Today in Physics 122: resistance Ohm s Law Resistivity and the physics behind resistance Resistors of different shapes and sizes, and how to calculate their resistance from their resistivity Resistor networks Resistance is futile. You will be assimilated. 4 October 2012 Physics 122, Fall 2012 1
Imperfect conductors Many materials contain electric charges that are free to move around, as in perfect conductors, but sustain electric fields and potential differences, as insulators can. If one subjects a chunk of such material to a potential difference, I one sees that it draws a current I that is linearly proportional to V this voltage: dq dt I V R Ohm s Law I charge per unit time entering or leaving the chunk. R is called the resistance of the chunk. 4 October 2012 Physics 122, Fall 2012 2
Units In the MKS system officially, and completely, called the MKSA or SI system the units of current and resistance are [ I] ( ) ampere A coul sec [ R] ohm ( ) Those honored in the names: André-Marie Ampère, best known for discovering the relation between electric current and magnetic field, which we will study this term as Ampère s Law. Georg Ohm, he who discovered Ohm s Law. 1 1 kg m Ω volt ampere 2 coul sec 2 4 October 2012 Physics 122, Fall 2012 3
Imperfect conductors (continued) The reasons for resistance (in solids): As free charges within the material are accelerated by E, they collide with immobile charges the material contains. Collisions transfer some of the energy and momentum the charges have picked up from E, to the immobile charges. The immobile charges absorb this energy and momentum through the normal modes of vibration of the material. The vibrational energy diffuses throughout the material; the free charge can t get it back in subsequent collisions. If collisions are very frequent, free charges never accumulate and cancel E, as in perfect conductors. This collisional transfer of energy heats the material. 4 October 2012 Physics 122, Fall 2012 4
Transport of one of the free electrons in a metal E -q +q 4 October 2012 Physics 122, Fall 2012 5
Imperfect conductors (continued) Materials which have free charges naturally have lots of fixed charges with which the free ones can collide. about one fixed charge per free charge. So perhaps a better question is, how there can be perfect conductors? The answer is There are none, except superconductors. Charge motion in superconductors escapes this collisional impediment by a subtle trick of quantum mechanics that only works in certain materials under special conditions. Materials exhibit a big range of collider density. Small pieces with small collider density have small enough resistance that perfect can be a good approximation. 4 October 2012 Physics 122, Fall 2012 6
Resistivity and resistance On the microscopic scale within materials, current density and electric field are proportional to one another: 1 Ohm s J E ρ Law where J is the current per unit area perpendicular to the direction 2 of the direction of E. Its MKS units are [ J] A m. ρ is the resistivity of the material, a scalar quantity (for now) which characterizes the density of colliders a free charge will face. Units: [ ρ ] Ω m. ρ is a property of the material: it is independent of the size and shape of the chunk. 4 October 2012 Physics 122, Fall 2012 7
Resistivity and resistance (continued) Resistors are usually made of materials with rather large values of the resistance ρ. If ρ is large enough, in a sense we will be able to quantify after we have learned a bit more about circuits, resistances can be calculates as follows. Start with an assumed charge on the electrodes, e.g. ±Q, and calculate E and V as if this were a capacitor. At either electrode, apply the microscopic form of Ohm s law: ρ J E. Multiply through by electrode area A; use expressions for E and V to substitute E out, and V in; and use I JA. The result will be of the form V IR, whence R V/I. 4 October 2012 Physics 122, Fall 2012 8
A wafer-like resistor Take the parallel-plate capacitor and fill the gap with material with resistivity ρ. What is its resistance? Electric field and potential difference are as before: Then σ Q A. E 4 πkσ, V 4 πkσd. 1 I 1 1 V J E 4πkσ ρ A ρ ρ d ( ρ ) V d A I RI R ρd A d E 4πkσ σ Q A. 4 October 2012 Physics 122, Fall 2012 9
