Solid conducting sphere with charge Q

A

E

r

204

1

r

QEAr

0 EAr

V

r

A

A

Q

04

1

r

QVAr

04

1

A

QVAr

04

1

Ionization and corona discharge

There is maximum potential to which a conductor in air can be raised because of ionization.

m

VEm

6103

mm E

R

V

Small potentials applied to sharp points in air produce sufficiently high fields just outside the point to ionize the surrounding air.

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.

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.

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.

dt

dQi

€

Ohm's Law : R =V

i; R = Const (independent of V )

A

lR

t

Qiave

dt

dQ

t

Qi

t

0

lim

Consti

VLawsOhm :'

,Ri

V where R is resistance

Resistance does not vary with the applied voltage

resistor

Ampere

VoltsR

Experimentally it is found that R depends on the material the wire is made of and its dimensions. Does not depend on the shape.

A

lR

areatheisAlengthisl ,

yresistivitis

tyconductiviis

1

In a wire of uniform resistivity and cross sectional area, the electric field is a constant for constant currents.

+

-V i

l

VA

l

VA

R

Vi

1

i

Exercise 1810

Given the resistivity of copper, about Ohm-m, what length of 0.5 cm diameter wire will yield a resistance of 10 Ohms?

Current Density

S

Sdji

Consider current flowing in a homogeneous wire with cross sectional area A.

jAdSjjdSSdjiA A A

A

ij

The Continuity Equation for Steady State Currents

0Sdj

Currents and current densities are constant in time – steady state. The flux of out of any closed surface must be zero.

j

Another form of Ohm’s Law

Ej

jE

For steady state situation

0 Sdj

0 rdE

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,

• What will be the current densities in the wires?• What will be the potential difference across each wire?• Will there be any charge on the surface where the wires

are connected?