Physical meaning of Neumann boundary condition

In summary: However, J does depend on V since σ is inversely proportional to V. So in the limit as V approaches 0 or goes to infinity, J will also approach 0.In summary, the PDE toolbox in Matlab allows you to solve Laplace's equation for various electrostatic geometries. You start by using the wrong end (or right end, depending on how you look at it) and slip down the rabbit hole to Maxwell's equations. The Neumann boundary condition equation for a given boundary is g ∝ ∇V. Ohm's Law is applied to argue that by setting g ∝ ∇V to zero in the normal direction on the boundary, the electric field component (and thus also J
  • #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?
 
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  • #2
gnurf said:
(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?
g ∝ E (Guass law) => g ∝ ∇V
 
  • #3
For the next part, (3) J = σE is defined for the bulk, & not a surface.
 

What is the physical meaning of Neumann boundary condition?

The Neumann boundary condition is a type of boundary condition in physics that specifies the derivative of a particular variable at the boundary of a system. It is used to model systems where the value of the variable is not fixed at the boundary, but rather the flux or flow of the variable is specified.

Why is the Neumann boundary condition important in physics?

The Neumann boundary condition is important in physics because it allows us to accurately model and solve many physical systems. It is commonly used in heat transfer, fluid dynamics, and electromagnetic problems, among others.

How is the Neumann boundary condition different from the Dirichlet boundary condition?

The Neumann boundary condition specifies the derivative of a variable at the boundary, while the Dirichlet boundary condition specifies the value of the variable at the boundary. In other words, the Neumann boundary condition describes the behavior of the system at the boundary, while the Dirichlet boundary condition describes the state of the system at the boundary.

Can the Neumann boundary condition be applied to all physical systems?

Yes, the Neumann boundary condition can be applied to a wide range of physical systems. It is a general boundary condition that can be used to model both linear and nonlinear systems, as well as systems with complex geometries.

How can the Neumann boundary condition be visualized in physical systems?

The Neumann boundary condition can be visualized as a surface or boundary where the flow or flux of a particular variable is specified. For example, in heat transfer problems, the Neumann boundary condition may represent the surface of an object where heat is flowing out or in, depending on the direction of the derivative specified.

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