Hi there,
I am trying to solve a physics problem but my knowledge of Math is not even close to sufficient for solving this problem:
Problem statement
On a thin infinite metal plate with thickness d and specific conductivity σ two electrodes are attached. The distance between the two electrodes is r and the potential is U.
Describe magnetic field anywhere in space, due to the currents in stationary state on the metal plate. (Ignore the wires which bring the current to the electrodes)
end of problem
Let me firstly describe the problem:
We have a metal plate on which two electrodes are attached. They are \(\displaystyle r\) appart. The origin of my coordinate system is exactly in the middle of electrodes, so the electrodes have coordinates \(\displaystyle (\pm \frac r 2 ,0)\). If needed, let's say that electrodes are not points, but have radius \(\displaystyle R<<r\). One electrode is at potential \(\displaystyle \frac U 2\) and the other at \(\displaystyle -\frac U 2\).
With my knowledge of physics and math everything stops with equation: \(\displaystyle \nabla ^2 \phi = 0\) where \(\displaystyle \phi \) is the electric potential. Now I need to solve this for this metal plate for which I can approximate as if it had no thickness - meaning I can solve \(\displaystyle \nabla ^2 \phi = 0\) in two dimensions \(\displaystyle x\) and \(\displaystyle y\) only.
Now I tried to solve it saying that \(\displaystyle \phi = X(x)Y(y)\) but this than brings me to only one solution: saying that \(\displaystyle U=0\). This is a trivial solution to everything and tells me nothing useful.
Now to get nontrivial solutions, I got a hint to solve the Laplace equation using Greens functions. My problem here is that I have never worked with them, therefore I have no idea how to use them. I did use a lot of google but I couldn't find an idiotproof explanation.
However, I did found out what Greens function look like in 2D but (Ok, I still don't know how to consider that there are two electrodes) but I have absolutely NO idea what do with now.
I would highly appreciate it if somebody could provide me idiotproof explanation on how to continue from this point on.
Thank you for all your help.
(I also sincerely apologize if this is not the right topic to post this problem).
I am trying to solve a physics problem but my knowledge of Math is not even close to sufficient for solving this problem:
Problem statement
On a thin infinite metal plate with thickness d and specific conductivity σ two electrodes are attached. The distance between the two electrodes is r and the potential is U.
Describe magnetic field anywhere in space, due to the currents in stationary state on the metal plate. (Ignore the wires which bring the current to the electrodes)
end of problem
Let me firstly describe the problem:
We have a metal plate on which two electrodes are attached. They are \(\displaystyle r\) appart. The origin of my coordinate system is exactly in the middle of electrodes, so the electrodes have coordinates \(\displaystyle (\pm \frac r 2 ,0)\). If needed, let's say that electrodes are not points, but have radius \(\displaystyle R<<r\). One electrode is at potential \(\displaystyle \frac U 2\) and the other at \(\displaystyle -\frac U 2\).
With my knowledge of physics and math everything stops with equation: \(\displaystyle \nabla ^2 \phi = 0\) where \(\displaystyle \phi \) is the electric potential. Now I need to solve this for this metal plate for which I can approximate as if it had no thickness - meaning I can solve \(\displaystyle \nabla ^2 \phi = 0\) in two dimensions \(\displaystyle x\) and \(\displaystyle y\) only.
Now I tried to solve it saying that \(\displaystyle \phi = X(x)Y(y)\) but this than brings me to only one solution: saying that \(\displaystyle U=0\). This is a trivial solution to everything and tells me nothing useful.
Now to get nontrivial solutions, I got a hint to solve the Laplace equation using Greens functions. My problem here is that I have never worked with them, therefore I have no idea how to use them. I did use a lot of google but I couldn't find an idiotproof explanation.
However, I did found out what Greens function look like in 2D but (Ok, I still don't know how to consider that there are two electrodes) but I have absolutely NO idea what do with now.I would highly appreciate it if somebody could provide me idiotproof explanation on how to continue from this point on.
Thank you for all your help.
(I also sincerely apologize if this is not the right topic to post this problem).