I1I1 R R R I2I2 I3I3 Lecture 11 Current & Resistance.

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Lecture 11Lecture 11Current & ResistanceCurrent & Resistance

Electric CurrentElectric Current

Definition:Definition: the current is the the current is the rate at which charge flows rate at which charge flows through this surface.through this surface.

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I Given an amount of charge, Given an amount of charge, Q, passing through the area A in a Q, passing through the area A in a

time interval time interval t, the current is the ratio of the charge to the time t, the current is the ratio of the charge to the time interval.interval.

QI

t

The SI units of current is The SI units of current is the ampere (A)the ampere (A)..

1 A = 1 C/s1 A = 1 C/s 1 A of current is equivalent to 1 C of charge passing through the 1 A of current is equivalent to 1 C of charge passing through the

area in a time interval of 1 s.area in a time interval of 1 s.

ExampleExample: The amount of charge that passes through the filament of a certain light bulb in 2.00 s is 1.67 c. Find the current in the light bulb.: The amount of charge that passes through the filament of a certain light bulb in 2.00 s is 1.67 c. Find the current in the light bulb.

Find no. of electrons?Find no. of electrons?

The current densityThe current density is electric current per unit area

Current and Drift SpeedCurrent and Drift Speed

Consider the current on a conductor of cross-sectional Consider the current on a conductor of cross-sectional area A. area A.

Avdq

vdt

Volume of an element of length Volume of an element of length x is : x is : V = A V = A x.x.Let n be the number of carriers per unit of volume.Let n be the number of carriers per unit of volume.The total number of carriers in The total number of carriers in V is: n A V is: n A x.x.The charge in this volume is: The charge in this volume is: Q = (n A Q = (n A x)q.x)q.

Distance traveled at Distance traveled at drift speed drift speed vvdd by carrier in time by carrier in time t: t: x = x = vvd d t.t.

Hence: Hence: Q = (n A Q = (n A vvd d t)q.t)q.

The current through the conductor: The current through the conductor:

I = I = Q/ Q/ t = n A t = n A vvd d qq..

The current density :

J = J = = n = n vvd d qq..

Example:

A copper wire of cross-sectional area 3.00x10-6 m2 carries a current of 10 A. Assuming that each copper atom contributes one free electron to the metal, find the drift speed of the electron in this wire. A = 3.00x10-6 m2 ; I = 10 A, q = 1.6 x 10-19 C.

n = 8.48 x 1022 electrons/ m3.

[Q] If the current density in a copper wire is equal to 5.8*106A/m2, calculate the drift velocity of the free electrons in this wire.

Drift speeds are usually Drift speeds are usually very smallvery small..

Drift speed much smaller than the average speed Drift speed much smaller than the average speed between collisions. between collisions.

Electrons traveling at 2.46x10Electrons traveling at 2.46x10-6-6 m/s would take 68 min to travel m/s would take 68 min to travel 1m.1m.

So why does light turn on so quickly when one flips a So why does light turn on so quickly when one flips a switch?switch?

The info (electric field) travels at roughly 10The info (electric field) travels at roughly 1088 m/s… m/s…

[Q] A silver wire 1 mm in diameter transfers a charge of 65 C in 1 hr, 15 min. Silver contains 5.80 x 1028 free electrons per cubic meter. a) What is the current in the wire? b) What is the magnitude of the drift velocity of the electrons in the wire?Ans. a) 0.0144 A; b) 1.98 x 10-6 m/s

Resistance and Ohm’s LawResistance and Ohm’s Law

When a voltage (potential difference) is applied across the ends of a When a voltage (potential difference) is applied across the ends of a metallic conductor, the current is found to be proportional to the metallic conductor, the current is found to be proportional to the applied voltage.applied voltage.

In situations where the proportionality is exact, one can In situations where the proportionality is exact, one can write.write.

The proportionality constant R is called resistance of the conductor.

• The resistance is defined as the ratio.

In SI, resistance is expressed in volts per ampere.In SI, resistance is expressed in volts per ampere.A special name is given: ohmsA special name is given: ohms

Example:Example: if a potential difference of 10 V applied across a conductor if a potential difference of 10 V applied across a conductor produces a 0.2 A current,produces a 0.2 A current,

then one concludes the conductors has a resistance of then one concludes the conductors has a resistance of 10 V/0.2 a = 50 10 V/0.2 a = 50 ..

