Basic Electrical Theory Questions- EASY

[Paul W. Abernathy]

Zero Hertz...ahhh........vibrators......they use batteries

String Theory.....the name of a fancy light fixture used in West Virginia...they dont have light switches...just strings on the keyless light fixtures in the living rooms....

Pythagoras could be called the first known string theorist. Pythagoras, an excellent lyre player, figured out the first known string physics -- the harmonic relationship. Pythagoras realized that vibrating Lyre strings of equal tensions but different lengths would produce harmonious notes (i.e. middle C and high C) if the ratio of the lengths of the two strings were a whole number.




Pythagoras discovered this by looking and listening. Today that information is more precisely encoded into mathematics, namely the wave equation for a string with a tension T and a mass per unit length m. If the string is described in coordinates as in the drawing below, where x is the distance along the string and y is the height of the string, as the string oscillates in time t,

then the equation of motion is the one-dimensional wave equation

where vw is the wave velocity along the string.
When solving the equations of motion, we need to know the "boundary conditions" of the string. Let's suppose that the string is fixed at each end and has an unstretched length L. The general solution to this equation can be written as a sum of "normal modes", here labeled by the integer n, such that

The condition for a normal mode is that the wavelength be some integral fraction of twice the string length, or

The frequency of the normal mode is then

The normal modes are what we hear as notes. Notice that the string wave velocity vw increases as the tension of the string is increased, and so the normal frequency of the string increases as well. This is why a guitar string makes a higher note when it is tightened.
But that's for a nonrelativistic string, one with a wave velocity much smaller than the speed of light. How do we write the equation for a relativistic string?
According to Einstein's theory, a relativistic equation has to use coordinates that have the proper Lorentz transformation properties. But then we have a problem, because a string oscillates in space and time, and as it oscillates, it sweeps out a two-dimensional surface in spacetime that we call a world sheet (compared with the world line of a particle).
In the nonrelativistic string, there was a clear difference between the space coordinate along the string, and the time coordinate. But in a relativistic string theory, we wind up having to consider the world sheet of the string as a two-dimensional spacetime of its own, where the division between space and time depends upon the observer.
The classical equation can be written as

where s and t are coordinates on the string world sheet representing space and time along the string, and the parameter c2 is the ratio of the string tension to the string mass per unit length.
These equations of motion can be derived from Euler-Lagrange equations from an action based on the string world sheet

The spacetime coordinates Xm of the string in this picture are also fields Xm in a two-dimension field theory defined on the surface that a string sweeps out as it travels in space. The partial derivatives are with respect to the coordinates s and t on the world sheet and hmn is the two-dimensional metric defined on the string world sheet.
The general solution to the relativistic string equations of motion looks very similar to the classical nonrelativistic case above. The transverse space coordinates can be expanded in normal modes as

[Paul W. Abernathy]

The string solution above is unlike a guitar string in that it isn't tied down at either end and so travels freely through spacetime as it oscillates. The string above is an open string, with ends that are floppy.
For a closed string, the boundary conditions are periodic, and the resulting oscillating solution looks like two open string oscillations moving in the opposite direction around the string. These two types of closed string modes are called right-movers and left-movers, and this difference will be important later in the supersymmetric heterotic string theory.
This is classical string. When we add quantum mechanics by making the string momentum and position obey quantum commutation relations, the oscillator mode coefficients have the commutation relations

The quantized string oscillator modes wind up giving representations of the Poincaré group, through which quantum states of mass and spin are classified in a relativistic quantum field theory.
So this is where the elementary particle arise in string theory. Particles in a string theory are like the harmonic notes played on a string with a fixed tension

The parameter a' is called the string parameter and the square root of this number represents the approximate distance scale at which string effects should become observable.
In the generic quantum string theory, there are quantum states with negative norm, also known as ghosts. This happens because of the minus sign in the spacetime metric, which implies that

So there ends up being extra unphysical states in the string spectrum.
In 26 spacetime dimensions, these extra unphysical states wind up disappearing from the spectrum. Therefore. bosonic string quantum mechanics is only consistent if the dimension of spacetime is 26.
By looking at the quantum mechanics of the relativistic string normal modes, one can deduce that the quantum modes of the string look just like the particles we see in spacetime, with mass that depends on the spin according to the formula

Remember that boundary conditions are important for string behavior. Strings can be open, with ends that travel at the speed of light, or closed, with their ends joined in a ring.
One of the particle states of a closed string has zero mass and two units of spin, the same mass and spin as a graviton, the particle that is supposed to be the carrier of the gravitational force.

