Lecture 7.1 : Current and ResistanceAnnouncements 2 •Homework #7 is now posted.Due next Tue., March 4, at 9am. •Optional review session for Exam #2 is scheduled for Tue., March

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Lecture 7.1 :Current and Resistance

Lecture Outline:Current and Current DensityConductivity and ResistivityResistance and Ohm’s Law

Feb. 25, 2014

Textbook Reading:Ch. 30.3 - 30.5

Announcements

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•Homework #7 is now posted. Due next Tue., March 4, at 9am.

•Optional review session for Exam #2 is scheduled for Tue., March 4, from 5-7pm in Stolkin. Bring your clickers.

Last Lecture...

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Conduction electrons are like free particles moving through a metal lattice.

No Electric FieldAverage Velocity is Zero

With Electric FieldAverage Velocity is NonZero

Last Lecture...

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Electrons are the charge carriers in metals!

electron current (ie) = number of electrons per second that pass through a conductor in a given time.

ie = neA�d =nee⇥A

mE

Clicker Question #1

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A wire carries a current. If both the wire diameter and the electron drift speed are doubled, the electron current increases by a factor of

A. 2.B. 4.C. 6.D. 8.E. Some other value.

ie ∝ Aνd

Current and Current Density

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I � dQ

dt

Current was known long before electrons were discovered, so conventional definition of current is in terms of Charge:

1 ampere = 1 A ≡ 1 coulomb per second = 1 C/s

I =Q

�t=

eNe

�t= eie

Conventional current and electron current are related:

Current and Current Density

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By convention, current is defined to flow in the direction in which

positive charges would move! (i.e. - in the direction of the Electric Field)

Current and Current Density

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We define current density (J) to arrive at a property intrinsic to a material, and not specific to the cross-sectional Area of a conductor.

J � I

A= nee�d

The current in a 100 Watt lightbulb is 0.85 A. The filament inside the bulb is 0.25 mm in diameter. What is the current density in the filament?

Current and Current Density

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Current is a conserved quantity! The current is the same at all points in a current-carrying wire.

Current is not “used up” by the paddle...but energy is required to turn the paddle.

Clicker Question #2

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A and B are identical lightbulbs connected to a battery as shown. Which is brighter?

A. Bulb A.B. Bulb B.C. The bulbs are equally bright.

Current and Current Density

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�Iin =

�Iout

Kirchoff’s Junction Law:

Current = ?

Conductivity and Resistivity

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J = nee�d = nee

�e⇥E

m

⇥=

nee2⇥

mE

J = �E

� = conductivity =nee2⇥

mDefine a new

property of materials

Conductivity describes how easily electrons can move through a material. (Also depends on temperature)

Clicker Question #3

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The current density in this wire is

A. 4 × 106 A/m2.B. 2 × 106 A/m2.C. 4 × 103 A/m2.D. 2 × 103 A/m2.E. Some other value.

Current Density: J = Current/Area = (8A)/(0.002m x 0.002m) = 2x106 A/m2

Which way does the current density vector point?

Clicker Question #4

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Both segments of the wire are made of the same metal. Current I1 flows into segment 1 from the left. How does current I1 in segment 1 compare to current I2 in segment 2?

A. I1 > I2.B. I1 = I2.C. I1 < I2.D. There’s not enough information to

compare them.

Clicker #5

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Both segments of the wire are made of the same metal. Current I1 flows into segment 1 from the left. How does current density J1 in segment 1 compare to current density J2 in segment 2?

A. J1 > J2.B. J1 = J2.C. J1 < J2.D. There’s not enough information to

compare them.

Smaller cross-section area

Conductivity and Resistivity

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� = resistivity =1

⇥=

m

nee2⇤

Inverse of conductivity (resistivity) tells how reluctant electrons are to move through a material.

Resistance and Ohm’s Law

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E =�V

�s=

�V

LI = JA = A⇥E =

A

�E

Potential Difference across conductor creates Electric field.

R = resistanceOhm’s Law: I =

A

�L�V � �V

R

Resistance and Ohm’s Law

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Resistance (R) characterizes ability of current to flow through specific piece of conductor with a specific geometry.

R =�L

A

1 ohm = 1 Ω ≡ 1 Volt/Ampere = 1 V/A

Wire 2 is twice the length and twice the diameter of wire 1. What is the ratio R2/R1 of their resistances?

Conductivity and Resistivity

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Superconductivity!

Some materials exhibit zero resistance to current if they are cooled to low enough temperatures.

Conductivity and Resistivity

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A 3.0 mm diameter wire carries a 12 A current when the electric field is 0.085 V/m. What is the wire’s resistivity?

Resistance and Ohm’s Law

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Ohm’s Law: Establishing a potential difference ΔV across the ends of a conductor with resistance R causes a current I through the current.

I =�V

R

Example: What resistor would have a 15 mA current if connected across the terminals of a 9.0 V battery?

Resistance and Ohm’s Law

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In this course, we will assume wires are ideal (R = 0 Ω ), so they don’t have any potential drop across them.

Real wires have some resistance, and use up some energy in a circuit.

Resistance and Ohm’s Law

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Resistors are placed in electrical circuits to control the voltage at different points in the circuit.

Resistors are made out of materials, such as carbon, that are poor conductors.

Reminders

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•Homework #7 is now posted.•Come see me if you have any questions or concerns.