phy1004w buffler m&ie&m1

Prof Andy Buffler

Room 503 RW James

[email protected]

PHY1004W 2010

Electricity and MagnetismPart 1

60 lectures: 3rd period, Monday to Friday

12 weekly problem sets

12 Tuesday afternoon tutorials and laboratories

2 class tests

1 final examination (November)

Class tutors: Maciej Stankiewicz and Michael Malahe

Use them!

Also check the course website regularly for resources

PHY1004W Second Semester 2008

These lecture notes are not a substitute for

check for significant errata files on website

My expectations of you

1. that if you come to lectures, then you will engage with what

is happening

2. that you read M&I daily (before and after lectures)

3. that you do what I ask you to do

4. that you will not copy another students work, but work

together, where appropriate.

(Collaboration becomes copying when both parties are not

gaining positive learning from the activity.)

5. spend enough time at home working on what you need to

what you can expect from me

1. the best course that I can deliver

2. a reasonable and appropriate homework load.

3. no mercy in the face of plagiarism

4. an open door policy

Real world (phenomena)

Physical model (shared, contextual)

idealization

Physical theories (shared, acontextual)

particularization

When making sense of the

ideas in this course, its

useful to think about both the

nature of physics and how

you learn physics yourself ...

Draw one or more pictures which show all the important objects, their

motion and any interactions.

Now ask What is being asked? Do I need to calculate something?

Think about what physics concepts and principles you think will be

useful in solving the problem and when they will be most useful.

Construct a mental image of the problem situation - do your friends

have the same image?

Specify a convenient system to use - circle this on your picture.

Identify any idealisations and constraints present in the situation -

write them down!

Specify any approximations or simplifications which you think will

make the problem solution easier, but will not affect the result

significantly.

An approach to solving physics problems

Step 1. Think carefully about the problem situation and draw a

picture of what is going on (Pictorial Representation).

Draw a coordinate axis (or axes) onto your picture (decide where to put the

origin and on the direction of the axes).

Translate your pictures into one or more diagrams (with axes) which only

gives the essential information for a mathematical solution.

If you are using kinematic concepts, draw a motion diagram specifying the

objects velocity and acceleration at definite positions and times.

If interactions or statics are important, draw idealised, free body and force

diagrams.

When using conservation principles, draw initial and final diagrams to

show how the system changes.

For optics problems draw a ray diagram.

For circuit problems, a circuit diagram will be useful.

Define a symbol for every important physics variable in your diagram and

write down what information you know (e.g. T1 = 30 N).

Identify your target variable? (What unknown must I calculate?).

Step 2. Describe the physics (Physics Representation).

Only now think about what mathematical expressions relate the physics

variables from your diagrams.

Using these mathematical expressions, construct specific algebraic

equations which describe the specific situation above.

Think about how these equations can be combined to find your target

variable.

Begin with an equation that contains the target variable.

Identify any unknowns in that equation

Find equations which contain these unknowns

Do not solve equations numerically at this time.

Check your equations for sufficiency... You have a solution if your plan has

as many independent equations are there are unknowns. If not, determine

other equations or check the plan to see if it is likely that a variable will

cancel from your equations.

Plan the best order in which to solve the equations for the desired variable.

Step 3. Represent the problem mathematically and plan a solution

(Mathematical Representation).

Do the algebra in the order given by your outline.

When you are done you should have a single equation with your target variable.

Substitute the values (numbers with units) into this final equation.

Make sure units are consistent so that they will cancel properly.

Calculate the numerical result for the target variable.

Step 4. Execute the plan

Step 5. Evaluate your solution

Do vector quantities have both magnitude and direction ?

Does the sign of your answer make sense ?

Can someone else follow your solution ? Is it clear ?

Is the result reasonable and within your experience ?

Do the units make sense ?

Have you answered the question ?

1 2 3 4 5In-class voting questions

Bring your 12345 sheets along with you to class every day

and have them ready !

A practice question:

I am really pleased to be back in PHY1004W because:

1. All vacation long I dreamed of physics

2. I missed the smell of this lecture theatre

3. Physics is my best course

4. I am a masochist

5. None of the above.

1 2 3 4 5

Which textbook do you have?

1.

2.

3. None, and I dont think I need one.

4. None, but I am planning to get one.

5. None, but I share with a friend.

11

The story so far.

