- #1
gnurf
- 370
- 8
I'm playing with the PDE toolbox in Matlab and solving Laplace's equation, ∇2V = 0, for various electrostatic geometries. I say 'playing' because I started in the wrong end (or right end, depending on how you look at it) by simple trial and error until the solutions looked like something reasonable. However, I quickly slipped down the rabbit hole, to the point where I had to find my old EM book and revisit Maxwell's equations.
Ok, so I'm posting this partly to 1) verify that I got the basics right, 2) to understand the physical implications of the Neumann and Dirichlet boundary conditions, and 3) to write this down somewhere so I can retrieve it if necessary. Also I have nothing better to do.
Since I'm only considering direct currents (DC), the magnetic field is static and thus according to Faraday's Law the electric field is irrotational ∇x E = 0. From basic vector identities we then know that the electric field vector can be be expressed as the gradient of the electric scalar function Φ, which has the same meaning as voltage V in this (static field) case:
(1) E = -∇V
Ok, so far so good, I think. Now, in Matlab's PDE toolbox the Neumann boundary condition equation for a given boundary is
(2) nε∇V+qV = g
where g and q = 0 is the surface charge and film conductance, respectively, and n is the normal vector to the boundary. Btw, I know that the charge density ρ ∝ ∇2V, but how can I show, and verify (2), that g ∝ ∇V?
Moving on: Can I apply Ohm's Law
(3) J = σE
and use (1) to argue that, for all non-zero values of g, I am effectively making that boundary a current source? And in the dual case, can I argue that by setting g ∝ ∇V to zero in the normal direction on the boundary, the electric field component (and thus also J) on the boundary is purely tangential. That means no current can cross the boundary, which again means that it must be an insulating boundary.
Is the Neumann condition in fact the only (or standard) way to construct an isolating boundary in these kind of problems? That is, if your problem has an isolating boundary you must use Neumann boundary condition on that boundary? Likewise, much in the same sense that a non-zero Dirichlet condition defines a voltage source, a non-zero Neumann condition defines a current source?
Ok, so I'm posting this partly to 1) verify that I got the basics right, 2) to understand the physical implications of the Neumann and Dirichlet boundary conditions, and 3) to write this down somewhere so I can retrieve it if necessary. Also I have nothing better to do.
Since I'm only considering direct currents (DC), the magnetic field is static and thus according to Faraday's Law the electric field is irrotational ∇x E = 0. From basic vector identities we then know that the electric field vector can be be expressed as the gradient of the electric scalar function Φ, which has the same meaning as voltage V in this (static field) case:
(1) E = -∇V
Ok, so far so good, I think. Now, in Matlab's PDE toolbox the Neumann boundary condition equation for a given boundary is
(2) nε∇V+qV = g
where g and q = 0 is the surface charge and film conductance, respectively, and n is the normal vector to the boundary. Btw, I know that the charge density ρ ∝ ∇2V, but how can I show, and verify (2), that g ∝ ∇V?
Moving on: Can I apply Ohm's Law
(3) J = σE
and use (1) to argue that, for all non-zero values of g, I am effectively making that boundary a current source? And in the dual case, can I argue that by setting g ∝ ∇V to zero in the normal direction on the boundary, the electric field component (and thus also J) on the boundary is purely tangential. That means no current can cross the boundary, which again means that it must be an insulating boundary.
Is the Neumann condition in fact the only (or standard) way to construct an isolating boundary in these kind of problems? That is, if your problem has an isolating boundary you must use Neumann boundary condition on that boundary? Likewise, much in the same sense that a non-zero Dirichlet condition defines a voltage source, a non-zero Neumann condition defines a current source?