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Chapter 17 Current and Resistance

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Electric Current Let us look at the charges flowing perpendicularly to a surface of area A The electric current is the rate at which the charge flows through this surface The SI unit of current is Ampere (A): 1 A = 1 C/s André-Marie Ampère 1775 – 1836

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Electric Current The conventional direction of the current is the direction positive charge would flow In a common conductor (e.g., copper), the current is due to the motion of the negatively charged electrons It is common to refer to a moving charge as a mobile charge carrier A charge carrier can be positive or negative

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Current and Drift Speed Charged particles move through a conductor of cross- sectional area A and a charge carrier density n The total number of charge carriers: n A Δx The total charge is the number of carriers times the charge per carrier, q: ΔQ = (n A Δx) q The drift speed, v d, is the speed at which the carriers move: v d = Δx / Δt ΔQ = (n A v d Δt) q I = ΔQ / Δt = n q v d A

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Current and Drift Speed If the conductor is isolated, the electrons undergo random motion (due to collisions with the atoms) When an electric field is set up in the conductor, it creates an electric force on the electrons and hence a current The zigzag line represents the motion of charge carrier in a conductor

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Current and Drift Speed The drift speed is much smaller than the average speed between collisions When a circuit is completed, the electric field travels with a speed close to the speed of light Therefore, although the drift speed is on the order of 10 -4 m/s the effect of the electric field is felt on the order of 10 8 m/s

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Meters in a Circuit – Ammeter, Voltmeter An ammeter is used to measure current in line with the bulb – all the charge passing through the bulb also must pass through the meter A voltmeter is used to measure voltage (potential difference) – connects to the two ends of the bulb

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Resistance In a conductor, the voltage applied across the ends of the conductor is proportional to the current through the conductor The constant of proportionality is the resistance of the conductor – it arises due to collisions between the electrons carrying the current with the fixed atoms inside the conductor SI unit of resistance is ohm (Ω): 1 Ω = 1 V / A Georg Simon Ohm 1787 – 1854

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Ohm’s Law For certain materials, including most metals, the resistance remains constant over a wide range of applied voltages or currents – this statement has become known as Ohm’s Law ΔV = I R Materials that obey Ohm’s Law are said to be ohmic (the relationship between current and voltage is linear) Non-ohmic

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Resistivity The resistance of an ohmic conductor is proportional to its length, L, and inversely proportional to its cross-sectional area, A ρ is the constant of proportionality and is called the resistivity of the material (See table 17.1) L R A

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Chapter 17 Problem 12 Suppose that you wish to fabricate a uniform wire out of 1.00 g of copper. If the wire is to have a resistance R = 0.500 Ω, and if all of the copper is to be used, what will be (a) the length and (b) the diameter of the wire?

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Temperature Variation of Resistivity For most metals, resistivity increases with increasing temperature – the atoms vibrate with increasing amplitude so the electrons find it more difficult to pass through the atoms For most metals, resistivity increases approximately linearly with temperature over a limited temperature range ρ 0 – resistivity at some reference temperature T 0 (usually taken to be 20° C); α – is the temperature coefficient of resistivity

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Temperature Variation of Resistance Since the resistance of a conductor with uniform cross sectional area is proportional to the resistivity, the effect of temperature on resistance is similar

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Chapter 17 Problem 52 A given copper wire has a resistance of 5.00 Ω at 20.0°C while a tungsten wire of the same diameter has a resistance of 4.75 Ω at 20.0°C. At what temperature will the two wires have the same resistance?

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Superconductors Superconductors – a class of materials whose resistances fall virtually to zero below a certain temperature, T C (critical temperature) The value of T C is sensitive to chemical composition, pressure, and crystalline structure Once a current is set up in a superconductor, it persists without any applied voltage (since R = 0) One application is superconducting magnets

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Electrical Energy and Power In a circuit, as a charge moves through the battery, the electrical potential energy of the system is increased by ΔQ ΔV (the chemical potential energy of the battery decreases by the same amount) The charge moving through a resistor loses this potential energy during collisions with atoms in the resistor (the temperature of the resistor increases) When the charge returns to A, the net result is that some chemical energy of the battery has been delivered to the resistor and caused its temperature to rise

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Electrical Energy and Power The rate at which the energy is lost is the power From Ohm’s Law, alternate forms of power are The SI unit of power is Watt (W) (I must be in Amperes, R in ohms and ΔV in Volts) The unit of energy used by electric companies is the kilowatt-hour (defined in terms of the unit of power and the amount of time it is supplied): 1 kWh = 3.60 x 10 6 J

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Chapter 17 Problem 62 In a certain stereo system, each speaker has a resistance of 4.00 Ω. The system is rated at 60.0 W in each channel. Each speaker circuit includes a fuse rated at a maximum current of 4.00 A. Is this system adequately protected against overload?

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Chapter 17 Problem 40 A certain toaster has a heating element made of Nichrome resistance wire. When the toaster is first connected to a 120-V source of potential difference (and the wire is at a temperature of 20.0°C), the initial current is 1.80 A but the current begins to decrease as the resistive element warms up. When the toaster reaches its final operating temperature, the current has dropped to 1.53 A. (a) Find the power the toaster converts when it is at its operating temperature. (b) What is the final temperature of the heating element?

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Answers to Even Numbered Problems Chapter 17: Problem 2 (a) 5.57 × 10 −5 m / s (b) the drift speed is smaller

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Answers to Even Numbered Problems Chapter 17: Problem 4 3.4 × 10 21 electrons

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Answers to Even Numbered Problems Chapter 17: Problem 10 8.89 Ω

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Answers to Even Numbered Problems Chapter 17: Problem 18 (a) 2.8 × 10 8 A (b) 1.8 × 10 7 A

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Answers to Even Numbered Problems Chapter 17: Problem 22 6.3 Ω

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Answers to Even Numbered Problems Chapter 17: Problem 34 (a) $0.29 (b) $2.6

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