Transcript Chapter 10: Simple Harmonic Motion

Chapter 27: Current and Resistance
 Current
dQ
I
dt
Q
 In terms of the current density

t
0
I (t )dt
 
I  J  dA

 For a uniform current density parallel to the area
element J  I / A

 
 Ohm’s Law
J  E  E / 
 Practical version
V  RI
where the resistance is
l
R
A
Ohmic material
Nonohmic material
Example Problem
Lightning strikes the ground with a current of
100 kA. A person and a cow are each a radial
distance D=60.0 m from the lightning strike. The
current spreads through the ground uniformly
over a hemisphere centered on the strike point.
The person’s feet are separated by radial
distance rper=0.50 m; the cow’s front and rear
hooves are separated by the radial distance
rcow=1.50 m. The resistivity of the ground is
gr=100 m. The resistance both across the
person, between left and right feet, and across
the cow, between front and read hooves, is
R=4.00 k. What are the currents through the
person and through the cow?
Microscopic Description of
Current and Ohm’s Law
x
Consider a conductor with cross sectional area A
and a segment length x
If there is no potential difference across it, the
electric field in the wire is zero and therefore the
current is zero
However, there are electrons
moving within the conductor
These conduction electrons move
in random directions, but at
high speeds ~106 m/s
No net displacement of the electrons  no
current  no electric field
However, if a V is applied then there is an
electric field in the conductor and a current
Considering the current at a microscopic level,
there is
 a volume element of Ax
 with n total number of charge carriers
per unit volume
 each with positive charge q
The total charge in the volume element is
Q  nAxq
They move with a constant speed, the drift
speed vd
In a vacuum with a uniform electric field,
electrons move in a straight line in the opposite
direction of the field lines
However, in a conductor, the electrons travel for
short distances (~40 nm), in random directions
until they encounter and atom, where the
electron is scattered in a random direction
Nevertheless, the electrons
move slowly in the direction
opposite the electric field at the
drift speed (~10-4 m/s)
The drift speed of electrical conduction can be
understood through the Drude model which
applies classical mechanics
Example Problem 27.25
If the magnitude of the drift velocity of
free electrons in a copper wire is
7.84x10-4 m/s, what is the electric field
in the conductor?
Variation of Resistivity and
Resistance with Temperature
The values of most physical properties of
materials vary with temperature
For a conductor, it is found, over a limited
temperature range, that this variation is linear
in temperature
 (T )   0 [1   (T  T0 )]
0 is the resistivity at room
temperature T0=20 C and  is
the temperature coefficient of
resistivity. It is given in Table
27.1 and has units of 1/C
Example Problem 27.29
A certain light bulb has a tungsten
filament with a resistance of 19.0 
when cold and 140  when hot.
Assume that the resistivity of tungsten
varies linearly with temperature even
over the large temperature involved
here, and find the temperature of the
hot filament. Assume the initial
temperature is 20.0 C.
Power in Electrical
Circuits
A circuit is a closed loop (usually consisting
of high conductivity wire) with a battery and
one or more circuit elements
We assume now that the resistance in the
wire is small and can be neglected
b
a
Mechanical Analogy
Example Problem 27.36
A toaster is rated at 600 W when
connected to a 120-V source. What
current does the toaster carry, and what
is its resistance?