Ohm’s LawOhm’s Law

Resistance in a conductor arises because of collisions between Resistance in a conductor arises because of collisions between electrons and fixed charges within the material.electrons and fixed charges within the material.

In many materials, including most metals, the resistance is constant In many materials, including most metals, the resistance is constant over a wide range of applied voltages.over a wide range of applied voltages.

This is a statement of Ohm’s law.This is a statement of Ohm’s law.

Ohm’s Law

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Linear or Ohmic MaterialNon-Linear or Non-Ohmic Material

Semiconductorse.g. devices called diodes

Most metals, ceramics

ResistivityResistivity

Electrons moving inside a conductor subject to an Electrons moving inside a conductor subject to an external potential constantly collide with atoms of the external potential constantly collide with atoms of the conductor.conductor.

They lose energy and are repeated re-accelerated by the They lose energy and are repeated re-accelerated by the electric field produced by the external potential.electric field produced by the external potential.

The collision process is equivalent to an internal friction.The collision process is equivalent to an internal friction.

This is the origin of a material’s This is the origin of a material’s resistanceresistance..

The resistance of an ohmic conductor is proportional to The resistance of an ohmic conductor is proportional to the its length, the its length, ll, and inversely proportional to the cross , and inversely proportional to the cross section area, section area, AA, of the conductor., of the conductor.

• The constant of proportionality is called the resistivity of the material.

Every material has a characteristic resistivity that depends on its electronic Every material has a characteristic resistivity that depends on its electronic structure, and the temperature.structure, and the temperature.

Good Good conductorsconductors have have low resistivitylow resistivity..

InsulatorsInsulators have have high resistivityhigh resistivity..

Resistivity - UnitsResistivity - Units

Resistance expressed in Ohms, Resistance expressed in Ohms,

Length in meter.Length in meter.

Area are mArea are m22, ,

Resistivity thus has units of Resistivity thus has units of mm..

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Material Resistivity (10-8  m) Material Resistivity (10-8  m)

Silver 1.61 Bismuth 106.8

Copper 1.70 Plutonium 141.4

Gold 2.20 Graphite 1375

Aluminum 2.65 Germanium 4.6x107

Pure Silicon

3.5 Diamond 2.7x109

Calcium 3.91 Deionized water

1.8x1013

Sodium 4.75 Iodine 1.3x1015

Tungsten 5.3 Phosphorus 1x1017

Brass 7.0 Quartz 1x1021

Uranium 30.0 Alumina 1x1022

Mercury 98.4 Sulfur 2x1023

Resistivity of various materials

Example

(a) Calculate the resistance per unit length of a nichrome wire of radius 0.321 m.

Cross section:

Resistivity (Table): 1.5 x 10 m.

Resistance/unit length:

(b) If a potential difference of 10.0 V is maintained across a 1.0-m length of the nichrome wire, what is the current?

[Q] A 2.4m length of wire that is 0.031cm2 in cross section has a measured resistance of 0.24W.  Calculate the conductivity of the material.

–The reciprocal of the resistivity is called the conductivity,

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[Q] Speaker wires: Suppose you want to connect your stereo to remote speakers. (a) If each wire must be 20m long, what diameter copper wire should you use to keep the resistance less than 0.1-W per wire? (b) If the current on each speaker is 4.0A, what is the voltage drop across each wire?

[Q] Stretching changes resistance: A wire of resistance R is stretched uniformly until it is twice its original length. What happens to its resistance? The resistance of the wire increases by a factor of four if the length increases twice

Temperature Variation of Resistance

• The resistivity of a metal depends on many (environmental) factors.

• The most important factor is the temperature.

• For most metals, the resistivity increases with increasing temperature.

• The increased resistivity arises because of larger friction caused by the more violent motion of the atoms of the metal.

For most metals, resistivity increases For most metals, resistivity increases approx. linearly with temperature.approx. linearly with temperature.

• is the resistivity at temperature T (measured in Celsius).

• is the reference resistivity at the reference temperature T (usually taken to be 20 oC).

• is a parameter called temperature coefficient of resistivity.

For a conductor with fixed cross section.For a conductor with fixed cross section.

TMetallic Conductor

TSuperconductor

Example:Example:A resistance thermometer, which measures temperature by measuring A resistance thermometer, which measures temperature by measuring the change in the resistance of a conductor, is made of platinum and the change in the resistance of a conductor, is made of platinum and has a resistance of 50.0 has a resistance of 50.0 at 20 at 20ooC. When the device is immersed in a C. When the device is immersed in a vessel containing melting indium, its resistance increases to 76.8 vessel containing melting indium, its resistance increases to 76.8 . . Find the melting point of Indium.Find the melting point of Indium.

