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 Topic  |
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ObsessiveMathsFreak
Infrared Wave
Burkina Faso (Upper Volta)
282 Posts |
 Posted - 02/27/2003 : 14:28:14
We've got a function in 3 dimensionsE = -x/|x|3 I'm leaving out any constants.
Now accoring to physicists ∇E =  /€ I have tried everything I can think of and I still keep getting ∇E = 0 the wierdest thing is that in one and two dimensions ∇E = 1/|x|3 does anyone know what's going on here?
"May the maths be with you"
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Edited by - ObsessiveMathsFreak on 02/27/2003 14:28:55
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HallsofIvy
Micro Wave
USA
163 Posts |
Posted - 02/27/2003 : 18:26:50
I'm not sure what you mean. The "physics" formula you give is for, I believe, the electric field where there is some charge density at each point. The formula E= x/|x| is for the electric field of a single charged particle at the origin. At every point other than the origin (where the charge DENSITY is infinite) the charge density is 0 so the result you get is the same as the formula.
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wavelength
Radio Wave
Canada
21 Posts |
Posted - 02/27/2003 : 18:39:59
I agree with Mr. Ivy here."There was a young lady named Bright,
Whose speed was far faster than light.
She left one day
In a relative way,
And returned home the previous night! "
--Anon
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lbooda
Gamma Wave
USA
2030 Posts |
Posted - 02/27/2003 : 20:43:45
ObsessiveMathsFreak, coming from "Upper Volta," you should know your E-M!Are you saying that dE/dx=?E Try working this problem backward by integrating. Refer to Gauss' Law. http://www.quantumdream.net
Phase reality!
{^,^}
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Edited by - lbooda on 02/27/2003 20:44:41
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alis
Radio Wave
USA
88 Posts |
Posted - 02/27/2003 : 21:24:00
Think delta function...---
The good Christian should beware of mathematicians and all those who make empty prophecies. The danger already exists that mathematicians have made a covenant with the devil to darken the spirit and confine man in the bonds of Hell. -St Augustine
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Hurkyl
Visible Light Wave
USA
723 Posts |
Posted - 02/27/2003 : 22:56:40
To answer the second part of your post, it's related to the number of dimentions of space.In 3-space, one can imagine that EM waves emitted by a point propagate along expanding spheres, so due to conservation laws the strength would be inversely proportional to the area of the sphere, so for the E-field of a point charge centered on the origin: |E| ~ 1 / (4  |x|2) And since the direction is given by x / |x|, the E-field is: E ~ x / |x|3 And if you check, you find that everywhere but the origin the divergence of this is 0.
If you carry out the same heuristic argument in other dimentional spaces, like 2-dimentions, one would observe that the strength should be proportional to the circumference of circles, so you get
E ~ x / |x|2 And you can check that the divergence of this in 2-space is indeed zero! (Incidentally, this argument also applies to cylindrical waves from infinite lines) In general, in n-space, the E-field of a point charge would be: E ~ x / |x|n Hurkyl
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ObsessiveMathsFreak
Infrared Wave
Burkina Faso (Upper Volta)
282 Posts |
Posted - 02/28/2003 : 12:58:08
OK, after much research I've found the answer to this problem. Once again another really good reason to use formal notation.I had thought ∇E = /€ was a constant everywhere. I should actually be written ∇E(x) = (x)/€ where (x)/€ = 0 everywhere but the origin. Sorry to have bothered everyone with by meaninless babble.
"May the maths be with you"
Alert Mentor now
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