Lesson 10The Fields of Accelerating

Charges

Class 29Today we will:•learn about threads and stubs of accelerating point charges.• learn that accelerating charges produce radiation (except in quantum mechanics).• learn the characteristics of radiation fields.

From Chapter 2…

motion of sourceU

θ

head line thread

x

y

P P

tail line

hr

tr

S

ray line

rr

ψ

shr

T

What happens when the source accelerates?

Let’s consider the case where a source initially at rest experiences a force in the +x direction.

A thread leaves at θ0 = 45○

02.0

4.06.0

8.0

0

Head lines, as a function of .

The Formula

0

0

cos

sintan

21

1

A thread leaves at 45○

A thread leaves at 45○

head line

A thread leaves at 45○

head line

tail line

head line

tail line

A little later…

As the thread moves out…

The thread length increases

rr

As the thread moves out…

The thread becomes perpendicular to .r

What about the stub?

The stub is

.ˆ

hrs

hr

What about the stub?

The stub is

.ˆ

hrs

The length of the stub is proportional to the length of the thread.

What about the stub?

The stub is

.ˆ

hrs

The stub is perpendicular to the thread and to the head line.

Accelerating Source

motion of source – if there were no acceleration!

U

θ

head line

thread

x

y

P P

tail line

hr

tr

S T

ray line

rr

ψ

shr

actual path of the source

The Final Result!

•The Electric “Velocity Field”:

•The Electric “Acceleration Field”:

•The Magnetic Field:

2/322320 )sin1(4 srs

rsv r

rqE

2/32232

0 )sin1(4 sr

rhsa rc

arrqE

Erc

B h

ˆ

1

The Acceleration Fields for Slow Particles

•The Electric “Acceleration Field”:

rc

arrq

rc

arrqE

rrr

rc

arrqE

ssa

hr

s

sr

rhsa

20

320

2/322320

ˆˆ

44

0

)sin1(4

The Acceleration Fields for Slow Particles

•In this limit, the magnetic field is given by

rc

arrrqEr

cB s

aa 30

ˆˆˆ

4ˆ

1

A Summary of the Important Points

•Acceleration fields drop off as 1/r rather than 1/r2.

•The electric field, the magnetic field, and are all mutually perpendicular.

•The vector points in the direction of .

•B is smaller than E by a factor of c in SI units.

r̂

r̂BE

Radiation

•Acceleration fields are also called “electromagnetic radiation.”

•Many kinds of electromagnetic radiation are due to oscillating sources.

What if a particle slows down?

motion of source – if there were no acceleration!

U

θ

head line

thread

x

y

P P

tail line

hr

tr

S T

ray line

rr

ψ

shr

actual path of the source

What if a particle slows down?

•The direction of the fields reverse when the direction of the acceleration reverses, but (far from the source) the field will always be perpendicular to

.r

Radiation

•Many kinds of electromagnetic radiation are due to oscillating sources.

•When sources oscillate, the direction of the fields oscillate, but remain perpendicular to .

•Let’s look at animations of the electric fields of accelerating charges.

r̂

Stationary Source

•The electric field of a stationary charge. We “turn the field on” at t = 0 and it propagates outward at the speed of light.

Velocity Field

•The electric field of a charge moving to the right at 70% of the speed of light. The field lines lie along the ray lines. Note how they bunch up the plane perpendicular to the motion. In what direction is the magnetic field?

Velocity Field

•The electric field of a charge moving to the right at 95% of the speed of light. Now the source almost catches up with the emitted threads. http://www.physics.byu.edu/faculty/rees/220/java/Rad3/classes/Rad3.htm

Acceleration Field

This time the charge is initially at rest. It accelerates to the right for a time and then continues at constant speed.

Acceleration Field

•Now let’s look at the field lines for this same acceleration. The dogleg in the field line is the acceleration field, or the radiation. Also note that the field lines are closer together in the dogleg region.

Acceleration Field

•We can understand these fields by comparing the threads emitted before acceleration and after acceleration.

Before After

Acceleration Field

•Now join the two sets of lines together without creating or destroying any field lines:

Acceleration Field

•Radiation – or the acceleration field – is the region where the doglegs join the two sets of lines.

Field Pulse from Acceleration

•The charge is again initially at rest. It accelerates to the right, remains at constant speed momentarily, then accelerates to the left until it comes to rest again. Note that this makes a square pulse in otherwise straight field lines.

