/s t s 6 s mi Thus, a light- rail train accelerates much slower than a car, but it can reach final speeds substantially faster than a car can sustain. So, typically light- rail tracks are very long and straight, to allow them to reach these faster final speeds without decelerating around sharp turns. 4

8 College Physics Student s Manual Chapter. ALTENATING CUENT ESUS DIECT CUENT 7. Certain heavy industrial equipment uses AC power that has a peak voltage of 679. What is the rms voltage? Using the equation rms, we can determine the rms voltage, given the peak 679 voltage: rms What is the peak power consumption of a - AC microwave oven that draws. A? Using the equation P ave Irms, we can calculate the average power given the rms rms values for the current and voltage: P I (. A)( ). kw ave rms rms Next, since the peak power is the peak current times the peak voltage: P I ( I ) Pave.4 kw 8. Find the time after t when the instantaneous voltage of 6- Hz AC first reaches the following values: (a) / (b) (c). (a) From the equation sin πft, we know how the voltage changes with time for an alternating current (AC). So, if we want the voltage to be equal to know that, we sin (.) sin π ft, so that: sin π ft, or t. Since we πf have a frequency of 6 Hz, we can solve for the time that this first occurs (remembering to have your calculator in radians):

9 College Physics Student s Manual Chapter t.6 rad.9 π (6 Hz) s.9 ms (b) Similarly, for : sin π ft, so that sin π / rad t 4.7 s 4.7 ms πf π (6 Hz) (c) Finally, for : sin π ft, so that π ft, π, π,..., or for the first time after t : π ft π, or t (6 Hz) 8. s 8. ms.6 ELECTIC HAZADS AND THE HUMAN BODY 89. Foolishly trying to fish a burning piece of bread from a toaster with a metal butter knife, a man comes into contact with - AC. He does not even feel it since, luckily, he is wearing rubber- soled shoes. What is the minimum resistance of the path the current follows through the person? From Table., we know that the threshold of sensation is I. ma. The minimum resistance for the shock to not be felt will occur when I is equal to this value. So, using the equation I,we can determine the minimum resistance for AC current:. Ω I. A 6