Electromagnetism fundamentals



My 1994 book Electromagnetism 1 is at http://www.ivorcatt.com/em.htm



Late 30may02 Ivor Catt

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