Chapter 24 Electric Current

The electric current I is the rate of flow of charge through some region of space The SI unit of current is Ampere (A): 1 A = 1 C/s Let us look at the charges flowing perpendicularly to a surface of area A The average current: The instanteneous current: André-Marie Ampère 1775 – 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, v d, is the speed at which the carriers move: v d = Δx / Δt ΔQ = (n A v d Δt) q I ave = ΔQ / Δt = n q v d 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 m/s the effect of the electric field is felt on the order of 10 8 m/s

Current Density The current density J of a conductor is defined as the current per unit area: If the current density is uniform and A is perpendicular to the direction of the current then this expression is valid: J = I / A = nqv d J has SI units of A/m 2 The current density is in the direction of the positive charge carriers

Conductivity A current density and an electric field are established in a conductor whenever a potential difference is maintained across the conductor For some materials, the current density is directly proportional to the field: J = σ E The coefficient of proportionality, σ, is called the conductivity of the conductor

Ohm’s Law Ohm’s law states that for many materials, the conductivity σ is a constant that is independent of the electric field producing the current Most metals obey Ohm’s law Ohm’s law is not a fundamental law of nature, but an empirical relationship valid only for certain materials Georg Simon Ohm 1787 – 1854

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

Resistivity The inverse of the conductivity is the resistivity of the material (see table 24.1): ρ = 1 / σ Resistivity has SI units of ohm-meters (Ω. m) The resistance of an ohmic conductor is proportional to its length, L, and inversely proportional to its cross-sectional area, A: L R A 

Ohmic and Nonohmic Materials Materials that obey Ohm’s Law are said to be ohmic (the relationship between current and voltage is linear, and the resistance is constant over a wide range of voltages) Not all materials follow Ohm’s law Materials that do not obey Ohm’s law are said to be nonohmic Non-ohmic

Resistance and Resistivity, Summary Every material has a characteristic resistivity that depends on the properties of the material and on temperature, i.e., resistivity is a property of substances The resistance of a material depends on its geometry and its resistivity, i.e., resistance is a property of an object An ideal conductor would have zero resistivity An ideal insulator would have infinite resistivity

Chapter 24 Problem 27 A uniform wire of resistance R is stretched until its length doubles. Assuming its density and resistivity remain constant, what’s its new resistance?

A Model for Electrical Conduction Treat a conductor as a regular array of atoms plus a collection of free electrons – conduction electrons In the absence of an electric field, the motion of the conduction electrons is random, and their speed is on the order of 10 6 m/s When an electric field is applied, the conduction electrons are given a drift velocity

A Model for Electrical Conduction We assume: 1) The electron’s motion after a collision is independent of its motion before the collision 2) The excess energy acquired by the electrons in the electric field is lost to the atoms of the conductor when the electrons and atoms collide (causing the temperature of the conductor to increase)

A Model for Electrical Conduction The force experienced by an electron is From Newton’s Second Law, the acceleration is Applying a motion equation Since the initial velocities are random, their average value is zero If  is the average time interval between successive collisions, then

A Model for Electrical Conduction The current density: Using Ohm’s Law: The conductivity and the resistivity do not depend on the strength of the field (characteristic of a conductor obeying Ohm’s Law)

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

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

Chapter 24 Problem 59 The resistivity of copper as a function of temperature is given approximately by ρ = ρ 0 [1 + α (T - T 0 )], where ρ 0 is Table 24.1’s entry for 20°C, T 0 = 20°C, and α = 4.3 × °C -1. Find the temperature at which copper’s resistivity is twice its room temperature value.

Residual Resistivity For some metals, the resistivity is nearly proportional to the temperature A nonlinear region always exists at very low temperatures, and the resistivity usually reaches some finite value as the temperature approaches absolute zero The residual resistivity near 0 K is caused primarily by the collisions of electrons with impurities and imperfections in the metal

Superconductors Superconductors – a class of materials whose resistances fall to virtually 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

Semiconductors Semiconductors are materials that exhibit a decrease in resistivity with an increase in temperature, i.e. α is negative The reason – an increase in the density of charge carriers at higher temperatures

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 10 6 J

Chapter 24 Problem 29 A 4.5-W flashlight bulb draws 750 mA. (a) At what voltage does it operate? (b) What’s its resistance?

Answers to Even Numbered Problems Chapter 24: Problem × 10 5 C

Answers to Even Numbered Problems Chapter 24: Problem kW

Answers to Even Numbered Problems Chapter 24: Problem km

Answers to Even Numbered Problems Chapter 24: Problem 54 (a) 8.70 kA (b) 15.1%