Today 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.

S
y head
 r h tail line r t
θ
T r
 h s motion of source
U
P ray line
ψ
 r r
P x


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

0
=
 
0
 
0 .
2



0 .
4

0 .
6
○
 
0 .
8

0

Head lines, as a function of

.

tan
 
   sin


0 cos

0

 
1
1
 
2

○

○ head line

○ head line tail line

A little later… head line tail line

The thread length increases
  r r


The thread becomes to r

.


 The stub is s

 r


ˆ  .
h
 r h

The stub is s

 r


ˆ  .
h
The length of the stub is proportional to the length of the thread.

The stub is s

 r


ˆ  .
h
The stub is perpendicular to the thread and to the head line.

y
P tail line head
 r h r t

θ ray line
ψ
 r r
S T r h
U s motion of source – if there were no acceleration!
P thread

 x actual path of the source

•The Electric “Velocity Field”:

E v
 q s
4

0
 s
2 r r
3
( 1
 
 r r s
2 sin
2

)
3 / 2
•The Electric “Acceleration Field”:

E a
 q s
4

0 c
2 r r
3
 r h
( 1



 s
 r r
2
 sin
 a

2

)
3 / 2
•The Magnetic Field:

B

1 c r
ˆ h


E

The Acceleration Fields for Slow Particles
•The Electric “Acceleration Field”:

E a
 q
4 s

0 c
2

 r r

E a s



0 r h

 q
4 s

0 r
 r

 r r
3 c
2
( 1
 r h



  r r

 a
 s
2 sin
2

 r


 a
 r
3
 q
4 s

0 r
ˆ
)
3 / 2

 r
ˆ 
 a
 c
2 r

The Acceleration Fields for Slow Particles
•In this limit, the magnetic field is given by

B a

1 r
ˆ 

E a c
 q
4 s

0 r
ˆ 
 r
ˆ 
 r
ˆ 
 a
  c
3 r

A Summary of the Important Points
•Acceleration fields drop off as
1/ r rather than
1/ r 2 .
r
ˆ all mutually perpendicular.



B r
ˆ
• B is smaller than E by a factor of c in SI units.

Radiation
•Acceleration fields are also called
“electromagnetic radiation.”
•Many kinds of electromagnetic radiation are due to oscillating sources.

y
P tail line
 r t head
 r h line
θ ray line
ψ
 r r
S T r h
U s motion of source – if there were no acceleration!
P thread

 x actual path of the source

•The direction of the fields reverse when the direction of the acceleration 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 r
ˆ
•Let’s look at animations of the electric fields of accelerating charges.

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:

Today 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:

E a
 q
4 s

0
 c
2
R

B a

1 c
R


E a

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:

E a
 
  
B a

R


E a

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

i
Current, Velocity, and Acceleration i

 di dt
Ne

T

L
  v
T
  a
L di

0 dt
 a

Finding the Electric Field
•Choose a field point P.
P i

Finding the Electric Field
•Consider only the part of the wire that contributes most to the fields.
P i

Finding the Electric Field
 a

R
P

R
 a i

Finding the Electric Field
•Find

 a
( Into the screen )
P

R
 a i

•Find

E
Finding the Electric Field

 
P

R
 a i

•Find
Finding the Magnetic Field


E

B
(out of the screen)

E
P

R
 a i

i

E
Finding

E


B

E


B

B

R
 a

Induced Current
•This can cause current to flow in an adjacent wire.

E
 a i

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 ---

1
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
2 i

1
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
2 i

A Wire with Increasing Current
•This variation of E with r is important.
2 1 i

Induced Current
… or if E is larger near the wire, current flows in an adjacent loop.

E i
 a

E 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 di

0 dt

R
 a

A Loop with Increasing Current
•If the current is increasing

E i di

0 dt

E 
R
 a

Induced Current in a Loop
•If the current is increasing

E i di

0 dt

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:
 

B
 
0 j


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:
 

E
 



B t

Faraday’s Law of Induction

 

E
 

B
 t
•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.

Faraday’s Law of Induction

 

E
 

B
 t
•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 
B .

The Integral Form of Faraday’s Law
 

E
 


B
 t
 
E
  d

B dt
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

E i

A Charging Capacitor
•The charge increases in time but the current decreases in time.
i

E i

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
 a

E i

A Charging Capacitor
•On the bottom plate, the current is inward toward the
•A charge (+) on the bottom experience an acceleration toward the “exit” wire. i

E
 a i

A Charging Capacitor
•Now let’s find the electric acceleration field from the charge on the top...

E i r

 a
P 
E a
 a i


E
A Charging Capacitor
B i i
 a r


B a
P 
E a
 a

A Charging Capacitor
•Now let’s find the electric acceleration field from the charge on the bottom...

E i

E a r

 a
P
 a i

screen.

E
A Charging Capacitor
B i i

E a r

 a
 a
P

B a

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:

B , acc
 d

E dt

Ampere’s Law
•Adding this to Ampere’s Law as we know it, we expect:

B
 
0
 i
 i dis

 
0 i

K

0 d

E dt
•The constant K can be determined either from the thread model or experimentally. Finally, we have:

B
 
0
 i
 i dis

 
0 i
 
0

0 d

E dt

Displacement Current
•Thus, the displacement current is a constant times the rate of change of the electric flux through an Amperian loop: i dis
  
0 d

E dt

Ampere’s Law Revised

B
 
0
( i
 i dis
)
 
0 i
 
0

0
 
 t
E

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.”
 

B
 
0
 j
 
0

0


E
 t

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

E
 

E

 d A
 q
 enc
0
•Gauss’s Law of Magnetism

B
 

B

 d A

•Ampere’s Law
•Faraday’s Law

B

E
 

B
 d

  
0 i
 
0 d

E dt
 

E
 d

   d

B dt

Maxwell’s Equations in Differential Form
•Gauss’s Law of Electricity
 

E


 enc
0
•Gauss’s Law of Magnetism

 
B

0
•Ampere’s Law
•Faraday’s Law
 
 

B

E




0




B
 j
 t
 
0


E
 t



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.