V. Electric Current

Dr. Bill Pezzaglia

Updated 2014Feb88

V. Current & Conduction

A. Current

B. Resistance & Ohms Law

C. Electric Power

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A. Current

1) Current as Flow of Charge

2) Conduction Model

3) Conservation of Charge

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1. Current is the Flow of Charge

a. Definition: the flux of (positive) charge

b. SI Units:Amp=Coulomb/secoften use mA or A.

• 2 mA threshold of feeling• 10 mA pain• 20 mA can’t let go• 100 mA DEAD• 100,000 A Lightning

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QI

NOTE: Electrons flow in opposite direction of “conventional current”

1b. Units of Current

• 1820 abampere (biot) defined as current which creates a force of 2 dynes/cm between wires 1 cm apart (based on Ampere’s experiment)

• 1834 (1832) Faraday equates current with amount of mass deposited in electrolysis. The “Faraday” is the charge of one mole of electrons. In modern units: 1 Faraday=96,485.3365 Coulombs

• 1865 Loschmidt estimates Avogadro’s number, hence can determine mass of one atom.

• 1893 international amp defined as depositing 1.118 mg/sec of Ag from AgNO3

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) 3365.485,96(Ag) of 8682.107(

)Ag of 001118.0(MoleCoul

molegmsecgm

Amp

1c. Current Density

1821 Davy shows current flows throughout interior of wire. Define:

• Current Density “J”, a vector,units of current per area

• Surface Current Density “j”units of Amp/meter

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AJI

jLI

2. Drude Model of Conduction

a) 1834 Wheatstone determines electric signals travel at speed of light.

b) However it is later found that the charge carriers move at a quite slow “drift velocity” vd, (takes hours to move one meter) which depends upon the strength of the electric field E and a constant “” called the “electron mobility”.

c) The current density is thus given to be:“-e” is the charge of the electron“n” is the valence electron density (the number of free electrons per unit volume which may be a function of temperature)

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Evd

dnveJ )(

3. Conservation of Current

a) Review: Continuity Equation of Fluids

This is based upon “conservation of mass”

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222111 AvAv

3b. Current Continuity Equation

This is based upon “conservation of current”. If no charged stored anywhere, the current in must equal the current out.

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outin IAJAJI 2211

+ + +

+ +

+

3c. Conservation of Charge

• Fundamental Law of Universe:

Charge is Conserved

• Current flowing out of a Leyden Jar (capacitor) must match the loss of charge in jar:

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B. Resistance

1) Conduction

2) Resistivity

3) Non-Ohmic devices

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1. Conduction

a) Electric Field “E” makes current flow:where “” is “conductance”

b) Conductance of materials: Units: sC2/(kgm3)• Insulator: = 0 (very small, 10-15)• Semiconductor 1• Conductor 108

• Superconductor

c) From Drude model of conduction, we can express the conductance in terms of the electron mobility:[units of mobiity: m2/(s-volt) ]

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Evd

dnveJ )(

EJ

en

2. Resistivity

a) Definition: Resistivity “” is the inverse of conductance. Units: ohm-meter

• Superconductor = 0• Copper: = 1.72x10-8

• Germanium = 0.46• Silicon = 250• Insulator

b) Ohm’s Law (1872) [1826?]• Current through a device is proportional to voltage• Resistance “R” is in units of “ohms”• Ohm=Volt/Amp=kgm2/(sC2)

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JE

1

IRV V R

2c. Resistance of a device

• Resistivity is an “intensive” quantity, while “Resistance” is “extensive” (macroscopic)

• Resistance of a macroscopic object depends upon its geometric properties.

• Derivation:

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EV JAI

AJA

E

I

VR

The historical definition of the “ohm was a 1 meter column of mercury with cross section area of 1 mm2. The unit system has been changed slightly since that time, such that the column of mercury would only be 0.96 ohms.

3. Non-Ohmic Behaviora) Resistance changes with

temperature (and the temperature changes with current).• Conductors: resistance increases with

temperature, =0.004• Semiconductors: resistance decreases

with temperature: =-0.05

b) Light Bulb: More current, gets hotter, resistance increases. Net result voltage is approximately proportional to square of current!

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00

1

exp)(

TT

TTT

2IV

3c. Vacuum TubeFor example, a “diode” tube Non linear (Child’s law)

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5

10

15

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30

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0 2 4 6 8 10 12

3d Diodes

Diode (e.g. LED)• Has low resistance in one direction• High resistance in other direction

• Behaves like a “one way street”. Current can only flow in direction of arrow.

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C. Electric Power

1) Source of Power (Batteries)

2) Electric Work

3) Joule Heating

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1. The Electric Battery(a) First Battery? 400 AD? The Baghdad Battery is the common name for a number of artifacts probably discovered in the village of Khuyut Rabbou'a (near Baghdad, Iraq) in 1936. These artifacts came to wider attention in 1938, when Wilhelm König, the German director of the National Museum of Iraq, found the objects in the museum's collections, and in 1940 (having returned to Berlin due to illness) published a paper speculating that they may have been galvanic cells, perhaps used for electroplating gold onto silver objects. -wikipedia

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1b. Luigi Galvani (1737-1798)

•1786 first battery cell (two different metals in contact)

•1791 Animalistic nature of electricity (frog legs jump from electric charge)•(Mary Shelly used this idea in her “Frankenstein” book)

•Did he also do work on corrosion? (galvanized nails?)

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http://www.corrosion-doctors.org/Biographies/VoltaBio.htm

1c. Voltic Pile

• 1786 Galvani creates first cell

• 1793 Volta shows cell creates an electric current

• 1800 makes 30 volt “Voltic Pile” from a column of cells

• 1826 Ohm determines voltage is the driver of current

• 1830 Faraday figures out the electrochemical reactions of a battery.

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http://www.corrosion-doctors.org/Biographies/VoltaBio.htm

Alessandro Volta(1745-1827)

2. Electric Work

(a) Battery is like a “pump” that increases energy of charge (current) passing through it

• The “voltage gain” is called EMF (electromotive force), measured in units of “volts”

• Change in potential energy of charge q passing through battery is: U=V q

• Often “EMF” is given the symbol “” or E

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2b. Electrical Power

• Power is units of Watts=Joule/Second

• 1841 Joule shows electrical work is equivalent to mechanical work

• Electric Power:

• Hence: Watt=VoltAmpOften we use Kilowatts or Megawatts

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qV

t

U

2c. Energy Usage

• Energy or work done is Power x Time

• Hence: Joule=Wattsecond

• However, the PGE uses the weird unit of:Kilowatt-Hour

• 1 kWh=3.6106 Joules

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3. Joule Heating

In a resistor, the electrical energy is converted into heat.

(a) Power lost in Resistor:

(b) Microscopically: Power lost per unit volume over a cylinder wire of length “L”, cross section area “A”:

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RIR

VVIP

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V

A

I

AL

VI

Volume

P

3c. Power Transmission Loss

• Consider transmission lines have resistance “R”and generator has EMF of

• Voltage delivered to house is:

• Power delivered “P” compared to generated P0:

• Hence, less power is lost if you transmit with high voltage and low current.

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IRV

RI

I

IIR

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VI

P

P1

)(

0

References

• Wheatstone: http://micro.magnet.fsu.edu/optics/timeline/people/wheatstone.html• Drude Model• http://en.wikipedia.org/wiki/Electron_mobility

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