description of electromagnet ic field evolution 91 charge velocities are
high or when electrical fields cancel out. For situations involving matter,
fields can indeed be distinguished with their sources. Up to the present
day, no particle with a magnetic
field is the definition of the electric potential as the energy per charge:
= (40) In other words, the potential () at a point is the energy
needed to move a unit charge to the point starting from a point where
the potential vanishes. The potential energ
a magnetic potential (), its momentum changes by ; it changes by
the difference between the potential values at the start and end points,
multiplied by its charge. Owing to this definition, the vector potential
has the property that = = curl (35) i.e., th
specific instant of time. F I G U R E 51 A plane, monochromatic and
linearly polarized electromagnetic wave, showing the evolution of the
electric field, the magnetic field, and again the electric field, in a further
visualization (Mpg films Thomas Weilan
Choosing a function is often called choosing a gauge; the 4-potential
is also called the gauge field. These strange terms have historic
reasons and are now common to all of physics. Ref. 46 * In the part
of the text on quantum theory we will see that the
motion. Motion is not only the change in state of objects and of spacetime, but also the change in state of fields. We therefore need, also for
fields, a complete and precise description of their state. The observations
using amber and magnets have shown
detailed parts of the energymomentum tensor are found to be =
( energy energy flow or density momentum density energy flow or
momentum momentum density flow density ) = ( / = )
=(02 + 2 /0)/2 0 0 0 /0
1/2(02 + 2 /0) ) (54) where = /0 is the Poynting vect
1/. Any other physical system does not obey conformal symmetry.
To sum up, electrodynamic motion, like all other examples of motion
that we have encountered so far, is deterministic, slower than ,
reversible and conserved. This is no big surprise. Nevert
the field. The precise amount depends on the observer. Two colliding
charged particles thus show us that electromagnetic fields have
momentum. If electromagnetic fields have momentum, they are able to
strike objects and to be struck by them. As we will sh
available at www.motionmountain.net the description of electromagnet
ic field evolution 83 Field lines imagined as water flow paddle-wheel F I
G U R E 46 Visualizing the curl of a vector field. Imagine the field to be
flowing air and check whether the sma
) d ( + )= ( ) , (43) which show that the changes
of generalized energy and momentum of a particle (on the * This is only
possible as long as the field is constant; since all fields drop again at
large distances because the energy of a field is always fi
) , (45) which means that the electromagnetic field is
completely specified by the 4-potential .* But as just said, the 4potential itself is not uniquely defined. Indeed, any other equivalent 4potential is related to by the gauge transformation = +
(46)
where is the magnetic vector potential. In summary, the
electromagnetic field has linear and angular momentum and energy, with
well-defined values. Nevertheless, for most everyday situations, the
actual values Challenge 91 e are negligibly small, as you m
tiny mass poses no special problems, and the corresponding Ref. 52
Lagrangian, the so-called Proca Lagrangian, has already been studied,
just in case. Strictly speaking, the photon mass cannot be said to vanish.
In particular, a photon with a Compton wave
complete for everyday life. Only quantum effects and the effects of
curved space-time are not included. Maxwells equations seem very
complex. But we should not forget that they contain only four basic
ideas. 1. Electric charges follow Coulombs rule. * In
will see, by electromagnetism. Both interactions share an important
property: substituting all coordinates in their equations by the negative
of their values leaves the equations unchanged. This means that for any
solution of these equations, i.e., for an
4 1 , (37) where is the magnetic flux inside the solenoid. We see
that, in general, the vector potential is dragged along by moving charges.
The dragging effect decreases for larger distances. This fits well with the
image of the vector potential as the
order to describe the energy momentum of the electromagnetic field
completely, we need to know the flow of energy and momentum at every
point in space, separately for each direction. This makes a description
with a tensor necessary, the so-called energymo
case of mechanics, using the variational method for the two Challenge
87 ny variables and , we recover the evolution equations for particle
position and fields = , = 0 , and
= 0 , (52) which we know already: they are the Lorentz relation and the
two fiel
danger for a misuse of the technique. There are good reasons to believe
that full thought reading will never be possible in this way, due to the
lack of localization of cognitive thought inside the brain and due to the
variations in cognitive processing f
animal experimentation have allowed deducing models that make
quantitative prePage 264 dictions. More on this will be told below. On
the experimental side, research into magnetoencephalography devices is
making rapid progress.The magnetic fields produced
momentum and angular momentum. Colliding charged particles
Electromagnetic fields move. A simple experiment clarifies the meaning
of motion for fields: When two charged particles collide, their total
momentum is not conserved. Let us check this. Imagine t
to two different values for the electric Ref. 42 fields, one at the position
of each particle. In other words, the system of the two particles is not in
inertial motion, as we would expect; the total momentum is not
conserved for Challenge 77 s this obser
CED = 2 d 1 40 , (48) which in index notation
becomes CED = d () d d () d dM ( 1
40 + ) d4 , (49) or, in 3-vector notation CED = 2
d + ( ) dd+(0 2 2 1 20 2 ) dd . (50) The new part
is the measure of the change or action due to the electromagnetic
field.
rightleft asymmetry. We will explore the issue later on. Motion
Mountain The Adventure of Physics copyright Christoph Schiller
June 1990October 2016 free pdf file available at
www.motionmountain.net 90 2 the description of electromagnet ic field
evolution
electromagnet ic field evolution v v 0 distance r m, q m, q F I G U R E
44 Charged particles after a collision. 2. Electric charges moves slower
than light. 3. Electric charges are conserved. 4. Magnetic charges do not
exist. If we want to be simplistic,
that electromagnetic fields move, because they carry energy and
momentum. What is contact? The exploration of collisions, together
with the result that matter consists of charged particles, allows us to
deduce Everyday contact is the exchange of electroma
in this domain. They fascinate many.* However, in this mountain
ascent we keep the discussion * This section can be skipped at first
reading. Challenge 78 s * What is the relation, for static fields, between
field lines and (equi-) potential surfaces? Can
energy and momentum values do not depend on gauge choices. The
energymomentum tensor, like the Lagrangian, shows that
electrodynamics is Challenge 88 e invariant under motion inversion. If
all charges change direction of motion a situation often confusing
pdf file available at www.motionmountain.net 84 2 the description of
electromagnet ic field evolution gin.* In the centre of the solenoid, the
potential vanishes. The analogy of the dragged honey gives exactly the
same behaviour. However, there is a catch