The Mathematical Universe
“ .... Dirac wasn’t interested in interpretation because he believed that the truth lay in the equations, and it was pointless to ask about their physical meaning.” – John Gribbin, “Erwin Schrödinger and the Quantum Revolution”, pub. Bantam 2012, p154
http://www.ivorcatt.co.uk/2_2.htm Malcolm Davidson has pointed out that since a capacitor is a transmission line, the model models a transmission line in terms of itself, which is absurd.
This is an interesting introduction to “The Mathematical Universe”, which has supplanted a physical universe in science. This can be seen by doing a Google search for “mathematics is the language of science”, which produces 200,000 hits.
The LCLCLC model for that transmission line is interesting because it illustrates the collision between these two universes. It could be argued that it is not absurd because it models the transmission line, which for this purpose is part of the physical universe, in terms of a capacitor from the mathematical universe. Thus, it does not commit the absurdity of modelling something in terms of itself. It is only a physical capacitor which is a transmission line, not a mathematical capacitor.
Far from being a transmission line, a mathematical capacitor has zero volume, and its capacitance is concentrated at a single point in space, where the dielectric constant is, of course, infinite, so that its capacitance can be more than zero.
A mathematical resistor also has zero volume. When a charged mathematical capacitor is connected to a mathematical resistor, the result is an exponential. In such a universe, the delay in the conductors connecting capacitor to resistor is zero. Thus, there is instantaneous action at a distance.
The question arises. Why, within the mathematical universe, is the transmission line ever considered, since it appears to breach a fundamental principle in the mathematical universe, that components have zero volume. This, of course, explains the incompetence shown by today’s professors and text book writers when addressing, or rather not addressing, the transmission line. It is not properly part of their mathematical world. http://www.electromagnetism.demon.co.uk/17136.htm
A mathematical capacitor (which has zero volume) has a stationary electric field because it has nowhere else to travel.
Ivor Catt 3 June 2013
Thank goodness I have now found http://www.ivorcatt.co.uk/x363.htm
In conventional circuit theory, a component has zero volume. There is instantaneous action at a distance along the wires connecting the components. In a circuit connecting various components in a loop, the electric current at every point is the same. This is the world of the exponential when a resistor discharges a capacitor. In http://www.ivorcatt.co.uk/x363.htm I call it “The Mathematical Universe”. The transmission line made up of two parallel conductors cannot be part of that world. Interestingly, since a capacitor has zero volume, its dielectric constant must be infinite. “Displacement Current” derives from this world. In the last diagram in http://en.wikipedia.org/wiki/Displacement_current , the electric charge/current does not have to spread out across the capacitor plate, because the plate has zero area, or perhaps the charge travels across the plate at infinite velocity. http://www.ivorcatt.com/3613.htm Certainly it instantaneously spreads out to be uniform in density across the plate. http://www.ivorcatt.org/icrwiworld78dec1.htm
Away from this Mathematical Universe is the Physical Universe, where components do not have zero volume. In this world we can have a transmission line. In the real, physical world, a capacitor plate has area. Now we get to the next point. The “electricity” does not enter the centre of a capacitor, as Wikipedia says. http://en.wikipedia.org/wiki/Displacement_current . Rather, as Professor Kip says, it enters the capacitor (or resistor) sideways, entering at one end, travelling at right angles to the direction indicated in Wikipedia.
Now let us consider a simple circuit with
battery and resistor. Two conductors
guide the energy current from battery to
resistor. It enters the resistor sideways
(Kip 1962)6. 'Electric current' is merely the
side of a wave of energy current.
Thus, once it enters a capacitor sideways, and advances across the capacitor, we see that the capacitor is a transmission line. But Displacement Current, invented to generate magnetic field in a capacitor, must not do so in a transmission line. http://www.ivorcatt.co.uk/x22j.pdf . The displacement current dD/dt at the front face of a TEM step must not generate magnetic field. The TEM wave travelling along in a transmission line is incompatible with displacement current. They both survive – displacement current in a capacitor and the TEM wave in a transmission line – so long as nobody notices that a capacitor is a transmission line. An editor who published; “a capacitor is a transmission line” will never again get a job as an editor, because he brings the whole structure tumbling down. http://www.ivorcatt.co.uk/x0309.htm . “A capacitor is a transmission line” does not appear in any peer reviewed journals, and will probably not appear during the next thirty years.
Ivor Catt. 20 June 2013