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Transcript
Lecture 08 Current & Resistance October 3 or 5, 2005 New Topic Current and Resistance Commercial Resistors Color Coded Conductors In the past we decided that in a conductor The Electric Field is ZERO because we postulated that no charges were to move. Static Situation Any electric field must be at the surface That field must be normal to the surface Let’s look at another situation. Consider a conductor V1 V2 Electric Field Walla ….. a CIRCUIT The mobile electrons can “move” under the influence of an electric field. We then have a “current” (to be defined) flowing in the wire. But WAIT … how can we????? Only if … Symbol From the Past Q Flows and then stops. Vo + - A Different Situation Vo circuit Charge will begin to flow through the short. Charge can flow back into the battery and discharge it. Wire can get warm, emit light or even burn our (fuse). The FLOW of charge is defined as a current. Current is a good thing Or a BAD thing Franklin’s Impact on Physics!!! ELECTRONS CURRENT Positive charge will leave the battery from the positive terminal and flow through an external circuit to the negative terminal. Electrons will go the other way. Current is defined as the flow of POSITIVE CHARGE. + charge does not normally flow in a wire. Definition Current is the total amount of charge that flows through a “wire” in one second. Current is measured in Coulombs per second. A current of one coulomb per second is defined as an AMPERE. (Amp.) CONCISE DEFINITION: CURRENT dq i dt dq idt t2 q i (t )dt t1 Current will flow throughout the cross-section of the wire (usually). Current through aa’ is the same as the current through bb’ and cc’. What is DIFFERENT between aa’ and bb’ with respect to current? What’s Different?? i=5 amps A=.1 m2 A=0.05 m2 CURRENT DENSITY current j unit area J=5 amps/.05 m2 = 100 amps/m2 J=5 amps / .1 m2 = 50 amps/m2 Question: The current density in a wire of radius R is given by I0r. What is the total current flowing through the wire? Overhead Sol. Current Current can’t “pile up” at a point n a circuit. Example For BOTH i0 i1 i2 First introduction to Kirchoff’s Node Equation Consider 3A ^ 5A 2A v 6A 8A Question: A 5 Amp current is set up in a circuit for 6 minutes by a 6 Volt Battery. How much chemical energy is provided by the battery? Vo + CIRCUIT OF SOME SORT WORK (Energy) per unit Charge = qV - coul 60 sec 5amps 6 min 5 6 min sec min 30 60coul 1800coulombs qV 1800coul 6Volts 10 KJ Definition wire i V High Low Current increases with Potential Difference (V) V R i Ohm’s Law Volt UNIT 1 1 OHM Ampere SYMBOL Observations Wires and Resistors are made from conducting materials. These materials have some fundamental properties associated with them. Electrons are attached to atoms. Outer electrons weakly bound Small Force (Applied Electric Field) can easily push them. They bump into things which retard their motion. The more things that retard their motion, the more difficult it is to push a current via an applied potential difference. Thus, the resistance goes up. What kinds of things cause resistance? Sudden Constrictions in the conductor including bends! Underlying structure Amorphous Crystalline Defects Impurities Thermal Collisions Temperature Fundamental Property RESISTANCE vs. RESISTIVITY Consider a wire made of some material. Resistance is a property of the wire itself … the material and the shape. New Quantity: RESISTIVITY is a property of the material itself regardless of its shape. Consider our wire: L A V What would happen to the current if we Increased the voltage? Increased the Area? Increased the