Right handed coordinate system:

Unit vectors kji 1 kji

, ,

( )

( , , )

( , , )

( , , )

( ) ( , , )

( )

d

x y z x y z

x y z x y z

x x y y z z

x y z

x x y y z z

x y z

x x y y z z

A ,A ,A A A A

B B B B B B

A B A B A B

A A A

A B A B A B

c cA cA cA

A B A B A B

A i j k

B i j k

A B

A

A B A B

A

A B

A B B A2 A

1 0

A A

i i j j k k i j j k k i

z

y

x

k j

i

Vector algebra

12

( ) 0

0 ; ;

A B B A A A

i i j j k k i j k j k i k i j

where and G G A G B

( ) + ( ) + ( ) y z z y z x x z x y y xA B A B A B A B A B A BA B i j k

easy to remember:always

x y z

x y z

A A A

B B B

i j k

In polar form in 2D:

and

where is the angle between tails of and .

cosABBA

kBA sinAB

B

A

The spherical polar coordinate system

2 2 2

cos sin

sin sin

cos

cos

tan

x y z

x

y

z

x y z

z

y

x

A A A

A A

A A

A A

A A A A

A

A

A

A

A i j k

Spherical coordinates: A, , :

z

y

x

k j

i

Ax

Az

Ay

A

The cylindrical polar coordinate system

2 2

cos

sin

tan

x y z

x

y

z

x y

y

x

z

A A A

A

A

A z

A A

A

A

z A

A i j k

Cylindrical coordinates: , , z :

z

y

x

k j

i

Ax

Az

Ay

A

z

( ) ( ) ( ) ( )

( )( ) ( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( )

x y z

yx z

t A t A t A t

dA tdA t dA tdt

dt dt dt dt

d d t d tt t

dt dt dt

d dc t d tc t t t c t

dt dt dt

d d t d tt t t

dt dt

A i j k

A i j k

A BA B

AA A

B AA B A ( )

( ) ( )( ) ( ) ( ) ( )

tdt

d d t d tt t t t

dt dt dt

B

B AA B A B

Differentiation of vector functions

Also:

then

If

If position m

then the instantaneous velocity m s-1

and the instantaneous acceleration m s-2

( ) ( ) ( ) ( )dx t dy t dz t

tdt dt dt

v i j k

( )( )

d tt

dt

va

( )( )

d tt

dt

rv

( )( ) ( ) ( )yx z

dv tdv t dv tt

dt dt dta i j k

( ) ( ) ( ) ( )t x t y t z tr i j k

Example of the time derivatives of a vector function

M&I

Chapter 13

Electric Field

Scalar and vector fields

A scalar or vector field is a distribution of a scalar or vector

quantity on a specified surface or throughout a specified

region of space such that there is a unique scalar or vector

associated with each position.

Fields may be time independent, e.g.

or time dependent

( , , )T x y z

( , , , )T x y z t

Examples of scalar fields:

Temperature, or

Potential, or

Examples of vector fields:

Electric field

Velocity

( )T r( , , )T x y z

( , , )V x y z ( )V r

( )E r

( )v r

Scalar fields

A scalar field can be represented by specifying a finite

number of scalar values at strategic positions in the region of

interest.

It is also possible to draw contour curves - continuous curves

joining points where the scalar values are the same.

In 3D space these contours are surfaces. Such representations

are always incomplete, since an infinite number of contours

or surfaces should really be drawn.

A third way of representing a scalar field is by a mathematical

function.

Scalar fields in 2D ....

... and 3D ...

A vector field is a vector function of position.

Vector fields may be represented

visually by field lines which

are everywhere parallel to the

local value of the vector function.

These lines are sometimes called

lines of force in mechanics and

stream lines in fluid mechanics.

A vector field may also be represented by lines which are

everywhere a tangent to the vectors. Although we lose track of the

lengths of the vectors, we can keep track of the strength of the field

by drawing lines far apart where the field is weak, and close where

it is strong.

Vector fields

Vector fields may also be represented mathematically, often

using differential equations.

We adopt the convention that the

number of lines per unit area at

right angles to the lines is

proportional to the field strength.

The electric field around a point charge q may be written as

where and

In Cartesian form

and

2

0

( ) 4

qE r r

r

r

rrr

0

2 2 2 2 2 2 3 22 2 20 0

( ) ( ) ( )

4 ( ) 4 ( )

q x y z q x y z

x y z x y zx y z

i j k i j kE r

kjir zyx

Example of a vector field function of position: the electric field

2 2 2

x y z

x y z

r i j kr

r

VPython scripts used in class can be found in the EM section of the

PHY1004W web site.

Also look at these PhET simulations from the University of Colorado:

Digital resources

Electric fieldElectric field

hockey

Charges

and fieldsTravoltage

A proton is at location < 0, 3, 2 > m.

An electron is at location m.

What is the relative position vector from the proton

to the electron?

