Resistance Is Futile! Physics 2113 Jonathan Dowling Physics 2102 Lecture 18: WED 08 OCT Current & Resistance II Georg Simon Ohm (1789-1854)
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. V R i and therefore : Ohm s laws V i = and V = ir R Georg Simon Ohm (1789-1854) Units :[R] = Volt Ampere Ohm (abbr. Ω) "a professor who preaches such heresies is unworthy to teach science. Prussian minister of education 1830 Devices specifically designed to have a constant value of R are called resistors, and symbolized by i dq = C dt s [ Ampere ] = [ A]
Resistivity ρ vs. Resistance R Metal field lines 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 is associated with a material, resistance with respect to a device constructed with the material. resistivity: E! ρ = or, as vectors, E = ρ J! J ( resistance: R=V/I ) ρ [ Ωm] = [ Ohm meter] Conductivi ty : σ = 1 ρ Example: A V E =, J = L i A ρ = V i L A = R A L R = L ρ A - L V + Makes sense! For a given material: Longer Thicker More resistance Less resistance
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:
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.
ICPP The copper wire has radius r. What happens to the Resistance R if you: (a) Double the Length? R 2R (b) Double the Area? R R / 2 (c) Double the Radius? R R / 4 A = πr 2 What happens to the Resistivity ρ if you: (a) Double the Length? ρ ρ (b) Double the Area? ρ ρ ρ ρ (c) Double the Radius?
Step I: The resistivity ρ is the same (all three are copper). Find the Resistance R=ρL/A for each case: R a = ρl A R a = R c < R b R b = ρ3l / 2 A / 2 = 3 ρl A = 3R a R c = ρl / 2 A / 2 = ρl A = R a Step II: Rank the current using V=iR or i=v/r i a = i c > i b Ranking is reversed since R is downstairs.
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 R A /R B, measured between their ends? A R=ρL/A B A A =π r 2 A B =π ((2r) 2 -r 2 )=3πr 2 R A /R B = A B /A A = 3 L A =L B =L Cancels
Example, A material has resistivity, a block of the material has a resistance.:
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:
Resistivity and Temperature Resistivity depends on temperature: ρ = ρ 0 (1+α (T T 0 ) ) 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?
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. V R i and therefore : Ohm s laws V i = and V = ir R Georg Simon Ohm (1789-1854) Units :[R] = Volt Ampere Ohm (abbr. Ω) "a professor who preaches such heresies is unworthy to teach science. Prussian minister of education 1830 Devices specifically designed to have a constant value of R are called resistors, and symbolized by i dq = C dt s [ Ampere ] = [ A]
26.5: Ohm s Law:
L A=πr 2 A Current Density: J=i/A Units: [A/m 2 ] Resistance: R=ρL/A Resitivity: ρ depends only on Material and Temperature. Units: [Ω m]
V = ir R = V / i = constant
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
R d = 1.0x10 5 Ω R w = 1.5x10 3 Ω i m = 1x10 3 A i m = V 1 = i m R d i 1 = V 1 /R w V 2 = i m R w
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