[Paul W. Abernathy]

lol....

At one time, string theorists believed there were five distinct superstring theories: type I, types IIA and IIB, and the two heterotic string theories. The thinking was that out of these five candidate theories, only one was the actual correct Theory of Everything, and that theory was the theory whose low energy limit, with ten dimensions spacetime compactified down to four, matched the physics observed in our world today. The other theories would be nothing more than rejected string theories, mathematical constructs not blessed by Nature with existence.
But now it is known that this naive picture was wrong, and that the the five superstring theories are connected to one another as if they are each a special case of some more fundamental theory, of which there is only one. These theories are related by transformations that are called dualities. If two theories are related by a duality transformation, it means that the first theory can be transformed in some way so that it ends up looking just like the second theory. The two theories are then said to be dual to one another under that kind of transformation.
These dualities link quantities that were also thought to be separate. Large and small distance scales, strong and weak coupling strengths -- these quantities have always marked very distinct limits of behavior of a physical system, in both classical field theory and quantum particle physics. But strings can obscure the difference between large and small, strong and weak, and this is how these five very different theories end up being related.
Large and small distance

The duality symmetry that obscures our ability to distinguish between large and small distance scales is called T-duality, and comes about from the compactification of extra space dimensions in a ten dimensional superstring theory.
Suppose we're in ten spacetime dimensions, which means we have nine space and one time. Take one of those nine space dimensions and make it a circle of radius R, so that traveling in that direction for a distance L=2pR takes you around the circle and brings you back to where you started.
A particle traveling around this circle will have a quantized momentum around the circle, and this will contribute to the total energy of the particle. But a string is very different, because in addition to traveling around the circle, the string can wrap around the circle. The number of times the string winds around the circle is called the winding number, and that is also quantized.
Now the weird thing about string theory is that these momentum modes and the winding modes can be interchanged, as long as we also interchange the radius R of the circle with the quantity Lst2/R, where Lst is the string length.
If R is very much smaller than the string length, then the quantity Lst2/R is going to be very large. So exchanging momentum and winding modes of the string exchanges a large distance scale with a small distance scale.
This type of duality is called T-duality. T-duality relates Type IIA superstring theory to Type IIB superstring theory. That means if we take Type IIA and Type IIB theory and compactify them both on a circle, then switching the momentum and winding modes, and switching the distance scale, changes one theory into the other! The same is also true for the two heterotic theories.
So T-duality obscures the difference between large and small distances. What looks like a very large distance to a momentum mode of a string looks, looks to a winding mode of a string like a very small distance. This is very counter to how physics has always worked since the days of Kepler and Newton.

