Electric Current Electrons will move between two oppositely-charged conductors when a conducting wire connects them Electric field exerts forces on conduction electrons, causing them to move Flow of electrons stops when the conductors are at the same potential and electric field is zero A net flow of charge is an electric current Charges of like sign move in one direction in an electric field E (negative charges move opposite E, positive charges with E) Current = rate of flow of electric charge past a point – + + – – + + – electrons – + + + – – + –

Electric Current Definition of current I DQ is amount of charge that passes through the cross-sectional area in time Dt SI unit of current is the ampere (A): 1 A = 1 C/s Small currents are measured in mA, mA, etc. Direction of current flow is taken as direction positive charges would flow Standard established by Benjamin Franklin This is opposite the flow of electrons (the mobile charge carriers) in a conductor Current flows in direction of the electric field present in the region Negative charge moving in one direction has same macroscopic effect as positive charge moving in other direction Moving electrons

CQ 1: Rank the magnitudes of the currents in the four regions shown below, from lowest to highest. d a c b a c b d c a b d d b c a a b c d

Circuits For currents to flow, a complete circuit is required Ammeters measure current while in series with circuit; voltmeter measures voltage across a device (from College Physics, Giambattista et al.)

Microscopic View of Current in a Conductor In the absence of an applied electric field, conduction electrons in a metal are in constant random motion at high speed (~ 106 m/s in copper) Their average velocity is zero due to many collisions with each other and with atoms In a uniform electric field, electrons have a uniform acceleration between collisions (opposite to the field) Result is a small but non-zero drift speed (vd) in the direction of the electric force (~0.1 mm/s for Cu wire) Think of air molecules in a gentle breeze Current and drift speed related by: n = number of mobile charge carriers / unit volume e– Drift A = cross-sectional area of current flow; q = charge / carrier (from College Physics, Giambattista et al.)

Example Problem #17.7 A 200-km-long high-voltage transmission line 2.0 cm in diameter carries a steady current of 1000 A. If the conductor is copper with a free charge density of 8.5  1028 electrons per cubic meter, how many years does it take one electron to travel the full length of the cable? Solution (details given in class): 27 years

Resistance and Ohm’s Law For many materials, I is proportional to DV The proportionality constant, electrical resistance, is defined as the ratio of DV to I: R is measured in ohms (W) Think of water flow down a river and the resistive effect of rocks and other obstructions (flow rate analogous to I) In ohmic materials, R is constant over a wide range of values of DV and I In that case, there is a linear relationship between DV and I Ohm’s Law: In non-ohmic materials, R changes with DV or I Semiconductor diodes Transistors ohmic non–ohmic

Resistivity Resistance in a circuit arises due to collisions between electrons and fixed atoms in a conductor Collisions inhibit movement of electrons (like friction) Resistance depends on the size and shape of the conductor Proportional to its length l and inversely proportional to its cross-sectional area A Think of flow of water through a pipe As l increases, there is more friction between water and walls of pipe and resistance to flow goes up As A increases, the pipe can transport more fluid in a given time interval, so the resistance goes down For an ohmic conductor: r = constant = resistivity of material (units of Wm) – see Table 17.1 in textbook

Resistors A resistor is a circuit element designed to have a known resistance They are commonly found in circuits as little cylinders with color-coded bands signifying their resistance Current flows through resistor in direction of electric field, which points from higher to lower potential As current moves through a resistor, voltage drops by an amount DV = IR due to drop in electrical potential energy Analogous to water flow: water flows downhill toward lower potential energy ↔ electric current flows toward lower potential Resistors indicated by zig–zag pattern in circuit (from College Physics, Giambattista et al.)

Example Problem #17.12 Suppose 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 W, and if all the copper is to be used, what will be (a) the length and (b) the diameter of the wire? Solution (details given in class): 1.82 m 0.280 mm

Temperature Variation of Resistance Two primary factors determine the resistivity of a metal Number of conduction electrons per unit volume Rate of collisions between an electron and an ion The second of these factors is sensitive to temperature At a higher temperature, atoms vibrate with larger amplitudes As a result, electrons collide more frequently with the atoms and acquire a smaller drift speed (current is smaller for a given electric field) As temperature goes up, resistivity goes up (conductors) Resistivity vs. temperature: Resistance vs. temperature: a = temperature coefficient of resistivity (see Table 17.1) (T0 = 20°C) (r0 = r @ 20°C) (R0 = R @ 20°C)

Electrical Energy and Power Consider the following circuit at right: As positive charge q moves from A to B (through battery), electrical potential energy of system increases by qDV From C to D (through resistor), this charge loses this energy through collisions with atoms in the resistor (no net KE gain after PE loss) This causes atoms in resistor to vibrate more, increasing the internal energy of the system (resistor temp. rises) The rate at which charges lose energy is DU / Dt = qDV / Dt = IDV which is rate at which resistor material gains internal energy Power = rate at which energy is delivered to resistor: Alternatively, from DV = IR for a resistor:

CQ 2: What is the energy required to operate a 60 W light bulb for 1 minute? 1 J 60 J 360 J 3600 J

CQ 3: Interactive Example Problem: Resistor Power! Which graphs show the correct relationship between the indicated quantities? Graphs A and D Graphs B and C Graphs A, D, and E Graphs B, C, and E (PHYSLET #9.5.3, copyright Prentice Hall, 2001)

Example Problem #17.30 A toaster rated at 1050 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? Solution (details given in class): 4.90  10–7 m2

Example Problem #17.37 The heating element of a coffeemaker operates at 120 V and carries a current of 2.00 A. Assuming that the water absorbs all of the energy converted by the resistor (heating element), calculate how long it takes to heat 0.500 kg of water from room temperature (23.0°C) to the boiling point. Solution (details given in class): 11.2 min

Example Problem #17.57 You are cooking breakfast for yourself and a friend using a 1200-W waffle iron and a 500-W coffeepot. Usually, you operate these appliances from a 110-V outlet for 0.500 h each day. (a) At 12 cents per kWh, how much do you spend to cook breakfast during a 30.0 day period? (b) You find yourself addicted to waffles and would like to upgrade to a 2400-W waffle iron that will enable you to cook twice as many waffles during a half-hour period, but you know that the circuit breaker in your kitchen is a 20-A breaker. Can you do the upgrade? Solution (details given in class): $3.06 No. The circuit needs at least 26.4 A.