Physics 2102 Lecture 18: WED 08 OCT

Current & Resistance II

Physics 2113 Jonathan Dowling

Georg Simon Ohm (1789-1854)

Resistance Is Futile!

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.

) (abbr. OhmAmpere

Volt [R] :Units Ω≡=

Georg Simon Ohm (1789-1854)

"a professor who preaches such heresies is unworthy to teach science.” Prussian minister of education 1830

iVR ≡ iRV

RVi == and : thereforeand

Ohm’s laws

Devices specifically designed to have a constant value of R are called resistors, and symbolized by

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.

Resistance is NOT Futile!

�

i ≡ dqdt

= Cs

⎡ ⎣ ⎢

⎤ ⎦ ⎥ ≡ Ampere[ ] = A[ ]

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.

Metal “field lines”

resistivity: JEJE !!

ρρ == vectors,as or,

Resistivity is associated with a material, resistance with respect to a device constructed with the material. ρ

σ 1 :tyConductivi =

Example: A

L V

+ -

AiJ

LVE == ,

LAR

AiL

V==ρ

ALR ρ=

Makes sense! For a given material: resistance LessThicker

resistance MoreLonger →→

Resistivity ρ vs. Resistance R

( resistance: R=V/I )

�

ρ ≡ Ωm[ ] = Ohm ⋅meter[ ]

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:

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.

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).

ICPP

The copper wire has radius r. What happens to the Resistance R if you: (a)  Double the Length? (b)  Double the Area? (c)  Double the Radius?

R→ 2RR→ R / 2R→ R / 4

A = πr2

What happens to the Resistivity ρ if you: (a)  Double the Length? (b)  Double the Area? (c)  Double the Radius?

ρ → ρρ → ρ

ρ → ρ

Step I: The resistivity ρ is the same (all three are copper). Find the Resistance R=ρL/A for each case:

Ra =ρLA

Rb =ρ3L / 2A / 2

= 3ρLA

= 3Ra Rc =ρL / 2A / 2

= ρLA

= Ra

Step II: Rank the current using V=iR or i=V/R

Ra = Rc < Rb

ia = ic > ib 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 RA/RB, measured between their ends?

B A

AA=π r2

AB=π ((2r)2-r2)=3πr2

R=ρL/A

RA/RB= AB/AA= 3

LA=LB=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

•  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?

Resistivity depends on temperature: ρ = ρ0(1+α (T–T0) )

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.

) (abbr. OhmAmpere

Volt [R] :Units Ω≡=

Georg Simon Ohm (1789-1854)

"a professor who preaches such heresies is unworthy to teach science.” Prussian minister of education 1830

iVR ≡ iRV

RVi == and : thereforeand

Ohm’s laws

Devices specifically designed to have a constant value of R are called resistors, and symbolized by

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.

Resistance is NOT Futile!

�

i ≡ dqdt

= Cs

⎡ ⎣ ⎢

⎤ ⎦ ⎥ ≡ Ampere[ ] = A[ ]

26.5: Ohm’s Law:

Current Density: J=i/A Units: [A/m2]

Resistance: R=ρL/A

A

L

A=πr2

Resitivity: ρ depends only on Material and Temperature. Units: [Ω•m]

V = iRR =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

V1 = imRd

i1 = V1/Rw

V2 = imRw

Rd = 1.0x105Ω Rw = 1.5x103Ω

im = im = 1x10–3A

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