Strong and weak coupling

What is a coupling constant? This is some number that tells us how strong an interaction is. Newton's constant is the coupling constant for the gravitational force, for example. If Newton's constant were twice the size it is measured to be now, then we would feel twice as much gravitational force from the Earth, and the Earth would feel twice as much from the Moon and the Sun, and so on. A larger coupling constant means a stronger force, and a smaller coupling constant means a weaker force.
Every force has a coupling constant. For electromagnetism, the coupling constant is proportional to the square of the electric charge. When physicists study the quantum behavior of electromagnetism, they can't solve the whole theory exactly, so they break it down to little pieces, and each little piece that they can solve has a different power of the coupling constant in front of it. At normal energies in electromagnetism, the coupling constant is small, and so the first few little pieces make a good approximation to the real answer. But if the coupling constant gets large, that method of calculation breaks down, and the little pieces become worthless as an approximation to the real physics.
This also can happen in string theory. String theories have a coupling constant. But unlike in particle theories, the string coupling constant is not just a number, but depends on one of the oscillation modes of the string, called the dilaton. Exchanging the dilaton field with minus itself exchanges a very large coupling constant with a very small one.
This symmetry is called S-duality. If two string theories are related by S-duality, then one theory with a strong coupling constant is the same as the other theory with weak coupling constant. Notice that the theory with strong coupling cannot be understood by means of expanding in a series, but the theory with weak coupling can. So if the two theories are related by S-duality, then we just need to understand the weak theory, and that is equivalent to understanding the strong theory. For a physicist, that is the proverbial two-for-one deal!
Superstring theories related by S-duality are: Type I superstring theory with heterotic SO(32) superstring theory, and Type IIB theory with itself.
What does it mean?

T-duality is something unique to string physics. It's something particles cannot do, because a particle cannot get wrapped around a circle like a string. If string theory is a correct theory of Nature, then this implies that one some deep level, the separation between large vs. small distance scales in physics is not a fixed separation but a fluid one, dependent upon the type of probe we use to measure distance, and how we count the states of the probe.
The same goes for S-duality, which teaches us that the strong coupling limit of one string theory can describe the weak coupling limit of a different string theory.
This sounds like it goes against all traditional physics, but this is indeed a reasonable outcome for a quantum theory of gravity, because Einstein's theory of gravity tells us that gravity is about how the sizes of objects and magnitudes of interactions are measured in curved spacetime.

[Patrick Bolliger]

Yes on the first question.. Anything that uses DC!! Batteries or DC power supply.. I asked a radio geek friend of mine once and he scratched his head because he was thinking radio frequency but it was a trick question so eventually got it...

Last answer on the "string Theory"... My third and fourth dimension is my bad one... ha ha ha ha Somewhere there are multipules of "us" ... Scary!! Einstein started this concept and others are trying to prove it if possible.. Think of the smallest building block of matter.. What is it's behavior with respect to time and energy, gravity... Scary.....

Can't believe you typed that off the top of your head! hmmm

I have too many PhD scientist friends... One is a high energy physicist working around the world in "big named labs"... Listening to these guys talk is like listening to aliens trying to communicate!!

[Paul W. Abernathy]

lol.....NO I did not type it off my head...I CHEATED...lol

[Paul W. Abernathy]

Gotta LOVE Google.........Im an electrician man....Not Einstein...lol

But the Battery Answer was all mine...thehehehe.......

[Marcel R. Cyr]

Boy, I am glad I am more into carpentry, foundations, structural, roofing, drywall, framing and anything that does not pertain to all these formulas, You have confused me.
I will stick to concrete items, once in place it is hard. Just make sure it is in the right place. ha. ha.

Marcel

[Paul W. Abernathy]

lol....heck Marcel........98% of electricians around the world don't even know these formula's....lol.....

Heck probably 99% don't even Add or Subtract anymore....we have software for that these days...lol

[Timothy J. Gardner]

Paul, You crack me up!

While reading the previous few posts, I was reminded of the time, during Reactor Core physics class, (5 Cr Hrs), of writing down the relationship of one particle to another and the resultant Fourier Statistical analysis of how a single neutron affected the core pile.

Several years later, 20 or so, I ran across my notes and calculations for my thesis, "Reactor Pile Fuel Element Failure Identifcation and Quantification System Design" for this class. Strikingly, I recognized my handwriting, that was ALL I recognized.

Regards, TG

[Paul W. Abernathy]

lol.....Been Their...lol.........We like to call it a Senior Moment...lol

[Patrick Bolliger]

I grad-e-ate-ed Col-age with Hon-er's Now I can spelz a Dam think!

90 % of my brain has died I think.. Glad to see the spelz checker is back!

[Paul W. Abernathy]

OK......relived for the NEWBIES...if you did it before....just ignore it