It's difficult to follow your notation :( So I can't quite agree or disagree on the math, but with regards to what is written: "We see that this energy density increases to infinity in proportion to 1/r^2 physical none sense because wave energy is limited." Why is that nonsense? You are absolutely correct that from a point source, the closer you get to the point, the higher the density of whatever flowing is. This is the case whenever there is a constant flux at a constant speed. This doesn't mean that there is an infinite amount of energy. You may be saying that yes this is true for an abstract point source, but for the formula for the energy associated with a magnetic field it doesn't work out to be a finite amount of energy and that is why the formula is wrong.
My bet is that if that is the result that you got, then you messed up on the math. As I can't quite follow your math, I can't point out where the fault lies.
Let's speak more abstractly: A single infinitely long wire with some AC current going through it exists at the origin. This would, as we know, radiation em waves outwards radially in a cylindrical fashion. We can calculate the changing magnetic field at any point near the wire just by knowing it's distance from the wire. In the ideal case, the wire is infinitely thin. Using maxwell's equations, the changing magnetic field results in a changing electric field as well, and these all propagate outwards at c. [btw, the fact that the speed of propogation predicted by the wave equation matches that of the measured speed of light was completely unintentional, but seeing as there are an infinite number of other values it could have been, philosophically it is evident that light and e/m waves are the same thing] Anyway, so using the equations we can come out with a wave equation that will give you the magnetic and electric field at any point at any time with respect to the wire current function. These field strengths are not infinite. They are finite values. If you count quantum physics then it gets a bit funky, but classicaly speaking (since you aren't discussing the validity of QED), you are dealing with finite B(r,t) and E(r,t) where r>0. So since the fields have certain values at certain r's than you can integrate and find the energy in that shell/ring/etcVolume easily.
You are asking though..what happens when you want to find the energy in a volume that starts at the center. That is just the integral of each shell/ring/etcVolume from 0 till r. The volume/surface area gets smaller as the energy density gets larger (since it's E/volume). This means that you will never get an infinite value for energy for a finite region unless the energy density (per volume) or energy density (per surface area / time) grows faster than the volume or surface area decreases.
It's a math situation, but if you just think about it as Energy through a volume = function1(magnetic flux through a region)=function2(electric flux through a region) [since the 2 are related] then you see that since the amount of flux through any region is finite, the energy must also be finite.
We can go over the math if you'd like..but we'd have to open a whiteboard or something for that.
It's difficult to follow your notation :(
ReplyDeleteSo I can't quite agree or disagree on the math, but with regards to what is written:
"We see that this energy density increases to infinity in proportion to 1/r^2
physical none sense because wave energy is limited."
Why is that nonsense? You are absolutely correct that from a point source, the closer you get to the point, the higher the density of whatever flowing is. This is the case whenever there is a constant flux at a constant speed. This doesn't mean that there is an infinite amount of energy.
You may be saying that yes this is true for an abstract point source, but for the formula for the energy associated with a magnetic field it doesn't work out to be a finite amount of energy and that is why the formula is wrong.
My bet is that if that is the result that you got, then you messed up on the math. As I can't quite follow your math, I can't point out where the fault lies.
Let's speak more abstractly:
A single infinitely long wire with some AC current going through it exists at the origin.
This would, as we know, radiation em waves outwards radially in a cylindrical fashion. We can calculate the changing magnetic field at any point near the wire just by knowing it's distance from the wire.
In the ideal case, the wire is infinitely thin.
Using maxwell's equations, the changing magnetic field results in a changing electric field as well, and these all propagate outwards at c.
[btw, the fact that the speed of propogation predicted by the wave equation matches that of the measured speed of light was completely unintentional, but seeing as there are an infinite number of other values it could have been, philosophically it is evident that light and e/m waves are the same thing]
Anyway, so using the equations we can come out with a wave equation that will give you the magnetic and electric field at any point at any time with respect to the wire current function. These field strengths are not infinite. They are finite values. If you count quantum physics then it gets a bit funky, but classicaly speaking (since you aren't discussing the validity of QED), you are dealing with finite B(r,t) and E(r,t) where r>0.
So since the fields have certain values at certain r's than you can integrate and find the energy in that shell/ring/etcVolume easily.
You are asking though..what happens when you want to find the energy in a volume that starts at the center. That is just the integral of each shell/ring/etcVolume from 0 till r. The volume/surface area gets smaller as the energy density gets larger (since it's E/volume). This means that you will never get an infinite value for energy for a finite region unless the energy density (per volume) or energy density (per surface area / time) grows faster than the volume or surface area decreases.
It's a math situation, but if you just think about it as Energy through a volume = function1(magnetic flux through a region)=function2(electric flux through a region) [since the 2 are related]
then you see that since the amount of flux through any region is finite, the energy must also be finite.
We can go over the math if you'd like..but we'd have to open a whiteboard or something for that.