 I1I1   R R R I2I2 I3I3

Today… Current and Current Density Devices –Capacitors –Batteries –Resistors Resistors in Series & Parallel Kirchhoff's Rules –Loop Rule ( V is independent of path) –Junction Rule (Charge is conserved) Superconductivity Text Reference: Chapter 26.1, 2, 4, & 5; 27.5 Examples: 25.4,5,6,7,8,9 and 10

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)

Devices Capacitors: Purpose is to store charge (energy). We have calculated the capacitance of a system We had to modify Gauss' Law to account for bulk matter effects (dielectrics) … C =  C 0 We calculated effective capacitance of series or parallel combinations of capacitors Batteries (Voltage sources, sources of emf): Purpose is to provide a constant potential difference 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". +- V + - OR

Current is charge in motion Charge, e.g. electrons, exists in conductors with a density, n e ( n e approx m -3 ) “Somehow” put that charge in motion: –effective picture - all charges move with a velocity, v e –real picture - a lot of “random motion” of charges with a small average equal to v e Current density, J, is given by J = q e n e v e –unit of J is C/m 2 sec or A/m 2 (A ≡ Ampere) and 1A ≡ 1C/s –current, I, is J times cross sectional area, I = J  r 2 –for 10 Amp in 1mm x 1mm area, J =10 +7 A/m 2, and v e is about m/s (Yes, the average velocity is only 1mm/s!)

Devices Resistors: Purpose is to limit current drawn in a circuit. 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”). The effective resistance of series and parallel combinations of resistors will be calculated using Kirchhoff's Laws (Notion of potential difference, current conservation). UNIT: Ampere = A = C/sNote:

Ohm’s Law Demo: Vary applied voltage V. Measure current I Does ratio remain constant? V I I R V I slope = R How to calculate the resistance? Include “resistivity” of material Include geometry of resistor

Resistance What about acceleration? V I I R Voltage -> E -field E -field -> constant force on electrons Constant acceleration -> very large and increasing currents These very large currents and “funny” I ( V ) are not observed –what’s wrong with this picture??? Constant force on electrons -> constant acceleration 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 For some reason simple constant acceleration isn’t happening….

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  = sec Charges ballistically accelerated for this time and then randomly scattered Average velocity attained in this time is v = eE  / m Current density is J = env so current is proportional to E which is proportional to Voltage OHM’s LAW J = (e 2 n  /m)E or J =  E  = conductivity

Resistance Resistance is defined to be the ratio of the applied voltage to the current passing through. 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... V I I R UNIT: OHM = 

Calculating Resistance To calculate R, we must calculate current which flows when voltage V is applied. Applying voltage V sets up an electric field in the resistor. What determines the current? Current is charge flowing past a point per unit time. Need to know velocity of charges. Is this determined by field? Field gives rise to force on charge carriers -> charge will be accelerated. Calculation splits here: –Part of this we bury as the “resistivity” ρ –Part depends on the geometry (length L and cross sectional area A ) V I I R

Resistivity where E = electric field and j = current density in conductor. L A E j Property of bulk matter related to resistance of a sample is the resistivity (  ) defined as: e.g., for a copper wire,  ~  -m, 1mm radius, 1 m long, then R .01  for glass,  ~  -m; for semiconductors  ~ 1  -m So, in fact, we can compute the resistance if we know a bit about the material, and YES, the property belongs to the material!, For uniform case:   where

Makes sense? 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 L A E j How thick? How big? What’s it made of? or

Lecture 9, ACT 1 The resistivity of both resistors is the same (  ). Therefore the resistances are related as: The resistors have the same voltage across them; therefore Two cylindrical resistors, R 1 and R 2, are made of identical material. R 2 has twice the length of R 1 but half the radius of R 1. –These resistors are then connected to a battery V as shown: V I1I1 I2I2 –What is the relation between I 1, the current flowing in R 1, and I 2, the current flowing in R 2 ? (a) I 1 < I 2 (b) I 1 = I 2 (c) I 1 > I 2

Lecture 9, ACT 1 1B A very thin metal wire patterned as shown is bonded to some structure. As the structure is deformed slightly, 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, is slightly increased. Also, because the overall volume of the wire is ~constant, increasing the length decreases the area A, which also increases the resistance. By carefully measuring the change in resistance, the strain in the structure may be determined (we’ll see later how to do this optically).

