Lecture10 Electric Current; Current Density; Resistance

8/3/2019 Lecture10 Electric Current; Current Density; Resistance

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http://www.nearingzero.net(nz105.jpg)


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Comment on exam scores

It just goes to show, you do nothave to be faster than themonsters you just have to be

faster than your slowest companion.

Titan Quest screenshot, just after escaping a monsterthat ate a slow-footed companion.


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Comment on exam scores

y FS2011 Exam 1 grade distribution (regrades not included)

In Physics 24, youd better be faster than the monsters.


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Todays agenda:

Electric Current.You must know the definition of current, and be able to use it in solving problems.

Current Density.You must understand the difference between current and current density, and be able touse current density in solving problems.

Ohms Law and Resistance.You must be able to use Ohms Law and electrical resistance in solving circuit problems.

Resistivity.You must understand the relationship between resistance and resistivity, and be able tocalculate resistivity and associated quantities.

Temperature Dependence of Resistivity.You must be able to use the temperature coefficient of resistivity to solve problemsinvolving changing temperatures.


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Electric Current

Definition of Electric Current

The average current that passes any point in a conductorduring a time (t is defined as

where (Q is the amount of charge passing the point.

One ampere of current is one coulomb per second:

av

QI

t(!(

The instantaneous current isdQ

I= .dt

.1C

1A=1s


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Heres a really simple circuit:

+-

current

Dont try that at home! (Why not?)

The current is in the direction of flow of positive charge

opposite to the flow of electrons, which are usually the chargecarriers.

Currents in battery-operated devices are often in the milliamprange: 1 mA = 10-3A.

m for millianother abbreviation to remember!


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+ -

current electrons

An electron flowing from to +

Conventional refers to our convention, which is always toconsider the effect of + charges (for example, electric fielddirection is defined relative to + charges).

An electron flowing from to + gives rise to the sameconventional current as a proton flowing from + to -.


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Hey, that figure you just showed me is confusing.

+ -

current electrons

Good question.

Hey, that figure you just showed me is confusing. Why dontelectrons flow like this?


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Chemical reactions (or whatever energy mechanism the batteryuses) force electrons to the negative terminal. The battery

wont let electrons flow the wrong way inside it. So electronspick the easiest paththrough the external wires towards the +terminal.

Of course, real electrons dont want anything.

Electrons want to get away from - and go to +.

+ -

current electrons


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Note!

Current is a scalar quantity, and it has a sign associated with it.

In diagrams, assume that a current indicated by a

symbol and an arrow is the conventional current. I1

If your calculation produces a negative value for the current,

that means the conventional current actually flows opposite tothe direction indicated by the arrow.


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Example: 3.8x1021 electrons pass through a point in a wire in 4minutes. What was the average current?

av

Q NeI

t t

(! !

( (

21 19

av

3.8 10 1.6 10I4 60

v v!v

avI 2.53A!


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Todays agenda:




Resistivity.You must understand the relationship between resistance and resistivity, and be able to

use calculate resistivity and associated quantities.



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Current Density

When we study details of charge transport, we use the conceptof current density.

Current density is the amount of charge that flows across a unit

of area in a unit of time.

Current density: charge per area per time.


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A current density J flowing through an infinitesimal area dAproduces an infinitesimal current dI.

dA

J

dI J dA! &&

The total current passing through A is just

surface

I J dA! &&

Current density is a vector. Itsdirection is the direction of thevelocity of positive charge carriers.

Current density: charge per area per time.

No OSEs on this page.Simpler, less-generalOSE on next page.


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surface surface

II J dA J dA JA J

A

! ! ! !

&&

If J is constant and parallel to dA (like in a wire), then

A

J

Now lets take a microscopic view of current.

Av

v(t

q

If n is the number of charges

per volume, then the number ofcharges that pass through asurface A in a time (t is

number volume n v t Avolume

v ! (


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The total amount of charge passing through A is the number ofcharges times the charge of each.

Av

v(t

qQ nqv t A( ! (

Divide by (t to get the current

QI nqv A

t

(! !

(and by A to get J:

J nqv .!


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To account for the vector nature of the current density,

J nqv!

&

&

and if the charge carriers are electrons, q=-e so that

eJ n e v.! &

&

The sign demonstrates that the velocity of the electrons isantiparallel to the conventional current direction.

Not quite

official yet.

Not quiteofficial yet.


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Currents in Materials

Metals are conductors because they have free electrons,which are not bound to metal atoms.

In a cubic meter of a typical conductor there roughly 1028 freeelectrons, moving with typical speeds of 1,000,000 m/s.

But the electrons move in random directions, and there is nonet flow of charge, until you apply an electric field...


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-

E electron drift velocity

The voltage accelerates the electron, but only until theelectron collides with a scattering center. Then the electronsvelocity is randomized and the acceleration begins again.

Some predictions made by this theory are off by a factor or 10or so, but it was the best we could do before quantummechanics. The scattering idea is useful.

We will see later in this course that the electron would follow curved trajectories,

but the idea here is still valid.

just oneelectronshown, forsimplicity

inside aconductor


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Even though the details of the model on the previous slide arewrong, it points us in the right direction, and works when you

take quantum mechanics into account.

In particular, the velocity that should be used in

J n q v.!&

&

is not the charge carriers velocity (electrons in this example).

Instead, we should the use net velocity of the collection ofelectrons, the net velocity caused by the electric field.

This net velocity is like the terminal velocity of a parachutist;we call it the drift velocity.

dJ n q v .!&

&

Quantum mechanics shows us how to deal

correctly with the collection of electrons.


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Its the drift velocity that we should use in our equations forcurrent and current density in conductors:

dJ n q v!&

&

dI nqv A!

d

Iv

nqA!


