Download Lecture 8 - UConn Physics

Physics 1502: Lecture 8
Today’s Agenda
• Announcements:
– Lectures posted on:
www.phys.uconn.edu/~rcote/
– HW assignments, solutions etc.
• Homework #3:
– On Masterphysics today: due next Friday
– Go to masteringphysics.com
Today’s Topic :
• Review Chapter 23:
– Definition of Capacitance
– Example Calculations
(1) Parallel Plate Capacitor
(2) Cylindrical Capacitor
(3) Isolated Sphere
– Energy stored in capacitors
– Dielectrics
– Capacitors in Circuits
Definitions & Examples
A
++++
d -----
a
b
L
a
C3
b
C1
C2

a
b
C
Capacitors in Parallel
a
a
V
Q1

Q2
C1
C2
V
b
b
Q
C

C = C1 + C2
Capacitors in Series
+Q
-Q
a
b
C1
C2

+Q
-Q
a
b
C

Energy of a Capacitor
• The total work W to charge a capacitor to Q is :
• In terms of the voltage V:
• The energy is stored in the Electric field itself.
• The energy density u in the field is :
Units: J/m3
• Dielectrics INCREASE the capacitance of a
capacitor: k dielectric constant of the material
e
R
I
e=RI
Overview
• Charges in motion
– mechanical motion
– electric current
• How charges move in a conductor
• Definition of electric current
Text Reference: Chapter 24
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 a
constant 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!
Charges in motion
• We know from experience that charges do move inside
conductors - this is the definition of a conductor
Is there a contradiction?
no
• Up to now we have considered isolated conductors in
equilibrium.
– 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 if
driven by a force such as a battery.
Current Definition
+
+
+
E
+
+
+

Consider charges moving down a conductor in which
there is an
electric
Note:
Thisfield.
definition assumes

If I take athe
cross
section
thedirection
wire, over
current
in of
the
ofsome amount of
time Dt I will count a certain number of charges (or total
the positive particles,
amount of charge) DQ moving by.

We define current as the ratio of these quantities,
NOT in the direction of the electrons!

Iavg = DQ / Dt or
I = Q/ t

Units for I, Coulombs/Second (C/s) or Amperes (A)
How charges move in a conducting material
E

Electric force causes gradual drift of bouncing electrons down
the wire in the direction of -E.

Drift speed of the electrons is VERY slow compared to the
speed of their bouncing motion, roughly 1 m / h !
(see example later)
Good conductors are those with LOTS of mobile electrons.
How charges move in a conducting material
E

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 * (vd Dt). Why ???

So, DQ = n A vd Dt q

And Iavg = DQ/Dt = n A vd q

More on this later …
Drift speed in a copper wire
• The copper wire in a typical residential building has a
cross-section area of 3.31e-6 m2. If it carries a current of
10.0 A, what is the drift speed of the electrons? (Assume
that each copper atom contributes one free electron to
the current.) The density of copper is 8.95 g/cm3, its
molar mass 63.5 g/mol.
• Volume of copper (1 mol):
• Because each copper atom contributes one free electron
to the current, we have (n = #carriers/volume)
Drift speed in a copper wire, ctd.
• We find that the drift speed is
with charge / electron q
• Thus
Resistance
R
• Resistance
Resistance is defined to be the
ratio of the applied voltage to
the current passing through.
I
I
V
UNIT: OHM = W
•
Is this a good definition?
i.e. does the resistance belong only to the resistor?
Recall the case of capacitance: (C=Q/V) depended on the geometry,
not on Q or V individually
Does R depend on V or I ?
Ohm's Law
• Vary applied voltage V.
I
R
I
• Measure current I
• Does ratio ( V/I ) remain
constant??
V
V
slope = R = constant
I
Resistivity
• Property of bulk matter related to
resistance of a sample is the
resistivity r defined as:
E
j
A
where E = electric field and
j = current density in conductor = I/A.
For uniform case:
n0 : carrier density (carriers/volume)
q : charge per carrier
material constant
v : carrier speed
h : viscosity
material
constant
L
Resistivity
E
j

A
L

So, in fact, we can compute the resistance if we know a bit about the
device, and YES, the property belongs only to the device !
eg, for a copper wire, r ~ 10-8 W-m, 1mm radius, 1 m long, then R  .01W
Make sense?
E
j
A
L
• 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
How thick?
or
How big?
What’s it made of?
Alternative Version of Ohm’s Law
• A related empirical observation
is that:
E
j
This is an alternative version of
Ohm’s Law. It can also be written as,
A
L
with
We can show this is also Ohm’s Law using the relations,
and
2
Lecture 8, ACT 1
• 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:
I1
I2
V
– 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
Current Idea
E

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.
Lecture 8, ACT 2
Consider a circuit consisting
of a single loop containing a
battery and a resistor.
1
e
4
-
Which of the graphs represents
the current I around the loop?
R
+
I
2
1
2
-
+
3
4
x
1
2
-
+
3
4
3
1
2
-
+
3
4
A more detailed model
E

Iavg = DQ/Dt = n A vd q

Difficult to know vd directly.

Can calculate it.
A more detailed model
E

Iavg = DQ/Dt = n A vd q

The force on a charged particle is,

If we start from v=0 (on average) after a collision then we
reach a speed,
t : average
collision-free
time

Substituting gives, (note j = I/A)
or
A more detailed model
E

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 a wave.
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)
Lecture 8, ACT 3
E
I am operating a circuit with a power supply and a resistor. I
crank up the power supply to increase the current. Which of
the following properties increases,
A) n
B) q
C) E
D) t
Conductivity versus Temperature
• In lab you measure the resistance of a light bulb
filament versus temperature.
• You find RT.
• This is generally (but not always) true for metals
around room temperature.
• For insulators R1/T.
• At very low temperatures atom vibrations stop. Then
what does R vs T look like??
• This was a major area of research 100 years ago –
and still is today.