1. < 1, 3, 8 > m

2. < 1, 3, 4 > m

3. < 1, 3, 4 > m

4. < 1, 3, 8 > m

5. < 1, 0, 6 > m

1 2 3 4 5

Electrostatics

Thales of Miletus (640-548 BC) basic phenomena of

charging on intimate contact (friction)

William Gilbert, 1574-1603, physician to Queen Elizabeth

amber, rubbed with cloth or fur, acquires the property of

attracting small bodies the amber has become electrified.

Dufay, 1733, originated (?) the two fluid theory of

electrification, calling the two sorts of electricity

vitreous (on glass) and resinous (on amber)

Benjamin Franklin (1747) introduced the terms positive and

negative

The Electrostatic Force

Can be either attractive or repulsive (gravity only attractive)

Can act through empty space

Very much stronger than gravitational force

From experiments we find that the electrostatic force decreases

with distance r as , and it is also proportional to the

product of the amount of charge on each of the charges:

Electrostatic forces are mutual forces of attraction

i.e. they obey Newtons 3rd law.

21

r

1 2F q q

Coulombs Law

Coulomb (1785) put a quantitative basis to

the observations that charged particles

attracted or repelled one another.

... measured the forces using a tensional

balance and found that ...

1 221 2

12

q q

rF 1 221 2

12

q qk

rF Coulombs Lawor

where: is the force on due to21F 2q 1q

9 2 -2

0

1 constant (from experiment) 9 10 N m C

4k

-12 2 -1 -2

0 permittivity of free space = 8.85 10 C N m

1 2

2

0 12

1

4

q q

r21 12

F r

Where is the unit vector (magnitude = 1) which

indicates the direction along which the force is acting

i.e. from q1 to q2.

12r

12 r12 12r r

i.e. but

12 21

12 21 12 21

F F

F F r r

Coulombs Law

So

Note:

1q 12r

21F

2q21r

12F

12r

+

+

+

Silicon atoms

Particle Mass Charge

electron 9.11 10-31 kg e = 1.60 10-19 C

positron 9.11 10-31 kg +e = 1.60 10-19 C

proton 1.67 10-27 kg +e = 1.60 10-19 C

antiproton 1.67 10-27 kg e = 1.60 10-19 C

muon 1.88 10-28 kg +e ( +) or e ( )

pion 2.48 10-28 kg +e (+) or e ( )

Some charged particles

Definition of electric field:

... where is the electric field at the location of charge q2

The electric field

a region in which a charge experiences a (mechanical) force is

called an electric field.

we assume that a charge creates a field of influence around it.

Any other charge present in that region will experience a force.

This force is described by Coulombs Law.

2q2 1F = E

1E

2q 2F

1E

1N CUnits:

( , , , )x y z tE E

M&I

13.3

The electric field around a point charge q may be written as

where and

In Cartesian form

and

The electric field of a point charge

2

0

( ) 4

qE r r

r

r

rrr

0

2 2 2 2 2 2 3 22 2 20 0

( ) ( ) ( )

4 ( ) 4 ( )

q x y z q x y z

x y z x y zx y z

i j k i j kE r

kjir zyx

2 2 2

x y z

x y z

r i j kr

r

This vector function is the sum of a 3 component vector:

where and are the three scalar components of

2 2 2 3 2

0

2 2 2 3 2

0

2 2 2 3 2

0

( , , ) 4 ( )

( , , ) 4 ( )

( , , ) 4 ( )

x

y

z

qxE x y z

x y z

qyE x y z

x y z

qzE x y z

x y z

).( , rE

zyx EEE

The electric field 2

2 2 2 3 2

0

( )( )

4 ( )

q x y z

x y z

i j kE r

A note on vector notation

In these notes, a vector in 3D such as

will be written

2

0

( ) 4

qE r r

r

2 2 2 3 2

0

( )( )

4 ( )

q x y z

x y z

i j kE r

The Matter and Interactions textbooks use a notation which is

similar to the notation use in VPython syntax:

2 2 2 3 2

0

, ,( )

4 ( )

q x y z

x y zE r

You should be comfortable with both!

E_pointcharge_drag.py

At location x there is an electric field in the direction shown

below, due to nearby charged particles.

If an electron were placed at location x, what would be the

direction of the force on the electron?

Ex

x

1

2

3

4

5 zero

1 2 3 4 5

What is the direction of the electric field at the different positions

below? Your arrows should be of the appropriate relative length.

Electric field lines

A

BC

D

E

F

G

+E

A charge as a projectile

A positive charge q of mass m initially moving at constant

velocity, enters and leaves a region where there is constant,

downward electric field.

0E 0E

(a)Draw the trajectory of the charge as it moves through each of

the three regions.

(b) Write down an expression for the acceleration of the charge in

each of the three regions.

(c) If the mass of the charge is doubled, then what will its

trajectory look like?

