Lecture 10-1 Physics 241 – Question 1 – September 22, 2011 Below, all capacitors are identical. Which combination has the smallest equivalent capacitance? a b d c e Lecture 10-2 Batteries • We use batteries as devices that provide direct currents in circuits • The voltage written on it is the potential difference that it can provide to a circuit • You will also find its rating in units of mAh – This rating provides information on the total charge that a single battery can deliver over its lifetime • The quantity mAh is another unit of charge: 3 1 mAh (10 A)(3600 s) 3.6 As 3.6 C Lecture 10-3 Circuits In this circuit, electrons flow around the circuit counterclockwise. (The conventionally defined current is clockwise; remember, electrons are negative charges.) The electrons can’t disappear so the current requires a whole loop! + Chemical action pumps electrons from the positive terminal (+) to the negative terminal (-) in the battery. The emf (electromotive force, or electric field) pushes electrons around the wire from (-) to (+). Direct electric current flows in one direction lightbulb Lecture 10-4 Microscopic View of Electric Current in Conductor All charges move with some velocity ve A random motion with high speeds (O(106)m/s) but with a drift in a certain direction on average if E is present Why random motion? • thermal energy • scattering off each other, defects, ions, … Drift velocity vd [O(10-4 m/s)] is orders of magnitudes less than the actual velocity of charges [O(106 m/s)]. Lecture 10-5 Drift Velocity (1) • Consider a conductor with cross sectional area A and electric field E • Suppose that there are n electrons per unit volume • The negatively charged electrons will drift in a direction opposite to the electric field • We assume that all the electrons have the same drift velocity vd and that the current density J is uniform. • In a time interval dt, each electron moves a distance vd dt • The volume that will pass through area A is then Avd dt; the number of electrons is dn = nAvd dt Lecture 10-6 Drift Velocity (2) The volume that will pass through area A is then Avddt; the number of electrons is dn = nAvd dt • Each electron has charge e so that the charge dq that flows through the area A in time dt is • So the current is dq I nevd A dt … and the current density is I J nevd A • For a positive charge, the current density and the drift velocity are parallel vectors, pointing in the same directions. As vectors: J nevd • For electrons: J nevd Lecture 10-7 Drift Velocity (3) • Consider a wire carrying a current • The physical current carriers are negatively charged electrons • These electrons are moving to the left in this drawing • However, the electric field, current density and current are directed to the right Comments Electrons are negative charges! On top of the coherent motion the electrons have random (thermal) motion. Lecture 10-8 The Ampere • The unit of current is coulombs per second, which has been given the unit ampere, named after French physicist AndréMarie Ampère, (1775-1836) • The ampere is abbreviated as A and is defined by 1C 1A 1s • Some typical currents are – – – – Flashlight - 1 A The starter motor in a car - 200 A iPod - 50 mA Lightning strike (for a very short time) – 100,000 A Lecture 10-9 Example: Current Through a Wire • The current density J in a cylindrical wire of radius R = 2.0 mm is uniform across a cross section of the wire and has the value J = 2.0 105 A/m2. Question: What is the current i through the outer portion of the wire between radial distances R/2 and R? Answer: • J = current per unit area = dI/dA Area A’ (outer portion) 2 3R 2 A R2 R / 2 9.424 10 6 m 2 4 Current through A’ I JA 2.0 105 A/m2 9.424 106 m2 1.9 A R Lecture 10-10 Ohm’s Law Current-Potential (I-V) characteristic of a device may or may not obey Ohm’s Law: I V ( J E) or V IR with R constant V V Resistance R I A (ohms) tungsten wire gas in fluorescent tube diode Lecture 10-11 Resistance and Resistivity Resistance (definition) V R I R I V Constant R Ohm’s Law R (in ) L R A resistivity (in m) Material (m) Ag 1.6 x10-8 Cu 1.7x10-8 Si 6.4x102 glass 1010 ~ 1014 wood 108 ~ 1011 Resistivity at 20C Lecture 10-12 Physics 241 – Question 2 – September 22, 20011 Which one(s) of the wires (made of the same material) below have the largest resistance in the vertical direction? A a) b) c) d) e) 1 and 3 2 only 3 and 4 4 only 2 and 4 L 1 2A 2 3 2L A/2 L 4 Lecture 10-13 Example: Resistance of a Copper Wire (1) • Standard wires that electricians put into residential housing have fairly low resistance. Question: What is the resistance of a length of 100 m of standard 12gauge copper wire, typically used in household wiring for electrical outlets? Answer: • The American Wire Gauge (AWG) size convention specifies wire cross sectional area on a logarithmic scale. • A lower gauge number corresponds to a thicker wire. • Every reduction by 3 gauges doubles the cross-sectional area. (36 AWG ) / 39 d 0.127 92 mm Lecture 10-14 Example: Resistance of a Copper Wire (2) • The formula to convert from the AWG size to the wire diameter is (36 AWG ) / 39 d 0.127 92 mm • So a 12-gauge copper wire has a diameter of 2.05 mm • Its cross sectional area is then A d 3.3 mm 1 4 2 2 • Look up the resistivity of copper in the table … L 100 m -8 R (1.72 10 m) 0.52 -6 2 A 3.3 10 m Lecture 10-15 Physics 241 – Question 3 – September 22, 2011 What is the resistance of a copper wire that has length L=70.0 m and diameter -8 1.72 10 m d=2.60 mm ? a) 0.119 b) 0.139 c) 0.163 d) 0.190 e) 0.227 Lecture 10-16 Demo - Resistance and Resistivity Resistance (definition) V R I R I V Ohm’s Law Constant R R (in ) L R A I new 2 I R new R 2 (m) Ag 1.6 x10-8 Cu 1.7x10-8 Si 6.4x102 glass 1010 ~ 1014 wood 108 ~ 1011 Resistivity at 20C resistivity (in m) L L 2 Material V I R I new V V 2 ( R / 2) R Lecture 10-17 • • • • • Resistors In many electronics applications one needs a range of resistances in various parts of the circuits For this purpose one can use commercially available resistors Resistors are commonly made from carbon, inside a plastic cover with two wires sticking out at the two ends for electrical connection The value of the resistance is indicated by four colorbands on the plastic capsule The first two bands are numbers for the mantissa, the third is a power of ten, and the fourth is a tolerance for the range of values Lecture 10-18 Resistor Color Codes Example: Colors (left to right) red, yellow, green, and gold Using our table, we can see that the resistance is 24×105 = 2.4 M with a tolerance of 5% Lecture 10-19 Temperature Dependence of Resistivity 0 1 (T T0 ) • Usually T0 is 293K (room temp.) • Usually > 0 (ρ increases as T ) Material 0 (m) (K-1) Ag 1.6x10-8 3.8x10-3 Cu 1.7x10-8 3.9x10-3 Si 6.4x102 -7.5x10-2 glass 1010 ~ 1014 sulfur 1015 Copper Lecture 10-20 Electric Current and Joule Heating electron gas • Free electrons in a conductor gains kinetic energy due to an externally applied E. • Scattering from the atomic ions of the metal and other electrons quickly leads to a steady state with a constant current I. Transfers energy to the atoms of the solid (to vibrate), i.e., Joule heating. Mean drift of electrons, i.e., current Lecture 10-21 Energy in Electric Circuits • Steady current means a constant amount of charge Q flows past any given cross section during time t, where I= Q / t. Energy lost by Q is V U Q (Va Vb ) I t V => heat So, Power dissipation = rate of decrease of U = dU P IV I 2 R V 2 / R dt