|
My 1994 book Electromagnetism 1 is at http://www.ivorcatt.com/em.htm
|
|
The Balloon Consider a balloon surface that is stretched in
both the north – south (n-s) and in the east - west direction (e-w). We will consider
a square of this material that is held along its four edges. The square is then stretched in the northerly
direction to form a rectangle with double the area. A certain amount of work
is done in the process, and that amount of extra potential energy is
delivered to the surface. (In the language of the TEM Wave, we say that the
aspect ratio of the stress has gone from unity to two.) At this point I will mention that the balloon is
only an illustrative model, and we will impose our own laws, which may not
match Hooke’s Law etc., but which will have a similar structure. The truth is, we are talking about a thin flat
cross-section of a Transverse Electromagnetic Wave (TEM Wave), which
Heaviside called “a slab of energy current”. The rectangle is now stretched in an east – west
direction to restore its shape to square, but with double the length of side. We have to decide how much more potential energy
will reside in the larger square, than in the smaller square. My first
response is to say four times as much. (But see 1aug02 below.) North – south is the E, or electric direction.
East – west is the H, or magnetic, direction. We further impose the requirement that the
original two independent movements were impossible. A square may only be
increased in size if it remains square throughout the process. We return to starting with a square and then
stretching it to a square with twice the side and four times the area. We now modify our picture and say that somehow,
four times the energy was pumped into a square while it retained the same
size throughout. That is, the space in which the slab resided was stressed,
and not the material. Putting it another way, the density of the energy was
increased by four times. (Similarly, the density of gas in a bottle can be
increased by four times while the volume of the bottle remains unchanged.) Thus, we have retreated to a square of fixed size
but with a variable amount of potential energy in it. [Actually, since the
slab of energy current has approaching zero thickness, it contains energy
density rather than energy. (In the same way, a brick does not have mass at a
point, only density. Even a vanishingly thin wafer of the brick has only
density, and no mass. Mass is only possible in a body with three non-zero
dimensions, l, w, h.)] Even when we realise that the slab has non-zero
dimension in two directions, n-s and e-w, it still has no volume, since it
has approaching zero thickness. This square of energy current can only travel in a
direction normal to its surface at a velocity of 300,000. This is its
inherent nature. You are sitting outside the balloon, and the surface of the
balloon approaches you at the speed of light, to reach every point on your
(flat) face at the same instant. We are interested in measuring the amount of
potential energy present. We have two types of measuring instrument, n-s and
e-w. The n-s instruments can measure the stress in the space in the n-s
direction. This is traditionally thought of as the electric stress E. The e-w
instrument can measure the e-w stress. This is conventionally thought of as
the magnetic stress H. Energy density is sometimes said to be proportional to
the product of these two stresses, E x H (called the Poynting Vector).
However, our more practical instruments are thought to measure only the E
stress or the H stress. The measurement E or the measurement H is then
squared to get a measure of the energy density. The correct value is reached
by multiplying by suitable (ad hoc) constants to get the numbers right and
consistent. [These ad hoc constants are called permittivity and permeability
of the space in which the energy (density) resides.] Ivor Catt
1aug02 The Balloon and the TEM Wave The square surface of a balloon does not like being
stretched in the north-south (ns) direction or in the east-west (ew)
direction. If it is stretched to double in one of these directions (ns) (and
it obeys Hooke’s Law), Hooke’s Law says that the potential energy in that
square surface of the balloon surface due to stretching goes up four times.
If the balloon is then stretched to double in the ew direction, the potential
energy goes up again, to sixteen times. Whereas the surface of a balloon dislikes an
increase of area, a TEM wave does not. (It only dislikes a change in aspect
ratio of the space presented to it.) Consider a pulse travelling down a
coaxial cable of characteristic impedance 50 ohms. Provided the change is not
sudden (and this proviso can be qualified in a complex way), it will happily enter
into a coaxial cable whose cross section is much greater, provided its
impedance is also 50 ohms; that is, provided the ratio of the two relevant
radii remains the same for the two cables. If a TEM wave is presented with an area ahead of
it which is thrice as high (ns) but unvarying in width (ew), then it does not
like it, in the same way as the balloon dislikes change of dimensions.
However, it is unable to adjust to the change in aspect ratio (as the balloon
did). When leaving an environment with impedance 377 ohms and required to
enter a new environment with impedance thrice 377 ohms, some of the energy
current reflects backwards to where it came from. What continues forward has
the same aspect ratio as the incident wave (but lower amplitude), as does the
reflected wave. It is easier to think of a pulse travelling down a
50 ohm coax cable and entering a 150 ohm coax cable. There is a 50% voltage
reflection. That is, one quarter of the incident energy (current) reflects.
Three quarters proceeds forwards into the new space, whose aspect ratio is
three times bigger; three squares, one below the other. Thus, the voltage
across each square of the new space is half of the incident signal, and the
energy is one quarter. Ivor Catt 1aug02 |
|
|