Chapter 17

Current and Resistance

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

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André-Marie Ampère1775 – 1836

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

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, vd, is the speed at which the carriers move: vd = Δx / Δt

ΔQ = (n A vd Δt) q

I = ΔQ / Δt = n q vd A

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

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 108 m/s

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

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

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Georg Simon Ohm1787 – 1854

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

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)

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A

Chapter 17Problem 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?

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

• ρ0 – resistivity at some reference temperature T0 (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 17Problem 22

A metal wire has a resistance of 10.00 Ω at a temperature of 20°C. If the same wire has a resistance of 10.55 Ω at 90°C, what is the resistance of the wire when its temperature is −20°C?

Superconductors• Superconductors – a class of materials

whose resistances fall to virtually zero below a certain temperature, TC (critical temperature)

• The value of TC 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

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

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 106 J

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Chapter 17Problem 60

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?

Chapter 17Problem 28

A toaster rated at 1 050 W operates on a 120-V household circuit and a 4.00-m length of nichrome wire as its heating element. The operating temperature of this element is 320°C. What is the cross-sectional area of the wire?

Answers to Even Numbered Problems

Chapter 17:

Problem 2

5.21 × 10−5 m / s

Answers to Even Numbered Problems

Chapter 17:

Problem 10

500 mA

Answers to Even Numbered Problems

Chapter 17:

Problem 18

(a) 2.8 × 108 A(b) 1.8 × 107 A

Answers to Even Numbered Problems

Chapter 17:

Problem 20

1.4 × 103 °C

Answers to Even Numbered Problems

Chapter 17:

Problem 32

(a) $0.29 (b) $2.6