Field of an Oscillating Source

•Now the charge oscillates along the x-axis, so it alternately accelerates to the right and to the left. Let’s first look at the field lines:

Class 30Today we will:•learn how accelerating charges affect circuits in significance ways•learn about induced electric fields•learn about induced magnetic fields and displacement current•learn Faraday’s Law•learn Maxwell’s Term of Ampere’s Law

Acceleration and Circuits

Circuits are affected by acceleration in two ways:

•From radiation – the part of the field that is proportional to acceleration.

•From retardation – the effects of finite propagation time on the velocity fields.

First, we’ll look at radiation ---

Radiation Fields Qualitatively

•If charges are moving slowly, the basic equations for the acceleration fields of point charges are:

Rc

aRRqE sa 2

0

ˆˆ

4

aa ERc

B

ˆ1

R

is the vector from the source to the field point, as in Chapter 8.

What We’re Going to Do

•To find quantitative results, we would have to slice sources into small regions and integrate over source distributions as we did in Chapter 8. (Except we have to be very careful about the time threads are emitted – these are the retardation effects!)

•Instead, we are going to qualitatively describe the radiation fields that are produced. For this, we’re mostly interested in directions: aRREa

ˆˆ

aa ERB

ˆ

The General Plan

•Find the part of the charge or current distribution that contributes most strongly to the fields at a point P.

•Find the direction of the electric and/or the magnetic field at P.

•Make flagrant generalizations.

Example 1: A Wire with Increasing Current

•In a long, cylindrical wire, current travels to the right. Current is increasing in time.

• When current increases, positive charge carriers experience an acceleration in the direction of the current.

L

i

Current and Velocity

•Assume the density of conduction electrons, λ, is known.

•Let T be the time it takes an electron to travel a distance L.

L

i

Current, Velocity, and Acceleration

L

i

adt

di

vT

L

T

Nei

a

0dt

di

Finding the Electric Field

i

•Choose a field point P.

P

Finding the Electric Field

i

•Consider only the part of the wire that contributes most to the fields.

P

Finding the Electric Field

ia

•Draw the vectors and

P

a

.R

R

Finding the Electric Field

ia

•Find

P

.ˆ aR

R(Into the screen)

Finding the Electric Field

ia

•Find

P

.ˆˆ aRR

R

E

Finding the Magnetic Field

ia

•Find

P

.ˆ ER

R

E

B

(out of the screen)

Finding

ia

R

E

B

BE

BE

Induced Current

ia

E

•This can cause current to flow in an adjacent wire.

Acceleration and Circuits

Circuits are affected by acceleration in two ways:

•From radiation – the part of the field that is proportional to acceleration.

•From retardation – the effects of finite propagation time on the velocity fields.

Now, we’ll quickly look at retardation ---

A Wire with Constant Current

•Consider the threads arriving at P at the same time•The threads produced at 1 and 2 came from sources moving with the same velocity•The total field is independent of velocity – and the same as for stationary charges• The net E of the wire is 0

i

PP

21

A Wire with Increasing Current

•Now assume that current is increasing•The threads produced at 1 were produced from sources moving more slowly than at 2•There is a net field in the –x direction that gets smaller as y increases

i

PP

21

A Wire with Increasing Current

•This variation of E with r is important.

i

PP

21

Induced Current

… or if E is larger near the wire, current flows in an adjacent loop.

E

iaE

i

Example 2: A Loop with Increasing Current

•A loop works much the same as a straight wire:

i

A Loop with Increasing Current

•If the current is increasing

i 0dt

di

R

a

A Loop with Increasing Current

•If the current is increasing

i

E

R

a

0dt

di

E

Induced Current in a Loop

•If the current is increasing

i

E

0dt

di

A Loop with Increasing Current

•The electric field we form in here is a new type of electric field that forms a loop. It resembles the magnetic field in this way.

E

The Curl of the Magnetic Field

•Magnetic fields are caused by a current. At a point in space where looping magnetic fields are formed, we found that the curl was proportional to the current density:

jB

0

The Curl of the Electric Field

•At a point in space where the electric field loops are formed, the only thing present is the magnetic field of the wire.

•The magnetic field itself isn’t the source of looping electric fields, as constant magnetic fields don’t produce any electric fields.