The net electric field at a location is the vector sum of the

individual electric fields contributed by all charged particles

located elsewhere.

The electric field contributed by a charged particle is

unaffected by the presence of other charged particles.

1 2netE E E

The superposition principle

1q

2q

1E2E

netE

M&I

13.4

The superposition principle: Example

The negative charge below has twice the magnitude of each positive

charge. Use graphical vector addition to estimate the direction and

relative magnitude of the electric field at each position.

++ _A

B

C

D

E

The superposition principle: an important worked example

A small object with charge Q1 = 6 nC is located at the origin.

A second small object with charge Q2 = 5 nC is located at

m. What us the net electric field at a location

m due to Q1 and Q2 ?

0.05,0.08,0

0.04,0.08,0

1Q

2Q1E

2E

netE

i

j

1 0.04,0.08,0 0,0,0r

0.04,0.08,0 m

11

2 2 21

0.04,0.08,0

( 0.04) (0.08) (0)

rr

r

0.447,0.894,0 m

9 9

11 12 2

0 1

3 3

(9 10 )(6 10 ) 0.447,0.894,0

(0.0894)4

3.02 10 ,6.04 10 ,0

QE r

r

N C-1

2 0.04,0.08,0 0.05,0.08,0r 0.09,0,0 m

22

2 2 22

0.09,0,0

( 0.09) (0) (0)

rr

r 1,0,0 m

9 9

22 22 2

0 2

3

(9 10 )( 5 10 ) 1,0,0

(0.09)4

5.56 10 ,0,0

QE r

r

N C-1

The superposition principle: an important worked example cont

3 3 3

net 1 2 3.02 10 ,6.04 10 ,0 5.56 10 ,0,0E E E

3 32.54 10 ,6.04 10 ,0 N C-1

or we can write a short VPython programme

Note that norm(A) gives the unit vector of

and mag (A) gives the magnitude of

from visual import *

Q1 = sphere(pos=(0, 0, 0), radius=.3e-2, color = (0,0,1), charge = 6e-9)

Q2 = sphere(pos=(0.05, 0.08, 0), radius=.3e-2, color = (1,0,0),

charge = -5e-9)

location = vector(-0.04, 0.08, 0)

k = 9e9

r1 = location - Q1.pos

E1 = k*Q1.charge*(r1/(r1.x**2+r1.y**2+r1.z**2)**0.5)/mag(r1)**2

r2 = location - Q2.pos

E2 = k*Q2.charge*norm(r2)/mag(r2)**2

Enet = E1 + E2

print Enet

Escale = 3e-6

Earrow = arrow(pos=location, color=(1,.6,0), axis=Enet*Escale,

shaftwidth = .5e-2)

A

A

A

A

compare

The electric field of a dipole

The electric field of a dipole

Along the x-axis:

2 21 10 2 2

1 2 4

qsx

x s x si

q

s

q E E

xE

x

i

1 12 2

2 22 21 11 10 02 22 2

1 1

4 4x

x s x sq q

x s x sx s x s

i iE E E

Along the y-axis:

12

s yr i j

s

E

E

yE

r

yr

j

i

12

s yr i j

Then

2 212

s yr2 21

2s yr

1 12 2

2 22 22 21 12 21 10 02 22 2

1 1

4 4

y

q s y q s y

s y s ys y s y

E E E

i j i j

322 210

2

1 4

qs

s y

i

Dipole field 2

Far along the x-axis:

If x >>s , then2 2 2 21 1

2 2x s x s x r

and3

0

1 2 4

x

qs

rE i

xE

x

i

Far along the y- and z-axis: 30

1 ( )4

y z

qs

rE E i

Dipole field 3

Interaction between a point charge and a dipole

q

s

q

dipoleE

d s

i+QF

dipole 3

0

1 2 ( )4

qsQ Q

dF E i

Hence force on dipole due to Q = ( )F i

qq

dipoleE

i+Q

on - on +F F F

pointE

on +F on -F

The dipole moment

Write p = qs for a dipole with in a direction from q to q

q

s

q

EE

p

p

p

See movie oscillation of an electric field in an external

electric field and try the challenge problem.

Electric field of a uniformly charged sphere of radius R

... see later ...

Q

r

+

+

+

+

+

++

++

+

+

+

2

0

1

4sphere

Q

rE r

for r > R 0sphereE

for r < R

r

A uniformly charged sphere acts

like a point charge, at locations

outside the sphere.

The dipole and charged ball worked example in M&I

qs

q

b

Q

R

C

a s

3 2 2 2 20 0

1 2 1 4 4

net dipole ball

qs Q b a

a b a b a

E E E

i jj

j

i

M&I

13.5Choice of system

Consider the following when making sense of things:

Split the Universe up according to:

the charges that are the source of the field

the charge that is affected by the field

... and there is the issue of retardation ...