A coaxial resistor What about filling the gap of the coaxial capacitor with resistive material? (A model for leakage in co-ax cable.) Again, E and V are the same as in the capacitance calculation: 2 2, 2 ln R E kλ r V kλ R1 whence I 12kλ I 1 V J A ρ R 2πRL ρ R ln R R ( ) 2 2 2 2 1 ρ R ρ R V I ln IR R L R L R 2 2 ln 1 1 4 October 2012 Physics 122, Fall 2012 10
Resistor and capacitor FAQs So in those last two examples, does the object have resistance or capacitance or both? Both: they are properly viewed as R and C in parallel. Usually, though, resistors are made with small-c electrodes, and capacitors with very good insulators in their gaps, unless this can t be avoided. What about more complicated shapes? You will learn how to deal with complex shapes analytically and numerically in courses like PHY 217. The numerical methods involve generally-useful techniques like finite-element differential-equation solvers; this is how one generally calculates non- lumped Rs and Cs in integrated circuits and printed-circuit boards. 4 October 2012 Physics 122, Fall 2012 11
Real resistors Real resistors are made in geometries that don t lend themselves to easy calculation, in order to minimize the capacitance that comes with them. Most of the ones you find around the lab are cylinders with coaxial leads and electrodes in between. The bulk of them can be either carbon, which in its graphite form has resistivity that lies in a range that produces kω-mω resistances in severalmm-size cylinder. Manufacturing tolerance of R is ~5%. or very thin metallic films on insulating ceramic cores. The same range of R is available, the manufacturing tolerance is better (1%) and they are correspondingly more expensive. 4 October 2012 Physics 122, Fall 2012 12
Resistors and power If a charge Q is transported through a voltage V, work in the amount QV is done on the charge. Therefore if charge is transported through a resistor at the rate I dq/dt, power in the amount dw d 2 P ( QV ) IV I R dt dt is deposited ( dissipated ) as heat in the resistor. But if the resistor has a steady temperature, then the power it gives off must be equal to the power deposited: the resistor transmits power 2 P I Rto its surroundings, by conduction and/or radiation. 4 October 2012 Physics 122, Fall 2012 13 V I R
Resistors in circuits Like capacitors, resistors are used ubiquitously in electrical circuits as energy-storage reservoirs. The appear in circuit diagrams as where the lines outside the zigzag are understood to be perfect conductors. Again like capacitors, there are two ways two resistors can be connected: series, and parallel. 4 October 2012 Physics 122, Fall 2012 14
Resistors in circuits (continued) By definition, resistors connected in parallel have the same voltage: V R1 R2 R 3 V Req The current drawn by resistor n is VR n, and the total current drawn by the parallel combination is I V V V + + + R R R 1 2 3 1 1 1 1 + + + V R1 R2 R3 R 4 October 2012 Physics 122, Fall 2012 15 eq V
Resistors in circuits (continued) Resistors in series have different voltages, but all the same current I, since they can t accumulate charge: R1 R2 R3 R eq I V 1 IR1 V2 V3 V V so ( ) V IR + IR + IR + I R + R + R + IR 1 2 3 1 2 3 eq 4 October 2012 Physics 122, Fall 2012 16
Combinations of Rs Thus the equivalent capacitances and resistances of series and parallel pairs exhibit a nice complementarity: C 1 C 2 1 1 1 + C C C C eq eq C 1 2 CC 1 2 + C 1 2 R 1 R 2 R R + R eq 1 2 C1 C2 Ceq C1 + C2 R1 R2 1 1 1 + R R R 1 2 1 2 4 October 2012 Physics 122, Fall 2012 17 R eq eq RR 1 2 R + R
Combinations of Rs (continued) So we can resolve networks of resistors that are composed purely of series and parallel combinations, in much the same way that we can for capacitors. Example. Replace each capacitor in the pyramid we used Tuesday with a resistor of value R. Find the equivalent resistance of the pyramid. 4 October 2012 Physics 122, Fall 2012 18
Combinations of Rs (continued) First resolve the parallel combinations: R 1 1 1 + R1 R R R 1 R 2 1 1 1 1 R + + R2 R2 R R R 3 1 1 1 1 1 + + + R3 R R R R R 3 R 4 4 October 2012 Physics 122, Fall 2012 19
Combinations of Rs (continued) Then the series combinations: R R/2 R/3 R R R 25 Req R+ + + R 2 3 4 12 R/4 4 October 2012 Physics 122, Fall 2012 20