Lecture 5 - Current

7/31/2019 Lecture 5 - Current

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ELECTRIC CURRENT, CHARGE, DENSITY &DRIFT VELOCITY

RESISTANCE & RELUCTANCE

EFFECT OF TEMPERATURE ON RESISTANCE

5th LECTURE

Semester 2 2011/2012

ELECTRIC & MAGNET


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Overview

Charges in motion mechanical motion electric current

How charges move in a conductor

Definition of electric current

Resistance vs. Temperature


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Charges in Motion

Up to now we have considered

fixed charges on isolated bodies

motion under simple forces (e.g. a single charge moving in aconstant electric field)

We have also considered conductors

charges are free to move

we also said that E=0 inside a conductor

If E=0 and there is any friction (resistance) present

no charge will move!


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Charges in motion

We know from experience that charges do move insideconductors - this is the definition of a conductor

Is there a contradiction?

no

Up to now we have considered isolated conductors inequilibrium.

Charge has nowhere to go except shift around on the body.

Charges shift until they cancel the E field, then come to rest.

Now we consider circuits in which charges can circulate ifdriven by a force such as a battery.


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Current Definition

Consider charges moving down a conductor in which there

is an electric field.

If I take a cross section of the wire, over some amount of

time Dt I will count a certain number of charges (or total

amount of charge) DQ moving by.

We define current as the ratio of these quantities,

Iavg = DQ /Dt or I = Q/t

Units for I, Coulombs/Second (C/s) or Amperes (A)

E

+

+

+

+

+

+

Note: This definition assumes

The current in the direction of

the positive particles,

NOT in the direction of the electrons!


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How charges move in a conducting material

Electric force causes gradual drift of bouncing electrons down the

wire in the direction of -E.

Drift speed of the electrons is VERYslow compared to the speed

of their bouncing motion, roughly 1 m / h !

(see example later)

Good conductors are those with LOTS of mobile electrons.

Eav

v


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How charges move in a conducting material

DQ is the number of carriers in some volume times the charge

on each carrier (q).

Let n be the carrier density, n = # carriers / volume.

The relevant volume is A * (vdDt). Why ???

So, DQ = n A vdDt q

And Iavg = DQ/Dt = n A vd q

More on this later

E avv


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Drift speed in a copper wire

Because each copper atom contributes one free electron

to the current, we have (n = #carriers/volume)

Volume of 1 mol copper:

The copper wire in a typical residential building has a

cross-section area of 3.31e-6 m2. If it carries a current of10.0 A, what is the drift speed of the electrons? (Assumethat each copper atom contributes one free electron to thecurrent.) The density of copper is 8.95 g/cm3, its molarmass 63.5 g/mol.


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Drift speed in a copper wire, ctd.

We find that the drift speed is

with charge / electron q

Thus

Then why a light turns on almost instantaneously when itsswitch flipped ??????


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Resistance

Resistance

Resistance is defined to be theratio of the applied voltage tothe current passing through.

If the resistance of a material is constant over a considerablerange of voltage, then the material is described to obeys Ohm'slaw.

V

I I

R

RV

I

UNIT: OHM =


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Ohm's Law

Vary applied voltage V.

Measure current I

Does ratio ( V/I) remainconstant??

V

I

slope = R

V

I IR

RV

I

= constant


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Resistivity

LA

E

j

Electrical resistivity is a measureof how strongly a materialopposes the flow of electriccurrent. A low resistivity indicatesa material that readily allows themovement of electrical charge.The SI unit of electrical resistivity

is the ohm meter. Resistivity or Rho is defined as:

where E = electric field and

j = current density in conductor = I/A.

j

E
http://en.wikipedia.org/wiki/Electrical_chargehttp://en.wikipedia.org/wiki/SIhttp://en.wikipedia.org/wiki/Ohmhttp://en.wikipedia.org/wiki/Ohmhttp://en.wikipedia.org/wiki/SIhttp://en.wikipedia.org/wiki/Electrical_charge


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Resistivity

LA

E

j

eg, for a copper wire, ~ 10-8 -m, 1mm radius, 1 m long, thenR .01

RL

A

So, in fact, we can compute the resistance if we know a bit about thedevice, and YES, the property belongs only to the device !

