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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 = k. 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 + OR 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 -

Current is charge in motion • Charge, e. g. free electrons, exists in conductors with a density, ne (ne approx 1029 m-3) • “Somehow” put that charge in motion: – effective picture -- all charge moves with a velocity, ve – real picture -- a lot of “random motion” of charges with a small average equal to ve • Current density, J, is given by J = qeneve – unit of J is C/m 2 sec or A/m 2 (A ≡ Ampere) and 1 A = 1 C/s – current, I, is J times cross sectional area, I = J pr 2 – for 10 Amp in 1 mm x 1 mm area, J = 10+7 A/m 2, and ve is about 10 -3 m/s (Yes, the average velocity is only 1 mm/s!)

Devices • Resistors: Purpose is to limit current drawn in a circuit. Note: UNIT: Ampere = A = C/s • 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).

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

Resistance • What about acceleration? I • V E-field constant force e. E on electrons constant acceleration R I V 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 (t = 10 -14 s) from things that get in the way – defects, lattice vibrations (phonons), etc. • Average velocity attained in this time is v = e. Et/m • Current density is J = env so current is proportional to E which is proportional to Voltage • OHM’s LAW J = (e 2 nt/m)E or J = s E s = conductivity

Resistance • Resistance is defined to be the ratio of the applied voltage to the current passing through. I UNIT: OHM = W R I V • 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 • Increase the length flow of electrons impeded • Increase the cross sectional area flow facilitated • What about r?

Resistivity • Property of bulk matter related to resistance of a sample is the resistivity (r) defined as: E J A L where E = electric field and J = current density in conductor. For uniform case: Þ , Þ where The resistivity varies greatly with the sort of material: e. g. , for copper r ~ 10 -8 W-m; for glass, r ~ 10+12 W-m; for semiconductors r ~ 1 W-m; for superconductors, r = 0 [see Appendix]

Lecture 9, ACT 1 1 A 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 I 1 I 2 – 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 1 B (b) I 1 = I 2 (c) I 1 > I 2 • 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 • 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 I 1 I 2 – 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 • The resistivity of both resistors is the same (r). • Therefore the resistances are related as: • The resistors have the same voltage across them; therefore

1 B Lecture 9, ACT 1 • 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

1 B Lecture 9, ACT 1 • 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).

Preflight 9: 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 2 d. 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

The Voltage “drops”: Resistors in Series I a R 1 b R 2 a c Whenever devices are in SERIES, the current is the same through both ! Reffective This reduces the circuit to: Hence: c

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

Preflight 9: Two resistors are connected in series to a battery with emf E. The resistances are such that R 1 = 2 R 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

The World’s Simplest (and most useful) circuit: Voltage Divider R 1 V 0 V R 2 By varying R 2 we can controllably adjust the output voltage!

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! e 1 I e + 1 R 1 - IR 1 R 2 - IR 2 e - 2 =0

Rules of the Road 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. e 1 I e + 1 R 1 - IR 1 R 2 - IR 2 e - 2 =0 Note: In the ECE convention, voltage drops enter with a + sign and voltage gains enter with a - sign.

Loop Demo R 1 e 1 b R 4 f a I I d c R 2 Þ KVL: Þ e 2 e R 3

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 12 V 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 12 V R a I R b (c) I 1 > I 0 12 V

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 12 V 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 • Write a loop law for original loop: 12 V +12 V - I 0 R = 0 I 0 = 12 V/R • Write a loop law for the new loop: 12 V - I 1 R = 0 I 1 = 12 V/R 12 V R a I R b (c) I 1 > I 0 12 V

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: • At low temperatures the resistance of some metals 0, 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 (UIUC!), 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)

Appendix: Superconductivity • 1986: “High” temp. superconductors (77 K) 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? ? . . . • 2003: 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”