Current and Resistance
February 22, 2006
Notes
New topic today – Current and Resistance
 Exam – Friday – March 3 Through
Chapter 27.
 No quiz on Friday.
 Watch for new WebAssign.

A capacitor is constructed from two square plates of sides ℓ and
separation d. A material of dielectric constant κ is inserted a
distance x into the capacitor, as shown in the figure. Assume that
d is much smaller than x. (a) Find the equivalent capacitance of
the device. (b) Calculate the energy stored in the capacitor, letting
ΔV represent the potential difference. (c) Find the direction and
magnitude of the force exerted on the dielectric, assuming a
constant potential difference ΔV. Ignore friction. (d) Obtain a
numerical value for the force assuming that ℓ = 5.00 cm, ΔV = 2
000 V, d = 2.00 mm, and the dielectric is glass (κ = 4.50).
Each capacitor in the combination shown in Figure P26.49 has a
breakdown voltage of 15.0 V. What is the breakdown voltage of the
combination?
Current and Resistance
Physical Resistors
What Happens?
IN TRUTH, THE ELECTRONS
“+”
“+”
ARE ACTUALLY MOVING THE
OTHER WAY!
-
“+”
“+”
DEFINITION


Current is the motion of CHARGE through a
circuit. Physically, it is electrons that move
but …
We define current as the motion of POSITIVE
charge! (Blame you know who!)
Conducting material
DQ,Dt
Conducting material
DQ,Dt
CURRENT
DQ
i
Dt
or
dq
i
dt
UNITS

A current of one coulomb per second is
defined as ONE AMPERE.
A small sphere that carries a charge q is
whirled in a circle at the end of an
insulating string. The angular frequency
of rotation is ω. What average current
does this rotating charge represent?
ANOTHER DEFINITION
current I
J

area
A
Figure P27.8 represents a section of a circular conductor of
non-uniform diameter carrying a current of 5.00 A. The radius
of cross section A1 is 0.400 cm. (a) What is the magnitude of
the current density across A1? (b) If the current density across
A2 is one-fourth the value across A1, what is the radius of the
conductor at A2?
Ohm





A particular object will
resist the flow of current.
It is found that for any
conducting object, the
current is proportional to
the applied voltage.
STATEMENT: DV=IR
R is called the resistance of
the object.
An object that allows a
current flow of one ampere
when one volt is applied to
it has a resistance of one
OHM.
Graph
A DIODE
Resistance Varies with Applied Voltage
(actually with current)
Let’s look at the atomic level ..




Conduction is via electrons.
They are weak and small and don’t exercise
much.
Positive charge is big and strong and doesn’t
intimidate easily.
It’s an ugly situation … something like ……
+
-
Consider a metal conductor

So far, we have said that a metal is an equipotential
because no charges were moving.



Hence, no electric field in the metal
You can move a charge freely in the metal BECAUSE there
is no electric field.
NOW we have a current.



This can only happen if we allow an electric field to push
the charges.
Thus, the metal is NO LONGER A TRUE
EQUIPOTENTIAL.
But almost …. as we shall see in the next chapter.
Vb  Va
E
l
The Current


Electrons are going the other way.
They probably follow a path like …
Average “drift”
speed - vd
Notation





vd average drift velocity of the electron
n number of electrons (mobile) per unit
volume.
Dt interval of time
Dx average distance the electron moves in
time Dt.
Q total amount of CHARGE that goes
through a surface of the conductor in time Dt.
DQ  (nAvd Dt )e
DQ
I avg 
 nAvd e
Dt
I avg
J
 nevd
A
J  nev d
Reference
The average drift velocity of an
electron is about 10-4 m/s
Conductivity
In metals, the bigger the electric field at a
point, the bigger the current density.
J  E
 is the conductivity of the material.
r=(1/) is the resistivity of the material
Going to the usual limit …
dI
J
dA
and
I   JdA
Example
A cylindrical conductor of radius R has
a current density given by
(a) J0 (constant)
(b) gr
Find the total current in each case.
r  r0 1   (T  T0 )
Range of r and 
Ye old RESISTANCE
DV  El
1 DV
1 DV I
J  E 
E

r El
r l
A
rl
DV  I
A
rl
R
A
DV  V  IR
Additional Topics


Power
Micro basis of conduction.