From other equations, it isfound that resistance R canbe expressed :

?


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Make sense?

LA

E

j

Increase the Length, flow of electrons impeded

Increase the cross sectional Area, flow facilitated The structure of this relation is identical to heat flow through

materials think of a window for an intuitive example

RL

A

How thick?

How big?

Whats it made of?

or


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Table 27-1, p.837

E l


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Example

Two cylindrical resistors, R1 and R2, are made of identical material. R2has twice the length of R1 but half the radius of R1.

These resistors are then connected to a battery V as shown:

V

I1 I2

What is the relation between I1, the current flowing in R1, and I2,the current flowing in R2?

(a) I1 < I2 (b) I1 = I2 (c) I1 > I2

Example


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Example Two cylindrical resistors, R1 and R2, are made of identical material. R2

has twice the length of R1 but half the radius of R1.

These resistors are then connected to a battery V as shown:

VI1 I2

What is the relation between I1, the current flowing in R

1, and I

2,

the current flowing in R2?

(a) I1 < I2 (b) I1 = I2 (c) I1 > I2 The resistivity of both resistors is the same . Therefore the resistances are related as:

R LA

LA

LA

R22

2

1

1

1

11

24

8 8 ( / )

The resistors have the same voltage across them; therefore

IV

R

V

RI2

2 11

8

1

8


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Table 27-2, p.838

Current Idea


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Current Idea

Current is the flow of charged particles through a path, at

circuit. Along a simple path current is conserved, cannot create or

destroy the charged particles

Closely analogous to fluid flow through a pipe.

Charged particles = particles of fluid

Circuit = pipes

Resistance = friction of fluid against pipe walls, with itself.

E avv

E ample


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Example

R

I

1

2 3

4

+

-

x1 2 3 4+-

1 2 3 4

+-

1 2 3 4

+-

Consider a circuit consistingof a single loop containing a

battery and a resistor.

Which of the graphs represents the

currentIaround the loop?

Example


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Example

Which of the graphs represents the

currentIaround the loop?

x1 2 3 4+-

1 2 3 4

+-

1 2 3 4

+-

The general rule for any component in a circuit

Works for conductors, batteries, resistors...

There is only one way in and one way out for eachcomponent in this circuit

Therefore the current everywhere must be the same

current in = current out

Example addendum


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Example, addendum

x1 2 3 4+-

1 2 3 4

+-

1 2 3 4

+-

Which of the graphs represents

the potential V around the loop?

The battery maintains a positive potential differencebetween its positive and negative terminals.

Current in all components must be the same.

Large electric fields are required to make current Iflow in a resistor, compared to the conductor.

Electric field in the conductor is very small so wecan consider that the potential is constant there.

A more detailed model


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Iavg = DQ/Dt = n A vd q

Difficult to know vd directly.

Can calculate it.

E avv



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Iavg = DQ/Dt = n A vd q

The force on a charged particle is,

E avv

If we start from v=0 (on average) after a collision then we

reach a speed,

or

Substituting gives, (notej = I/A)

t: averagecollision-free

time



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This formula is still true for most materials even for the most

detailed quantum mechanical treatment.

In quantum mechanics the electron can be described as awave. Because of this the electron will not scatter off of atoms

that are perfectly in place in a crystal.

Electrons will scatter off of

1. Vibrating atoms (proportional to temperature)

2. Other electrons (proportional to temperature squared)

3. Defects in the crystal (independent of temperature)

E avv


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Conductivity versus Temperature

Over a limited temperature range, the resistivity of a

conductor varies approximately linearly withtemperature.

This implies R T.

But R will generally not decreases to 0 at low

temperature. For insulators R 1/T.

For some special material R decreases to 0 belowcertain temperature Tc (critical temperature). Thismaterial is known as Superconductor.

This was a major area of research 100 years ago andstill is today.

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Lecture 5 - Current