Solution:Solution:

Using Using =3.92x10=3.92x10-3-3((ooC)C)-1-1 from table. from table.

RRoo=50.0 =50.0 ..

TToo=20=20ooC.C.

R=76.8 R=76.8 ..

1) A resistance thermometer using a platinum wire is used to measure the temperature of a liquid. The resistance is 2.42 ohms at 0oC, and when immersed in the liquid it is 2.98 ohms. The temperature coefficient of resistivity of platinum is 0.0038 . What is the temperature of the liquid?

Solution:

SuperconductivitySuperconductivity

19111911: H. K. Onnes, who had figured out how to make liquid helium, used it : H. K. Onnes, who had figured out how to make liquid helium, used it to cool mercury to 4.2 K and looked at its resistance:to cool mercury to 4.2 K and looked at its resistance:

At low temperatures the resistance of some metalsAt low temperatures the resistance of some metals0, measured to be less 0, measured to be less than than 1010-16-16••ρρconductorconductor (i.e., (i.e., ρρ<<1010-24 -24 ΩΩmm)!)!

Electrical energy and powerElectrical energy and power

In any circuit, battery is used to induce electrical currentIn any circuit, battery is used to induce electrical current chemical energychemical energy of the battery is transformed into of the battery is transformed into kinetic energykinetic energy

of mobile charge carriers (electrical energy gain)of mobile charge carriers (electrical energy gain)

Any device that possesses resistance (resistor) present Any device that possesses resistance (resistor) present in the circuit will transform electrical energy into heatin the circuit will transform electrical energy into heat

kinetic energykinetic energy of charge carriers is transformed into of charge carriers is transformed into heatheat via via collisions with atoms in a conductor (electrical energy loss)collisions with atoms in a conductor (electrical energy loss)

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Electrical energyElectrical energy

Consider circuit on the right in detailConsider circuit on the right in detail

AB: charge gains electrical energy AB: charge gains electrical energy form the batteryform the battery

(battery looses chemical energy)(battery looses chemical energy)

CD: electrical energy lost (transferred CD: electrical energy lost (transferred into heat)into heat)

Back to A: same potential energy Back to A: same potential energy (zero) as before(zero) as before

Gained electrical energy = lost Gained electrical energy = lost electrical energy on the resistorelectrical energy on the resistor

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PowerPower

Compute rate of energy loss (power dissipated on the resistor)Compute rate of energy loss (power dissipated on the resistor)

Use Ohm’s lawUse Ohm’s law

Units of power: SI: watt Units of power: SI: watt

delivered energy: kilowatt-hours delivered energy: kilowatt-hours

3 61 kWh 10 3600 3.60 10W s J

[Q] Calculate Determine the total current drawn by all the devices in the circuit in the figure.

ExampleExample

A high-voltage transmission line with resistance of 0.31 A high-voltage transmission line with resistance of 0.31 /km carries 1000A , /km carries 1000A , starting at 700 kV, for a distance of 160 km. What is the power loss due to starting at 700 kV, for a distance of 160 km. What is the power loss due to resistance in the wire? resistance in the wire?

Observations: 1. Given resistance/length, compute total resistance2. Given resistance and current, compute power loss

Now compute power

(1) An aluminum wire carrying a current has a diameter 0.800 mm. The electric field in the wire is 0.640 V/m. What is: a) the current carried by the wire? b) the potential difference between two points in the wire 12.0 m apart? C) the resistance of a 12.0 m length of the wire?Ans. a) 12.2 A; b) 7.68 V; c) 0.628 Ω

(2) A copper wire has resistance 5 Ohms. Given that the resistivity of silver is 85 percent of the resistivity of copper, what is the resistance of a silver wire three times as long with twice the diameter?

(3) A current of 5A exists in a 10 W resistor for 4min. (a) How many coulombs, and (b) how many electrons pass through any cross section of the resistor in this time?

(4) What is the resistance of a device that operates with a current of 7A when the applied voltage is 110V?

(5) Thermal energy is developed in a resistor at a rate of 100W when the current is 3.0A. What is the resistance in ohms?

(6) A 1250W radiant heater is constructed to operate at 115V. (a) What will be the current in the heater? (b) What is the resistance of the heating coil?