 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) Appendix: Superconductivity

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. free electrons, exists in conductors with a density, n e ( n e approx m -3 ) “Somehow” put that charge in motion: –effective picture -- all charge moves 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  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’sOhm’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 V  E -field  constant force eE on electrons  constant acceleration  ever-increasing currents These very large currents and “funny” I ( V ) do not occur. Charges are ballistically accelerated, but scatter in a very short time (  = s) from things that get in the way –defects, lattice vibrations (phonons), etc. 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. How do we calculate it? Recall the case of capacitance: ( C = Q / V ) depended on the geometry (and dielectric constant), not on Q or V individually Similarly, for resistance –part depends on the geometry (length L and cross-sectional area A) –part depends on the “resistivity” ρ of the material V I I R UNIT: OHM =  Increase the length  flow of electrons impeded Increase the cross sectional area  flow facilitated What about  ?

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 copper  ~  -m;  for glass,  ~  -m; for semiconductors  ~ 1  -m; for superconductors,  = 0 [see Appendix] The resistivity varies greatly with the sort of material:, For uniform case:   where

Lecture 9, ACT 1 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 1A 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

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

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).

Two cylindrical resistors are made from the same material, and they are equal in length. The first resistor has diameter d, and the second resistor has diameter 2d. 2) Compare the resistance of the two cylinders. a) R 1 > R 2 b) R 1 = R 2 c) R 1 < R 2 3) If the same current flows through both resistors, compare the average velocities of the electrons in the two resistors: a) v 1 > v 2 b) v 1 = v 2 c) v 1 < v 2 Preflight 9:

Resistors in SeriesSeries 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...

Two resistors are connected in series to a battery with emf E. The resistances are such that R 1 = 2R 2. 5) Compare the current through R 1 with the current through R 2 : a) I 1 > I 2 b) I 1 = I 2 c) I 1 < I 2 6) What is the potential difference across R 2 ? a) V 2 = E b) V 2 = 1/2 E c) V 2 = 1/3 E Preflight 9:

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 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 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!

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” temp. superconductors (77K) discovered –Important because liquid N (77 K) is much cheaper than liquid He –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 lines, magnetic levitation trains, quantum computers?? : UIUC Prof. Tony Leggett shares Nobel prize for helping to explain the related phenomenon of superfluids. –Superconductivity arises from quantum correlations between pairs of electrons in the metal, resulting a total loss of “friction” –In some materials at low temperature, a similar effect allows them to flow with no viscosity  “superfluidity”