Why bother with a field?

... knowing the field at a location means that we know the

force acting on any charge q placed at that location...

... no matter how that field was produced.

q

( , , , )x y z tE E

E

Take q away

E

For how long will

you still detect ?E

A real example (e e+ annihilation): e + e+ +

... so is the electric field real, or only a construct?

M&I

13.6

M&I

Chapter 14

Matter and

Electric Fields

M&I

14.1

Charged particles

net charge of an object sum of all the

charges of all its constituent particles

conservation of charge

the net charge of a system and its surroundings

cannot changee+ + e +

M&I

14.2Electric interactions between charged particles

Do it yourself experiments with U and L tapes

M&I

14.3Interaction of charges and neutral matter

The electron cloud around the

nucleus of an atom is described by

a probability distribution :

bring another charge close to atom

the electron cloud will be distorted by

the electric field

average location of the electron no

longer at the centre where the nucleus is

located the atom is polarized

+

can be represented simply: +

E

Such polarized atoms are induced dipoles return back to

original state when external electric field is removed.

Write: p E

where is the dipole moment, is the external field, and

the constant is the polarizability

which is characteristic of the particular material (measured)

p E

A neutral atom and a point charge

+1E r

q1

r

+1E

q1

r

+

s

q2 q2 2E

1F2F

Charge polarizes

the atom

2 1p E

which makes electric

field at q12E

2

1 1 12 3 3 3 2 5

0 0 0 0 0

2 21 2 1 1 2 1 1 4 4 4 4 4

E q qpE

r r r r r

2 22

1 11 1 2 1 25 5

0 0

2 21 1 =

4 4

q qq q

r rF E r r F

M&I

14.4Conductors and insulators

Conductors: contain mobile charges that can

move through the material.

Insulators: have no mobile charges

Insulators can be polarized:Eapplied

+ +

+ +

+ +

+ +

+ + so can conductors

such as ionic solutions

Ionic solutions

++

+

++

Eapplied

+

+

+

+

+

Eapplied

Epolarization

Enet

+

+

++

+

Eapplied

Epolarization

Enet= 0

+ + + + + + + + + ++ + + + + + + + +

+ + + + + + + + + ++ + + + + + + + +

Model of a metal

Metal: atoms arranged in regular 3D geometric lattice,

most electrons tightly bound, one or two outer electrons per atom

free to move within the metal (sea of electrons) but are not

easily removed from the metal.

+ + + + + + + + + ++ + + + + + + + +

+ + + + + + + + + ++ + + + + + + + +

+ + + + + + + + + ++ + + + + + + + +

+ + + + + + + + + ++ + + + + + + + +

+ + + + + + + + + ++ + + + + + + + +

+ + + + + + + + + ++ + + + + + + + +

Epolarization

+

+

+

+

+

+

EappliedEpolarization

static equilibrium

Eapplied

conductor insulator

mobile charges yes no

polarization entire sea of mobile individual atoms or

charges move molecules polarize

static equilibrium Enet = 0 inside Enet nonzero inside

location of only on surface anywhere on or

excess charge inside material

distribution of spread out over located in patches

excess charge entire surface

A negatively charged ion is located to the left of a

neutral molecule. Which diagram correctly shows

the polarization of the neutral molecule?

1 2 3 4 5

A point charge is brought near a neutral molecule.

(There is nothing else nearby).

Is it possible for the point charge and the neutral molecule to

repel each other?

1. Yes. The molecule can polarize so that it repels

the point charge.

2. No. The molecule can only polarize in a way

that will attract the point charge.

1 2 3 4 5

In a region of space there is an electric field upward (in the +y

direction), due to charges not shown in the diagram. A neutral

copper block is placed in the region.

Which diagram best describes the charge distribution on the block?

1 2 3 4 5

A negatively charged iron block is placed in a region where

there is an electric field downward (in the y direction) due

to charges not shown.

Which diagram best describes the charge distribution in

and/or on the iron block?

1 2 3 4 5

M&I

14.5Charging and discharging

An object is charged when its net charge is non-zero

and may be discharged by contact or grounding

An object may be charged by induction

Try it yourself

1. 2.+

+

++

++

++3. 4. 5.

++

++

+

++

+

A and B are identical metal blocks.

What is the final charge of block B?

1. +6 nC

2. +3 nC

3. 0 nC

4. 3 nC

5. 6 nC

1 2 3 4 5

What happens?

1. protons move from A to B

2. positrons move from A to B

3. electrons move from B to A

4. both protons and electrons move

5. no charges move

1 2 3 4 5

You neutralize a positively charged tape by running your

finger across it.

