Transcript Lect18
e1 R I1 I2 e2 R I3 e3 R V n loop 0 I in I out Today… • Current and Current Density • Devices – Batteries – Resistors • Read Fishbane Chapter 26 • Remember: Quiz on Thursday and Friday. – Covers chapters 24-25-12, – Potential, Capacitors, Gravity (and everything before) Devices and Circuits • We are now finished with electrostatics: the study of fields and potentials produced by static charge distributions. • Next topic: Devices and Circuits • We have studied one device so far: the capacitor. For the next week we will investigate circuits composed of the following devices: • Capacitors • Batteries • Resistors } and combinations in “DC” circuits, (Direct Current circuits) Current is charge in motion • Electrons exists in conductors with a density, ne (ne approx 1029 m-3) and are constantly in random motion – In general only electrons move, the heavy nucleii remain fixed in the material lattice • In the absence of electric fields there is no net motion of the charge, electrons bounce around like atoms in a gas • When an electric field is applied: – a small average velocity, ve ,is added to the random motion (an electric current) + E • NOTE that the current direction is defined as the direction of the field BUT the electrons move in the opposite direction Current is charge in motion - ne electrons/m3 + Area S E Velocity ve • Current density, J, is given by J = qeneve • unit of J is C/m2sec or A/m2 (A ≡ Ampere) and 1A ≡ 1C/s • Current, I, is J times cross sectional area, I = J S – for 10 Amp in 1mm x 1mm area, J=10+7 A/m2, – ve is about 10-3 m/s – (Yes, the average velocity is only 1mm/s!) Devices: Batteries • Batteries (Voltage sources, sources of emf): Purpose is to provide a constant potential difference and source of current between two points. + - • Cannot calculate the potential difference from first principles... chemical electrical energy conversion. Non-ideal batteries will be dealt with in terms of an "internal resistance". OR + • Positive terminal has the higher potential • Current is defined as flowing from the positive to the negative terminal • Inside the battery chemical processes return the charge from the negative to positive terminals • emf is the term for the electrical potential provided by the battery E dW dq V - Devices: Resistors • Resistors: • Resistors limit the current drawn in a circuit. Note: I dQ dt UNIT: Ampere = A = C/s • Resistance is a natural property of almost all materials which opposes the motion of charge through the material • Resistance can be calculated from knowledge of the geometry of the resistor AND the “resistivity” of the material out of which it is made (often “conductors”). Ohm’s Law • Set up this circuit • Vary applied voltage V. I R I • Measure current I • Ratio V remains constant I V • Resistance R V slope = R I V R I Resistance • What is happening in the resistance? I R • Voltage means Potential Difference -> E-field • E-field -> constant force on electrons • Constant force on electrons -> constant acceleration V • Constant acceleration -> very large and increasing currents • This does not happen large increasing currents are not observed –what’s wrong with this picture??? • Simple constant acceleration isn’t happening…. • Electrons undergo a lot of rapid and random scattering • No constant acceleration (acceleration proportional to Voltage) • Instead velocity of electrons is proportional to Voltage velocity proportional to current -> I=V/R I What gives rise to non-ballistic behavior? • E-field in conductor (resistor) provided by a battery • Charges are put in motion, but scatter in a very short time from things that get in the way – it’s crowded inside that metal – defects, lattice vibrations (phonons), etc • Typical scattering time t = 10-14 sec • Charges ballistically accelerated for this time and then randomly scattered What gives rise to non-ballistic behavior? • Newton’s 2nd Law says F=ma • So the acceleration of the electron is eE/m • Average velocity attained between scatters is given by v=at or v = eEt/m • Current density is J = env so current is proportional to E which is proportional to Voltage • OHM’s LAW J = (e2nt/m)E or J = s E • Or V I R s = conductivity Resistance R • Resistance Resistance is defined to be the ratio of the applied voltage to the current passing through. R V I UNIT: OHM = W I I V • Is this a good definition? i.e., does the resistance belong only to the resistor? Recall the case of capacitance: (C=Q/V) depended on the geometry, not on Q or V individually Does R depend on V or I ? It seems as though it should, at first glance... Calculating Resistance • To calculate R, must calculate current I which flows when voltage V is applied. • Applying voltage V sets up an electric field in the resistor. What determines the current? I R I V R V I • Current is charge flowing past a point per unit time, which depends on