Download Document

Solid conducting sphere with charge Q
A
E
r A E0
rA E
V
A
r
1
Q
40 r 2
1 Q
rA V 
40 A
1
Q
40 A
r
rA V 
1 Q
40 r
Ionization and corona discharge
There is maximum potential to which a conductor in air
can be raised because of ionization.
Em  3106
V
m
Vm
 Em
R
Small potentials applied to sharp points in air produce
sufficiently high fields just outside the point to ionize the
surrounding air.
A lightning rod has a sharp end so that lightning bolts will pass through a
conducting path in the air that leads to the rod; a conducting wire leads from the
lightning rod to the ground.
The metal mast at the top of the Empire State
Building acts as a lightning rod. It is struck by
lightning as many as 500 times each year.
Benjamin Franklin
"For my own part I wish the Bald Eagle had not been chosen the Representative of
our Country. He is a Bird of bad moral Character. He does not get his Living
honestly. You may have seen him perched on some dead Tree near the River,
where, too lazy to fish for himself, he watches the Labour of the Fishing Hawk; and
when that diligent Bird has at length taken a Fish, and is bearing it to his Nest for
the Support of his Mate and young Ones, the Bald Eagle pursues him and takes it
from him.
"With all this Injustice, he is never in good Case but like those among Men who live
by Sharping & Robbing he is generally poor and often very lousy. Besides he is a
rank Coward: The little King Bird not bigger than a Sparrow attacks him boldly and
drives him out of the District. He is therefore by no means a proper Emblem for the
brave and honest Cincinnati of America who have driven all the King birds from our
Country....
"I am on this account not displeased that the Figure is not known as a Bald Eagle,
but looks more like a Turkey. For the Truth the Turkey is in Comparison a much
more respectable Bird, and withal a true original Native of America... He is besides,
though a little vain & silly, a Bird of Courage, and would not hesitate to attack a
Grenadier of the British Guards who should presume to invade his Farm Yard with
a red Coat on."
--Benjamin Franklin, in a letter to his daughter
Current, Ohm’s Law, Etc.
dQ
i
dt
V
Ohm ' s Law : R  ; R  Const (independent of V )
i
l
R
A
iave 
Q
t
Q dQ
i  lim

t 0 t
dt
V
Ohm ' s Law :
 Const
i
V
 R,
i
where R is resistance
Resistance does not vary with the applied voltage
resistor
Volts
R 

Ampere
Experimentally it is found that R depends on the material
the wire is made of and its dimensions. Does not depend
on the shape.
l
R
A
l is length,
A is the area
 is resistivit y

1

is conductivity
In a wire of uniform resistivity and cross sectional area,
the electric field is a constant for constant currents.
V
+
-
i
V VA 1 VA
i 


R
l 
l
  i 
Exercise 1
8
10
Given the resistivity of copper, about
Ohm-m, what
length of 0.5 cm diameter wire will yield a resistance of 10
Ohms?
Can you make a light bulb work
with a battery and a wire?
“Minds of Our Own”
by Dr. Matthew H. Schneps and Dr. Philip M. Sadler
Harvard-Smithsonian
How can students graduate from
prestigious schools like Harvard or
MIT and not know even some of the
most basic ideas in science taught in
grade school?
Current Density
 
i   j  dS
S
Consider current flowing in a homogeneous wire with cross sectional
area A.
 
i   j  dS   jdS  j  dS  jA
A
A
i
j
A
A
The Continuity Equation for Steady
State Currents
Currents and current densities
are constant in time –

steady state. The flux of j out of any closed
surface must be zero.
 
 j  dS  0
Another form of Ohm’s Law


j  E


E  j
For steady state situation
 
j

d
S

0

 
 E dr  0
Resistivity and temperature

T
Metal: ρ increases with increasing T
Semiconductors: ρ decreases with increasing T

T
Superconductor
1911 Dutch physicist Heike Kamerlingh
Onnes - Tc  4.2 K
Hg

0
2003
Tc
Tc  160K
T
  0 for T  Tc
Once a current has been established in a superconducting ring, it
continues indefinitely without the presence of any driving field.
Water
Nitrogen
Boils
100 C (212 F)
-196 C (-322 F)
-183 C (-297 F)
Freezes
0 C (32 F)
-210 C (-346 F)
-223 C (-369 F)
Oxygen
Our air is ¾ Nitrogen and ¼ Oxygen
Superconductivity
• 1908- liquefied helium produced
• First discovered in mercury by
Kamerlingh-Onnes in 1911.
• Critical temperature 4.21K.
• Nobel Prize in 1913.
High-Tc Superconductivity
Complex ceramic materials
were discovered in 1986. They
exhibit superconductivity at
much higher temperatures –
above LN temperature!
Muller and Bednortz, Nobel
Prize 1987
Liquid nitrogen temperature 77 K
Meissner effect and magnetic levitation
Current, Ohm’s Law, Etc.
dQ
i
dt
V
Ohm ' s Law : R  ; R  Const (independent of V )
i
l
R
A


j  E


E  j
The Continuity Equation for Steady
State Currents
Currents and current densities
are constant in time –

steady state. The flux of j out of any closed
surface must be zero.
 
 j  dS  0
For steady state situation
 
j

d
S

0

 
E

d
r

0

Problem 4
Two wires having different resistivities ρ1 and ρ2 and
equal cross sections, a, are connected end to end. Their
lengths are l1 and l2. If a battery is connected to this
system such that a potential difference of V is
maintained between the ends,
a) What will be the current densities in the wires?
b) What will be the potential difference across each wire?
c) Will there be any charge on the surface where the wires
are connected?
Exercise 5
Consider a cylindrical shell, inner radius a and outer radius
b. It is made of material with resistivity ρ. Suppose a current
can be made to flow out from the inner surface to the outer.
What would the resistance be for this current?
For steady state situation
 
j

d
S

0

 
 E dr  0
1.Kirchhoff’s junction rule: The algebraic sum
of the currents into any junction is zero.
2.Kirchhoff’s loop rule: The algebraic sum of
the potential differences in any loop must be
zero.