Lecture 5 - Current
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Transcript of 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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A more detailed model
Iavg = DQ/Dt = n A vd q
Difficult to know vd directly.
Can calculate it.
E avv
A more detailed model
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A more detailed model
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
A more detailed model
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A more detailed model
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.