Resistance and Resistivity in Physics

1. Physics 2113 Jonathan Dowling Resistance Is Futile! Physics 2102 Lecture 20: WED 04 MAR Current & Resistance II Georg Simon Ohm (1789-1854)

2. Ohm’s laws Georg Simon Ohm (1789-1854) "a professor who preaches such heresies is unworthy to teach science.” Prussian minister of education 1830 Resistance is NOT Futile! Electrons are not “completely free to move” in a conductor. They move erratically, colliding with the nuclei all the time: this is what we call “resistance”. The mechanical analog is FRICTION. The resistance is related to the potential we need to apply to a device to drive a given current through it. The larger the resistance, the larger the potential we need to drive the same current. Devices specifically designed to have a constant value of R are called resistors, and symbolized by

3. Resistivity ρ vs. Resistance R Metal “field lines” Example: A L - + V These two devices could have the same resistance R, when measured on the outgoing metal leads. However, it is obvious that inside of them different things go on. resistivity: Resistivity is associated with a material, resistance with respect to a device constructed with the material. ( resistance: R=V/I ) Makes sense! For a given material:

4. Fluid Flow: An Analogy for the PETEs! • Amount of Water = Charge • Pressure = Potential = Voltage • Flow Rate = Current = Amps • The pressure at the end of the hose represents voltage. • The amount of water in the tank represents charge. • The rate of flow gallons/minute out the hose represents current or amperage. https://learn.sparkfun.com/tutorials/voltage-current-resistance-and-ohms-law

5. Fluid Flow: An Analogy for the PETEs! • Amount of Water = Charge • Pressure = Voltage • Flow Rate = Current = Amps Decrease hose width, decrease A, increase resistance R. Increase hose length L, increase resistance R. Put pebbles in the hose, increase resistivity ρ, increase resistance R. https://learn.sparkfun.com/tutorials/voltage-current-resistance-and-ohms-law

6. 26.4: Resistance and Resistivity: The resistivity ρ of a resistor is defined as: The SI unit for ρ is Ω.m. The conductivity σ of a material is the reciprocal of its resistivity: Put pebbles in the hose, increase resistivity ρ, increase resistance R.

7. 26.4: Resistance and Resistivity, Calculating Resistance from Resistivity: Think pumping water through a long hose. It is easier if L is short and A is big (small R). It is harder if L is long or A is small (big R). If the streamlines representing the current density are uniform throughout the wire, the electric field E and the current density J will be constant for all points within the wire.

8. ICPP • The copper wire has radius r. What happens to the ResistanceR if you: • Double the Length? • Double the Area? • Double the Radius? • What happens to the Resistivityρ if you: • Double the Length? • Double the Area? • Double the Radius?

9. Step I: The resistivity ρ is the same (all three are copper). Find the Resistance R=ρL/A for each case: Step II: Rank the current using V=iR or i=V/R with V constant! Ranking is reversed since R is downstairs.

10. B Example Two conductors are made of the same material and have the same length. Conductor A is a solid wire of diameter r=1.0mm. Conductor B is a hollow tube of outside diameter 2r=2.0mm and inside diameter r=1.0mm. What is the resistance ratio RA/RB, measured between their ends? R=ρL/A A AA=π r2 AB= π (2r)2 - πr2 =3πr2 RA/RB= AB/AA= 3 LA=LB=L Cancels

11. Example, A material has resistivity, a block of the material has a resistance.:

12. 26.4: Resistance and Resistivity, Variation with Temperature: The relation between temperature and resistivity for copper—and for metals in general—is fairly linear over a rather broad temperature range. For such linear relations we can write an empirical approximation that is good enough for most engineering purposes:

13. Resistivity and Temperature Resistivity depends on temperature: ρ = ρ0(1+α (T–T0) ) • At what temperature would the resistance of a copper conductor be double its resistance at 20.0°C? • Does this same "doubling temperature" hold for all copper conductors, regardless of shape or size?

14. Ohm’s laws Georg Simon Ohm (1789-1854) "a professor who preaches such heresies is unworthy to teach science.” Prussian minister of education 1830 Resistance is NOT Futile! Electrons are not “completely free to move” in a conductor. They move erratically, colliding with the nuclei all the time: this is what we call “resistance”. The mechanical analog is FRICTION. The resistance is related to the potential we need to apply to a device to drive a given current through it. The larger the resistance, the larger the potential we need to drive the same current. Devices specifically designed to have a constant value of R are called resistors, and symbolized by

15. 26.5: Ohm’s Law:

16. L A=πr2 A Current Density: J=i/A Units: [A/m2] Resitivity: ρ depends only on Material and Temperature. Units: [Ω•m] Resistance: R=ρL/A

18. Example An electrical cable consists of 105 strands of fine wire, each having r=2.35 Ω resistance. The same potential difference is applied between the ends of all the strands and results in a total current of 0.720 A. (a) What is the current in each strand? i=I/105=0.720A/105=[0.00686] A(b) What is the applied potential difference?V=ir=[0.016121] V(c) What is the resistance of the cable?R=V/I=[.0224 ] Ω

19. Rw = 1.5x103Ω Rd = 1.0x105Ω im = 1x10–3A im= V1 = imRd  i1 = V1/Rw  V2 = imRw 

20. Example A human being can be electrocuted if a current as small as i=100 mA passes near the heart. An electrician working with sweaty hands makes good contact with the two conductors he is holding. If his resistance is R=1500Ω, what might the fatal voltage be?(Ans: 150 V) Use: V=iR

21. i=100 mA? i=100 μA? i=100 MA? http://www.phys.lsu.edu/~jdowling/PHYS21132-SP15/lectures/NRL13.pdf