the average velocity of the charges. • Field gives rise to force on the charge carriers which reach a terminal velocity. L • Resistance calculation splits into two parts R A – Part depends on the “resistivity” ρ, a property of the material – Part depends on the geometry (length L and cross sectional area A) Resistivity • Resistivity is a property of bulk matter related to the resistance of a sample. • E The resistivity () is defined as: E j 1 m 2 s e nt j A L • Where E = electric field and j = current density in conductor. • For the case of a uniform material I j A V EL Resistivity E j j I A V EL I ρL V EL jL L I A A V IR where R E j A L L A So YES, the property belongs to the material and we can calculate the resistance if we know the resistivity and the dimensions of the object e.g., for a copper wire, ~ 10-8 W-m, 1mm radius, 1 m long, then R .01W; for glass, ~ 10+12 W-m; for semiconductors ~ 1 W-m Makes sense? E L R A j A L • Increase the length, flow of electrons impeded • Increase the cross sectional area, flow facilitated • The structure of this relation is identical to heat flow through materials … think of a window for an intuitive example How thick? or How big? What’s it made of? Question 1 I1 • Two cylindrical resistors, R1 and R2, are made of identical material. R2 has V twice the length of R1 but half the radius of R1. – The resistors are then connected to a battery V as shown – What is the relation between the currents I1 and I2 (a) I1 < I2 (b) I1 = I2 (c) I1 > I2 I2 Question 1 1. a 2. b 3. c Question 1 I1 • Two cylindrical resistors, R1 and R2, are made of identical material. R2 has V twice the length of R1 but half the radius of R1. – The resistors are then connected to a battery V as shown – What is the relation between the currents I1 and I2 (b) I1 = I2 (a) I1 < I2 (c) I1 > I2 •The resistivity of both resistors is the same (). •Therefore the resistances are related as: R2 L2 2 L1 L 8 1 8R1 A2 ( A1 / 4) A1 •The resistors have the same voltage across them; therefore V V 1 I2 R2 I1 8R1 8 I2 Question 2 • A very thin metal wire patterned as shown is bonded to some structure. • As the structure is deformed this stretches the wire (slightly). – When this happens, the resistance of the wire: (a) decreases (b) increases (c) stays the same Question 2 1. a 2. b 3. c Question 2 • A very thin metal wire patterned as shown is bonded to some structure. • As the structure is deformed this stretches the wire (slightly). – When this happens, the resistance of the wire: (a) decreases (b) increases (c) stays the same •Because the wire is slightly longer, R ~ L A is increased. •Because the volume of the wire is ~constant, increasing the length, decreases the area, which increases the resistance. •By carefully measuring the change in resistance, the strain in the structure may be determined Is Ohm’s Law a good law? • Our derivation of Ohm’s law ignored the effects of temperature. – At higher temperatures the random motion of electrons is faster, – time between collisions gets smaller – Resistance gets bigger – Temperature coefficient of resistivity () – Typical values for metals 410-3 1 m 2 s e nt 0 1 T T0 Is Ohm’s Law a good law? • Our derivation of Ohm’s law ignored quantum mechanical effects • Many materials, only conduct when sufficient voltage is applied to move electrons into a “conduction band” in the material • Examples are semiconductor diodes which have very far from linear voltage versus current plots Is Ohm’s Law a good law? Superconductivity • At low temperatures (cooled to liquid helium temperatures, 4.2K)the resistance of some metals0, measured to be less than 10-16•ρconductor (i.e., ρ<10-24 Ωm)! –Current can flow, even if E=0. –Current in superconducting rings can flow for years with no decrease! • 1957: Bardeen, Cooper, and Schrieffer (“BCS”) publish theoretical explanation, for which they get the Nobel prize in 1972. – It was Bardeen’s second Nobel prize (1956 – transistor) Is Ohm’s Law a good law? Superconductivity • 1986: “High” temperature superconductors are discovered (Tc=77K) – Important because liquid nitrogen (77 K) is much cheaper than liquid helium – Highest critical temperature to date ~140K • Today: Superconducting loops are used to produce “lossless” electromagnets (only need to cool them, not fight dissipation of current) for particle physics. [Fermilab accelerator, IL] • The Future: Smaller motors, “lossless” power transmission lines, magnetic levitation trains, quantum computers?? ... Is Ohm’s Law a good law? • Answer NO • Ohm’s Law is not a fundamental law of physics • However it is a good approximation for metallic conductors at room temperature as used in electrical circuits Summary •Ohm’s Law states V I R •Ohm’s Law is not a physical law but an approximation which works well enough in normal conditions •Read Chapter 27 for tomorrow •Remember the Quiz on Thursday and Friday.