length? Summary VA i L or V L A is the resistivity i iR R is called the RESISTANCE and is measured in OHMS () Define CONDUCTIVITY s s 1 m 1 Resistivity – Ohm-Meters Silver Copper Aluminum Tungsten Platinum Silicon-Pure Glass Fused Quartz 1.62 x 10-8 1.69 2.65 5.25 10.6 2.5 x 103 1010 to 1014 ~1016 WIRES American Wire Gauge Diameter - mm 30 gauge 0.255 24 0.511 18 (typical household) 1.024 14 1.628 12 2.05 BACK TO OHM V=iR i=V/R Not EVERYTHING is a resistor The Semiconductor Diode Effect of Temperature Temperature ( 0 ) 0 (T T0 ) 0 0 (T T0 ) 0 (1 T ) Linear over a limited temperature range. QuestionFor an 18 gauge wire of length L to have a resistance of 1 ohm what must L be if the material is copper? L R A R 1 .0 L A d 2 / 4 1.69 10 6 8 d 1mm 10 m 2 2 2 (d for 18 gauge is about 1 mm) A current of 6.5 Amps exists in a 9 Ohm resistor for 5 minutes. How many coulombs and how many electrons pass through the resistor in this time? Coulombs: 6.5 COULOMBS per Second for 5 minutes 6.5C x 5 min X 60 sec/min = 1950 coulombs Number of electrons = # coulombs / electron charge = 1950 / 1.6 x 10-19 = 1.22 x 10 +22 Microscopic Theory J and E i JA i V V 1V J A RA L L A A V J s sE L let A 0 and J sE Consider a wire N n d tA Ne current ne d A t J ne d Micro-View “Resistivity” depends on the material and is the mean time between collisions ease of motion – mobility resistance to motion - scattering eE a a m eE d a m ne 2 1 J nevd E sE E m ne 2 s m Power Battery supplies energy to the resistor which, in turn, dissipates it in the form of heat. + V i - E L E C T R O N s Work done on charge Q = Q x V Work / time POWER P QV Q 2 P V iV i iR i R t t REMEMBER: P=iV and P=i2R The Spectrum of Conductors ENGINEERED MATERIALS!!! Semiconductors When 105 volts are applied across a wire that is 12 meters long and has a 0.30 mm radius, the current density us 1.7 x 10 4 A/m2. What is the resistivity of the wire?? L R A so that From the current density we can find the CURRENT. RA L I = 4.8 ma i=JA = 1.7 x 104 amp/m2 X ( X 0.32) mm2 X (1m/1000mm)2 R= V/i = 105 volts / (4.8 x 10-3) amps = 2.18 x 104 ohms = 2.18 x 104 ohms x 3 x 10-7 m2 / 12m = 0.005 ohm-meters The figure below gives the electrical potential V(x) along a copper wire carrying a uniform current, from a point at higher potential (x=0m) to a point at a lower potential (x=3m). The wire has a radius of 2.45 mm. What is the current in the wire? What does the graph tell us?? *The length of the wire is 3 meters. *The potential difference across the wire is 12 m volts. *The wire is uniform. Let’s get rid of the mm radius and convert it to area in square meters: A=r2 = 3.14159 x 2.452 x 10-6 m2 or A=1.9 x 10-5 m 2 copper 12 volts 0 volts Material is Copper so resistivity is (from table) = 1.69 x 10-8 ohm meters We have all we need…. 8 L 1.69 x10 ohm - m 3.0 m R 2.67 m 5 A 1.9 x10 From Ohm' s Law : 6 V 12 10 volts i 4.49 ma 3 R 2.67 10 ohms i i Series Combinations R1 V1 V1 iR1 V2 V V2 iR2 and V V1 V2 iR iR1 iR2 R R1 R2 general : R ( series ) Ri i R2 Parallel Combination?? R1, I1 V iR V V V i i1 i2 R1 R2 R R2, I2 V so.. 1 1 1 R1 R2 R general 1 1 R i Ri What’s This??? In Fig. 28-39, find the equivalent resistance between points (a) F and H and [2.5] (b) F and G. [3.13] The current density across a cylindrical conductor of radius R varies according to the equation J = J0(1 - (r/R)), where r is the distance from the central axis. Thus the current density is a maximum J0 at the axis (r = 0) and decreases linearly to zero at the surface (r = R). Calculate the current in terms of J0 and the conductor's cross-sectional area A = R2. That’s it for Resistance Next Time Electric Circuits