• The source is not the magnetic field, but the change in the magnetic field:

t

BE

Faraday’s Law of Induction

•This is Faraday’s Law of Induction in differential form. It means that at any point in space where a magnetic field is changing, there must an exist a looping electric field.

t

BE

Faraday’s Law of Induction

•This is Faraday’s Law of Induction in differential form. It means that at any point in space where a magnetic field is changing, there must an exist a looping electric field.

•The loops form around lines that are in the direction of

t

BE

.B

The Integral Form of Faraday’s Law

dt

d

t

BE B

E

This says the line integral of the electric field around an Amperian loop is minus the time derivative of the magnetic flux through the Amperian loop.

Faraday’s Law

In other words:

If the number of magnetic field lines through a loop is changing, we produce a looping electric field.

Example 3: A Charging Capacitor

•A capacitor with circular plates (for symmetry) is charging.

i

i

A Charging Capacitor

•A “normal” electric field between the plates increases in time.

i

i

E

A Charging Capacitor

•The charge increases in time but the current decreases in time.

i

i

E

A Charging Capacitor•On the top plate, the current is outward from the center. Since this current decreases, the acceleration is toward the center.

i

i

E

a

A Charging Capacitor

•A charge (+) on the bottom experience an acceleration toward the “exit” wire.

i

i

E

a

•On the bottom plate, the current is inward toward the center. Since this current decreases, the acceleration is away from the center.

A Charging Capacitor

•Now let’s find the electric acceleration field from the charge on the top...

i

i

E

a

P

a

r

aE

A Charging Capacitor

•…and the magnetic acceleration field from the charge on the top. comes out of the screen. i

i

E

a

P

a

r

aE

aB

aB

A Charging Capacitor

•Now let’s find the electric acceleration field from the charge on the bottom...

i

i

E

a

P

a

raE

A Charging Capacitor

•…and the magnetic acceleration field from the charge on the bottom. comes out of the screen.

aB

i

i

E

a

P

a

raE

aB

A Charging Capacitor

•By integrating all the magnetic fields produced by the current in the capacitor plates, we find there are magnetic field lines going around in circles inside the capacitor, just as if real current were passing between the capacitor plates.

Displacement Current

•No real charges pass between the plates of the capacitor, but we say that “displacement current” between the plates of the capacitor causes the magnetic field.

Displacement Current

•The only “real” thing between the plates is an electric field.

•But a constant electric field can’t cause the displacement current, because there is no magnetic field between the capacitor plates when the plates are fully charged.

Displacement Current

•We might guess that the displacement current is related to a changing electric field.

•Guided by Faraday’s Law, we might expect:

dt

d EaccB

,

Ampere’s Law

•Adding this to Ampere’s Law as we know it, we expect:

dt

dKiii E

disB

000

•The constant K can be determined either from the thread model or experimentally. Finally, we have:

dt

diii E

disB

0000

Displacement Current

•Thus, the displacement current is a constant times the rate of change of the electric flux through an Amperian loop:

dt

di Edis

0

Ampere’s Law Revised

tiii E

disB

0000 )(

Ampere’s Law

In other words:

If either 1) a current is passing through a loop or 2) the net number of electric field lines passing through a loop is changing, we produce a looping magnetic field.

Maxwell’s Term

•The part of Ampere’s Law that comes from displacement current is called “Maxwell’s Term of Ampere’s Law.”

t

EjB

000

Maxwell’s Term

•We won’t do much with Maxwell’s term in class, but be sure to look over the example in the text where we use Maxwell’s term to find the magnetic field inside a charging capacitor.

Maxwell’s Equations

•In the 1860s, James Clerk Maxwell added his term to Ampere’s Law and organized the known relations about electric and magnetic fields together in a mathematical form.

Maxwell’s Equations in Integral Form

•Gauss’s Law of Electricity

•Gauss’s Law of Magnetism

•Ampere’s Law

•Faraday’s Law

0enc

E

qAdE

dt

didB E

B 00

dt

ddE B

E

0AdBB

Maxwell’s Equations in Differential Form

•Gauss’s Law of Electricity

•Gauss’s Law of Magnetism

•Ampere’s Law

•Faraday’s Law

0encE

0 B

t

EjB

00

t

BE

Maxwell’s Equations

•We’ll learn how to use these new equations in coming chapters. For now, you simply need to see how accelerating charges lead to electric and magnetic fields with curl.