Resistors in Series a c R effective a b c R1R1 R2R2 I The Voltage “drops”: Hence: Whenever devices are in SERIES, the current is the same through both ! This reduces the circuit to:

Another (intuitive) way… Consider two cylindrical resistors with lengths L 1 and L 2 V R1R1 R2R2 L2L2 L1L1 Put them together, end to end to make a longer one...

The World’s Simplest (and most useful) circuit: Voltage Divider By varying R 2 we can controllably adjust the output voltage! V0V0 R1R1 R2R2 V

Kirchhoff’s First Rule “Loop Rule” or “Kirchhoff’s Voltage Law (KVL)” "When any closed circuit loop is traversed, the algebraic sum of the changes in potential must equal zero." KVL: This is just a restatement of what you already know: that the potential difference is independent of path!  R1R1  R2R2 I     IR 1  IR 2     0 0

Rules of the Road Note: In the ECE convention, voltage drops enter with a + sign and voltage gains enter with a  sign.  R1R1  R2R2 I     IR 1  IR 2     0 0 Our convention: Voltage gains enter with a + sign, and voltage drops enter with a  sign. We choose a direction for the current and move around the circuit in that direction. When a battery is traversed from the negative terminal to the positive terminal, the voltage increases, and hence the battery voltage enters KVL with a + sign. When moving across a resistor, the voltage drops, and hence enters KVL with a  sign.

Loop Demo a d b e c f  R1R1 I R2R2 R3R3 R4R4 I    KVL:

Lecture 9, ACT 2 (a) I 1 < I 0 (b) I 1 = I 0 (c) I 1 > I 0 The key here is to determine the potential ( V a - V b ) before the switch is closed. From symmetry, V a - V b = +12V. Therefore, when the switch is closed, NO additional current will flow! Therefore, the current before the switch is closed is equal to the current after the switch is closed. Consider the circuit shown. –The switch is initially open and the current flowing through the bottom resistor is I 0. –Just after the switch is closed, the current flowing through the bottom resistor is I 1. –What is the relation between I 0 and I 1 ? R 12V R I a b

Lecture 9, ACT 2 Consider the circuit shown. –The switch is initially open and the current flowing through the bottom resistor is I 0. –After the switch is closed, the current flowing through the bottom resistor is I 1. –What is the relation between I 0 and I 1 ? (a) I 1 < I 0 (b) I 1 = I 0 (c) I 1 > I 0 Write a loop law for original loop: 12V  I 1 R = 0 I 1 = 12V/R Write a loop law for the new loop: 12V +12V  I 0 R  I 0 R = 0 I 0 = 12V/R R 12V R I a b

Summary When you are given a circuit, you must first carefully analyze circuit topology –find the nodes and distinct branches –assign branch currents Use KVL for all independent loops in circuit –sum of the voltages around these loops is zero! Preflight 10 is a survey (on preflights and the course) and is due on Monday (10/8). Reading assignment: Ch. 26.3, 5 & 6 Examples:

Appendix: Superconductivity 1911: H. K. Onnes, who had figured out how to make liquid helium, used it to cool mercury to 4.2 K and looked at its resistance: 1957: Bardeen (UIUC!), Cooper, and Schrieffer (“BCS”) publish theoretical explanation, for which they get the Nobel prize in – It was Bardeen’s second Nobel prize (1956 – transistor) –Current can flow, even if E=0. –Current in superconducting rings can flow for years with no decrease! At low temperatures the resistance of some metals  0, measured to be less than ρ conductor (i.e., ρ< Ωm)!

Appendix: 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 138 K (-135˚ C = -211˚ F) 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??...