Conducting surface - equipotential. Potential varies across the conducting surface. Lecture 9: Electrical Resistance.

Transcription

1 Lecture 9: Electrical Resistance Electrostatics (time-independent E, I = 0) Stationary Currents (time-independent E and I 0) E inside = 0 Conducting surface - equipotential E inside 0 Potential varies across the conducting surface

2 Ordered (crystalline) materials Conductors Electrolytes Liquid and Solid Ionic Conductors Dielectrics Semiconductors Metals diamond silicon Large concentration of mobile ions Large concentration of mobile electrons

3 Electrons in Metals Electrons in random motion, colliding with static (= defects) and dynamic (= vibrations) imperfections of crystal lattice.

4 E 0: Electron Drift E = 0 E 0 Under a gentle breeze of electric field, the electron mosquito cloud drifts slowly inside the wire. v d = μe μ - mobility v d Why the drift velocity, not constant acceleration??? V Because scattering transfers energy and momentum from electrons to the lattice - compare to air friction. time in el The terminal velocity is reached when the rate of energy gained from the electric field becomes equal to the rate of energy loss. drift velocity terminal velocity

5 Current E = 0 E 0 disregard random motion I dq dt j I A Current: the charge carried through a wire cross section in unit time. - current density [A/m 2 ] cross sectional area Units: Amperes 1A = 1C 1s A I = ne v d 1s A 1s n concentration of charge carriers e their charge A v d 1s the volume that passes through cross section in 1s I I = nev d A j = I A = nev d

6 Current as Fluid Flow Current resembles a steady flow of incompressible fluid: Charge is neither created nor destroyed. Charge does not accumulate in currentcarrying wires (Except: at capacitor plates).

7 iclicker Two copper wires of different diameter are joined end-to-end, and a current flows in the wire combination. I = nev d A None of the above

8 iclicker Two copper wires of different diameter are joined end-to-end, and a current flows in the wire combination. When electrons move from the larger-diameter wire into the smaller-diameter wire, A. their drift speed increases. B. their drift speed decreases. C. their drift speed stays the same. D. not enough information given to decide

9 iclicker Electrons in an electric circuit pass through a resistor. The wire has the same diameter on each side of the resistor. Compared to the drift speed of the electrons before entering the resistor, the drift speed of the electrons after leaving the resistor is A. faster. B. slower. C. the same. D. not enough information given to decide

10 Drift Velocity Estimate A copper wire (n m -3, cross section 1 mm 2 = 10-6 m 2 ) carries current 1 A. Find the drift velociy v d. v d = I nea 1A m C 10 6 m 2 = 10 4 m/s Compare with the average velocity of an electron: v 10 6 m/s (quantum mechanics) The drift velocity v d v

11 Electrons and Holes Despite what you know about current being carried by electrons, we may think of negative charge carriers moving from to +, or positive charge carriers moving form + to -. These two pictures are to some extent equivalent. However, when magnetic fields are present, these two pictures are no longer equivalent: We can perform measurements (details later) which tell us the sign of the charge carriers! Although in some conductors the charge carriers are indeed negative ( electrons ), in others they are positive ( holes, i.e., missing electrons)! In both cases, the direction of current flow is defined as the direction a positive charge would move. Thus, current always flows from higher potential to lower potential: j = nev d = n e ( v d )

12 Ohm s Law For metals I V over a broad range of V (experimental observation) Ohm s Law V = RI R V I - the coefficient of proportionality the resistance Units: Ohms, Ω = Georg Ohm For metals, the resistance is V- and I-independent (in the linear regime, where there is no appreciable heating).

13 Voltmeters and Ammeters A good voltmeter should have a very high R in and should be connected in parallel with the circuit element across which the potential difference being measured. A good ammeter should have a very low R in and should be connected in series with the circuit element through which the current flows.

14 Resistivity A A E v d = μe E I ja = nev d A = neμea = neμ V L A = V R R = 1 neμ L A R = ρ L A ρ = 1 neμ - resistivity Ω m Ohm s Law in terms of the resistivity: j = E ρ - current density A/m 2

15 iclicker Consider two wires. Wire A is 10 cm long, and wire B is 5 cm long. Both wires are otherwise identical, and both have the same electric field acting in them. How do the currents in these wires compare? A. I A = 4I B B. I A = 2I B C. I A = I B D. I A = 0.5I B E. I A = 0.25I B

16 Typical Resistivities Resistivity at room temperature metals semi-metal semiconductor dielectric Resistance of a copper wire: cross section 1 mm 2, length 1 m: R = ρ L A 10 8 Ω m 1m 10 6 m 2 = 0.01Ω

17 Temperature Dependence of Resistivity for Metals Metallic resistivity typically increases with temperature because the mobility decreases (thermal scattering). Platinum resistance thermometer

18 Temperature Dependence of Resistivity for Semiconductors Semiconductors resistivity typically decreases with increasing temperature because the carrier density increases. Use for temperature measurement: Thermistor.

19 Superconductors 19