What happens?

1. electrons move from skin to tape

2. Cl- ions move from skin to tape

3. protons move from tape to skin

4. + ions move from tape to skin

5. no charges move

1 2 3 4 5

Two aluminum blocks, A and B, are initially neutral. They have

insulating handles, which are not shown. This sequence occurs:

At a time after t4, what is the net charge of A?

1. positive 2. negative 3. neutral

1 2 3 4 5

Example problem:

What force does the charged ball exert on the neutral wire?

R

Q

r

L

the ball polarizes the wire

which becomes a dipole with +q and q on either ends

wire 3

0

1 2

4

qLE

r

on ball on wireF F

on ball ball wireF Q E

wireE

2 2

0 0

1 1 2 0

4 42wire sphere

q Q

rLE E

2

8

Q Lq

r

Then

22 2 3

on ball 3 3 5

0 0 0

1 2 2 1

4 4 8 4 4

qL Q L L Q LF Q

r r r r

Example problem 2

Inside the wire

at static equilibrium:

net 0E

Putting in some sensible numbers:2 3

-9 -3

9 2 -2 -11

on ball 5

10 C 4 10 m9 10 N m C 1.4 10 N

4 0.1 mF

M&I

14.7Sparks in air

... air is an excellent insulator,

consisting mainly of neutral N2 and O2 during a spark, these molecules are

ionized N2+ and O2

+

How can electric charge move through air?

Take two charged balls, closely located, but not touching

More charge here

(polarized)

-- -

-

--

- -

- ++

++

+

+

+

+

+

+

-- -

-

--

- -

- ++

++

+

+

+

+

+

+

-

Join 2 balls with wire, and free

electrons move onto positive ball

For a 1 m long wire, there are about 1023 free electrons.

The balls are charged about 10-9 coulombs (1010 e)

So in a fraction 1010/1023 (=10 13) of the 1 m long wire are

enough electrons to neutralize the positive ball

i.e. the electron sea shifts about 10-13 m!

What happens in the case when there is only air between the balls?

? Electrons jump between the balls ?

how far (mean free path) does an electron travel in air

before colliding with a gas molecule ?

Mean free path d of electrons in air

A

de

No. of molecules in cylinder 1

No. of molecules/m3 volume of cylinder 1

At STP, one mole

of air occupies

0.0224 m3

2310 26 10 (1.5 10 ) 1

0.0224d

giving d 5 10-7 m

? Positive ions and electrons move in ionized air ?

If the oxygen and nitrogen molecules in air become ionized (how?)

. Then we have a gas of charged particles (a conductor)

-- -

-

--

- -

- ++

++

+

+

+

+

+

++

+

+

+-

-

-

-

No particle moves further than one mean free path

(no electrons move between the balls)

what happens here?

E

The spark!

electrons drift towards the positive ball and positive ions drift

(more slowly) to the negative ball.

As the electrons from the air move onto the positively charged

ball, the electric field between the balls decreases slightly.

electrons also move off the negative ball to neutralize

positively charged molecules.

The spark only lasts a short time, unless the charge on the balls

is replenished, since the excess charge on the balls will not be

sufficient to maintain an electric field large enough to keep the

air ionized.

A photon of light is given off when a free

electron re-combines with a positive ion

the energy of the photon is equal to the

difference between the high energy unbound

state and the lower energy bound state.

+-

? How does the air become ionized ?

Need E = 3 106 N C-1 (experimentally determined) to

maintain the air in an ionized state

What is the electric field between the atomic core

(nucleus + inner electrons) and a single outer electron

-199 2 -2 11 -1

22-10

0

1 1.6 10 C 9 10 N m C 1.4 10 N C4 10 m

eE

r

which is much larger than the experimentally observed

E = 3 106 N C-1

so if its not having a strong electric field, what ionizes the air ?

fast moving charged particles knock electrons out of atoms

muons from cosmic rays, particles from radioactive sources,

Once there is a single free electron, which is then accelerated in

an electric field, a chain reaction can start, as 2, 4, 8,

electrons can be knocked out of molecules

and the air becomes ionized.

need about to knock one electron from a molecule 2.4 10-18 J

2-19

9 2 -2

-100

1.6 10 C1 ( )( ) 9 10 N m C4 10 m

e eU

r

Then 18critical 2.4 10 JeE d

-18

6 -1

critical -19 -7

2.4 10 J30 10 N C

1.6 10 C 5 10 mE

close enough?

1 2

3

Drift speed of free electrons in a spark

2 1812

2.4 10 Jmv

-18

6 -1

-31

2 2.4 10 J2 10 m s

9 10 kgv

The magnitude of the electric field is not uniform between

the two charged metal balls, and is largest near the balls

Why is this the case?