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Example: the 12-gauge copper wire in a home has a cross-sectional area of 3.31x10-6 m2 and carries a current of 10 A.

The conduction electron density in copper is 8.49x1028

electrons/m3. Calculate the drift speed of the electrons.

d

Iv

nqA!

d

Iv

neA!

d 28 -3 19 6 2

10C/sv (8.49 10 m )(1.60 10 C)(3.31 10 m ) ! v v v

4

dv 2.22 10 m/s

! v


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Quiz time (maybe for points, maybe just for practice!)


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Todays agenda:








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Resistance

The resistance of a material is a measure of how easily acharge flows through it.

Resistance: how much push is needed toget a given current to flow.

VR

I!

The unit of resistance is the ohm:1V

1 .

1 A

; !

Resistances of kilohms and megohms are common:

3 61 k 10 , 1 M =10 .; ! ; ; ;


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This is the symbol we use for a resistor:

All wires have resistance. Obviously, for efficiency in carrying acurrent, we want a wire having a low resistance. In idealizedproblems, we will consider wire resistance to be zero.

Lamps, batteries, and other devices in circuits have resistance.

Every circuit component has resistance.


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Resistors are often intentionally used incircuits. The picture shows a strip of fiveresistors (you tear off the paper andsolder the resistors into circuits).

The little bands of color on the resistors have meaning.H

ereare a couple of handy web links:http://www.dannyg.com/examples/res2/resistor.htmhttp://xtronics.com/kits/rcode.htm


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Ohms Law

In some materials, the resistance is constant over a wide rangeof voltages.

For such materials, we write and call the equation

Ohms Law.

V IR,!

In fact, Ohms Law is not a Law in the same sense asNewtons Laws

Newtons Laws demand; Ohms Law suggests.

and in advanced classes you will write something other thanV=IR when you write Ohms Law.


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Materials that follow Ohms Law are called

ohmic materials, and have linear I vs. Vgraphs.

I

V

slope=1/R

I

V

Materials that do not follow Ohms Law arecalled nonohmic materials, and havecurved I vs. V graphs.


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Materials that follow Ohms Law are called

ohmic materials, and have linear I vs. Vgraphs.

I

V

slope=1/R

I

V

Materials that do not follow Ohms Law arecalled nonohmic materials, and havecurved I vs. V graphs.

Demo (if time allows):ohmic and nonohmic conductors.


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Demo:temperature dependence of resistivity.

Demo:resistive heating.

I may wait and do these demos next lecture.


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Todays agenda:








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This makes sense: a longer wire or higher-resistivity wireshould have a greater resistance. A larger area means morespace for electrons to get through, hence lower resistance.

It is also experimentally observed (and justified by quantummechanics) that the resistance of a metal wire is well-describedby

Resistivity

LR ,

A

V!

where V is a constant called the resistivity of the wirematerial, L is the wire length, and A its cross-sectional area.


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L

The longer a wire, the harder it is to push electrons through

it.

R =VL / A,

V

The greater the resistivity, the harder it is to push electronsthrough it.

The greater the cross-sectional area, the easier it is to pushelectrons through it.

A

Resistivity is a useful tool in physics because it depends on theproperties of the wire material, and not the geometry.

units ofV

are ;m


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Resistivities range from roughly 10-8 ;m for copper wire to1015 ;m for hard rubber. Thats an incredible range of23orders of magnitude, and doesnt even include superconductors

(we might talk about them some time).

R =VL / A

A =VL / R

A =T (d/2)2 geometry!

T (d/2)2=VL / R

Example (will not be worked in class): Suppose you want toconnect your stereo to remote speakers.(a) If each wire must be 20 m long, what diameter copper wireshould you use to make the resistance 0.10 ; per wire.


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(d/2)2=VL / TR

d/2= ( VL / TR ) dont skip steps!

d =2 ( VL / TR )

d =2 [ (1.68x10-8) (20) / T (0.1) ]

d = 0.0021 m =2.1 mm

V =I R

(b) If the current to each speaker is 4.0 A, what is the voltagedrop across each wire?


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V = (4.0) (0.10)

V = 0.4 V


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Homework hint you can look up the resistivity of copper in a

table in your text.


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Ohms Law Revisited

The equation for resistivity I introduced five slides back is asemi-empirical one. Heres how we define resistivity:

E.

J

V !

Our equation relating R and V follows from the above equation.

We define conductivity W as the inverse of the resistivity:

1 1, or .W ! V !

V W

NOT an officialstarting equation!


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With the above definitions,

E J,!V& &

J E.! W& &

The official Ohms law, valid for non-ohmic materials.

Cautions!

In this context:V is not volume density!W is not surface density!


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Example: the 12-gauge copper wire in a home has a cross-sectional area of 3.31x10-6 m2 and carries a current of 10 A.

Calculate the magnitude of the electric field in the wire.

IE J

A!V !V

86 2

(1.72 10 m) 10C/sE

(3.31 10 m )

v ; !

v

2E 5.20 10 V/m

! v

Homework hint (not needed in this particular example): in this chapter it is safe to use (V=Ed.


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Todays agenda:








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Temperature Dependence of Resistivity

Many materials have resistivities that depend on temperature.We can model* this temperature dependence by an equationof the form

0 01 T T ,V ! V E -

where V0 is the resistivity at temperature T0, and E is thetemperature coefficient of resistivity.

*T0 is a reference temperature, often taken to be 0 C or 20 C. This approximationcan be used if the temperature range is not too great; i.e. 100 C or so.


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The result is very sensitive to significant figures in resistivityand E.

-10.0005 CE ! r

RA(R)L

V !

0

0

1T(R) T 1

V!

E V-

T(0.030) 122.6 C! r

Documents

Lecture10 Electric Current; Current Density; Resistance