How can a region of ionization propagate though space?

General approach:

1. Think about the distribution ... draw it! ... are there

any symmetries?

2. Cut up the distribution into pieces and consider the

electric field from a single piece.

3. Write down an expression for the electric field from

that one piece.

4. Repeat for all other pieces and sum (integrate) over

the entire distribution.

5. Check your result.

M&I

Chapter 15

Electric Field of Distributed Charges

The electric field of an uniformly charged rod

... with total length L and total charge Q

Magnitude of : r22

0r x y y

Then

0

22

0

x y y

r x y y

i jrr

Magnitude of : E 22 20 0 0

1 1

4 4

Q QE

r x y y

E

yr

j

i

y

0

0y

x

Q

Q Q

y L 0 x y yr i j

M&I

15.2

A uniformly charged rod 2

22 20 0 0

1 1

4 4

Q QE

r x y y

Then 022 22

0 0 0

1

4

x y yQ

x y y x y y

i jE

3 32 22 22 2

0 00 0

1 1

4 4x

x Q Q xE y

Lx y y x y y

3 32 2

0 0

2 22 20 0

0 0

1 1

4 4y

y y Q y yQE y

Lx y y x y y

0zE

1E

2E

1 2E = E E

... now sum up the contributions of all the pieces

The y-components of all the sum to zeroE

322 2

0

1

4x x

Q xE E y

L x y

As 0y 2

32

2

2 20

1 1

4

L

L

x

QE x dy

L x y

The x-components:

and set y0 = 0



2

32

2

2 20

1 1

4

L

L

x

QE x dy

L x y

Evaluate the integral ... try it yourself ... or look it up ...

2

2

2 2 20

1

4

L

L

x

Q yE x

L x x y

220

1

4 2x

QE

x x L

220

1

4 2

QE

r r L

Write or22

0

1

4 2

Q

r r LE r


220

1

4 2

Q

r r LE r

Check the result ... units? ... direction?

Special cases and :r L L r2

0

1

4

Q

rE r

Another special case: r L

Then 2 22 2 2 2r r L r L r L

0

21

4

Q L

rE rand

from visual import *

scene.x = scene.y = 0

scene.height = 800

scene.width = 600

kel = 9e9

Q = 1e-8

N = 50.

L = 1.0

dl = L/N

Escale = 3e-5

rod = []

for y in arange (-(L/2.)+(dl/2.), (L/2.), dl):

a = sphere(pos=(0,y,0), color=color.red, radius=0.01, q=Q/N)

rod.append(a)

obs = []

dy = L/4.

r = 0.05

for y in arange (-(L/2.), (L/2.)+dy, dy):

for theta in arange(0,2*pi,(2*pi/6.)):

pt = vector(r*cos(theta), y, r*sin(theta))

obs.append(pt)

for pt in obs:

E = vector(0,0,0)

ar = arrow(pos=pt, color=(1,.5,0), axis=Escale*E, shaftwidth=0.01)

for source in rod:

r = pt - source.pos

E = E + norm(r)*kel*source.q/mag(r)**2

ar.axis = Escale*E

if pt.y == 0:

print '%e' %mag(E)

Erod.py

Electric field along the axis of a uniformly charged thin ring

E

j

i

Q

2

Q

Q

... with radius R and total charge Q

k

Magnitude of : r 2 2r R z

0 0 cos sin 0

cos sin

z R R

R R z

r i j k i j k

i j kr

Magnitude of : E 2 2 20 0

1 1

4 4

Q QE

r R z

z

M&I

15.4

A uniformly charged thin ring 2

2 2

0

1

4

QE

R z

Then2 2 2 2

0

1 cos sin

4 2

Q R R zE

R z R z

i j kE r

2Q Q

From thinking about the symmetry, = 0x yE E

and322 2

0

1

4 2z

Q zE

R z

3 32 2

2

2 2 2 20 0

0

1 1

4 2 4z

Q z QzE d

R z R zThen

A uniformly charged thin ring 3

322 2

0

1

4

QzE

R zAlong the axis of the ring:

Special cases:

0E

Another special case: z R

Then 3 32 22 2 2 3R z z z

3

0

2

0

1

4

1

4

QzE

z

Q

z

and

Exact centre of the ring, z = 0:

(point charge)

zE

z

Ering.py

Electric field at a few other positions:

Electric field along the axis of a uniformly charged disc

zE

j

i

R

2

area of ring 2

area of disc

Q

Q R


k Magnitude of : zE

32

2 2 2 2 2 20 0

1 1

4 4z

Q z Q zE

z z z

z

r

M&I

15.5

Again, only is nonzerozE

uniformly charged disc 2

322 2

0

1

4z

Q zE

z

2

2Q

Q R

3 32 2

2

2 2 2 20 0

2

1 1 4 4

z z

Q zQ z R

Ez z

E k k k

32

22 2

0

1

2z

Q zE

R z

3 122

2 22 22 2

0 0 0

1 1 1

2 2

R

z

Q Q zE z d

R R R zz

uniformly charged disc 3

Write

0

12

Q A zE

R

2A R

then

Special case: z R

and

122 2

0

12

Q A zE

R z

If is extremely small, then /z R 1 1z

R

0

2

Q AE

which is true near any large uniformly charged plate

Edisk.py

Edisk_add_rings.py

A charged disk viewed edge on:

The capacitor

consider two uniformly charged metal disks,

of area A, a close distance s apart, carrying

charges Q and Q

s

+

+

+

+

+

Q Q

+

+

1 2 3x x x

k

0 z s

At location 2:

What are the directions of the electric fields at

locations 1, 2 and 3?

2

2

0 0

0 0

( )1 1

2 2

2 1

Q A z Q A s zE

R R

Q A s Q A

R

2

0

( ) Q A

E k

M&I

15.6

The capacitor 2

At location 3 (fringe field):3

3

0 0

0

1 12 2

2

Q A z s Q A zE

R R

Q A s

R

3 1

0

( ) 2

Q A s

RE k E

+

+

+

+

+

sQ Q

+

+

1 2 3x x x

k

0 s z

3 0

2

0

2

2

Q A s

E R s

Q AE R very small, if s R

Electric field of a uniformly charged spherical shell

r

+

+

+

+

+

+

+

++

+

+

+

i 2Every 0point on sphere

1

4sphere

Q

rE E r

For r > R:

i

Every point on sphere

0sphereE E

For r < R:

r


12

3

4

5

6

M&I

15.7

Esphere_outside_rings.py

Esphere_rings.py

1 2 3 4 5

A negatively charged hollow plastic sphere is near a negatively

charged plastic rod. What is the direction of the net electric field at

location P, inside the sphere?

1

4

3

2

5 zero magnitude

1 2 3 4 5

You stand at location A, a distance d from the origin, and hold a small

charged ball. You find that the electric force on the ball is 0.008 N.

You move to location B, a distance 2d from the origin,

and find the electric force on the ball to be 0.004 N.

What object located at the origin might be the source of the field?

0. A point charge

1. A dipole

2. A uniformly charged rod

3. A uniformly charged ring

4. A uniformly charged disk

5. A capacitor

1 2 3 4 5


charged ball. You find that the electric force on the ball is

0.08 N. You move to location B, a distance 2d from the origin,



0. A point charge

1. A dipole




5. A capacitor

1 2 3 4 5


charged ball. You find that the electric force on the ball is 0.009 N.

You move to location B, a distance 2d from the origin,



0. A point charge

1. A dipole




5. A capacitor


2

0

1

4sphere

Q

rE r

For r > R:

For r < R:


(think of a series of concentric spherical

shells, all uniformly charged)

r E

Contribution to at r due to all

concentric spherical shells

between r and R is zero

Contribution to at r due to all

concentric spherical shells

between 0 and R is

E

E

2

0

1

4

Q

r

Electric field of a uniformly charged solid sphereM&I

15.8

343

343

volume of inner shells

volume of sphere

Q r

Q R

Therefore, inside the sphere:

343

2 2 3 3430 0 0

1 1 1

4 4 4

rQ Q QrE

r r R R

( )E r

rR

Electric field of a uniformly charged solid sphere 2

Try it yourself

M&I

15.9The hollow 3/4 cylinder

What is the direction of the electric field due to the two charged

rods at each of the positions shown?

_ _ _ _ _ _ _ _ _

1. 2. 3. 4. 5. zero

1 2 3 4 5

A

+ + + + + + + + +

+ + + + + + + + +

+ + + + + + + + +

B

C

DE

F

G

H

What is the direction of the electric field at the centre of the ring

in each case?

_ __

_

_

_

_

_

_

_

_

_ _

__

_

_ __

_

__

_

_

_

+++

+

+

+

+_

_

_

_

_

_

__

++

+

++

+

1. 2. 3. 4. 5. zero

1 2 3 4 5

A B C

What is the direction of the electric field due to the charged ring

at the position shown in each case?

_ __

_

_

_

_

_

_

_

_

_ _

__

_

_ __

_

__

_

_

_

+++

+

+

+

+_

_

_

_

_

_

__

++

+

++

+

1. 2. 3. 4. 5. zero

1 2 3 4 5

A B C

phy1004w buffler m&ie&m1

Documents