Boundary conditions

  • Thread starter dorist84
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  • #1
Hello hello,

I cannot for the life of me wrap my head around the idea of a boundary condition. I understand the idea (at least I think I do) of solving a differential equation with given initial conditions. But is solving for a magnetic field or electric field while enforcing boundary conditions sort of the same thing? If so, why is it solved in terms of its perpendicular and parallel components?

I read at one point that the purpose of a boundary condition is so specify a "point in space" or to obtain a sense of direction/position when solving for a magnetic or electric field when given a specific geometry. Makes sense to me: it would do no good to just say a magnetic field is n-amount Gauss or Teslas. But isn't that why we use coordinate systems? I guess I just don't understand the idea of "enforcing boundary conditions in order to solve a problem." ....

The best I can make of it is perhaps, hypothetically, if a current or charge density is specified upon a surface with a known geometry, then do you use that to extrapolate information for the B-Field and E-field....(i.e. field will be 0 here....will also flow in this direction...etc ..) But if so, how does all the parallel and perpendicular stuff come in?

I apologize for the convuluted way I've asked this question. I think the problem is more that I'm confused to the extent that I don't even really know HOW to ask the question. So hopefully, if someone is patient enough with me...I can weed through this boundary conditions business.

Thanks for everything. You guys are great!


Answers and Replies

  • #2
I think you're on the right track. When you're solving some sort of situation (generally a differential equation) you need initial condition or boundary conditions to completely specify the solution. Otherwise you can only come up with the form of the solution - i.e. inverse square law, or sine function etc.
The difference between an initial condition and a boundary condition is generally that boundary conditions are always. for instance, with magnetic / electric fields - generally there is a boundary condition that the field is zero at infinity, or the field is continuous at some boundary. An initial condition is used when something is changing, so perhaps if you are describing a dynamic field (changing in time) you would specify its initial form (i.e. maybe it starts as zero everywhere, then grows etc).

Initial and boundary conditions due serve the exact same purpose --> to fill in the details of a solution.
  • #3
Andy Resnick
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Boundary conditions are indeed similar to initial conditions, in terms of coming up with a specific solution instead of a general solution containing "constants of integration".

Boundary conditions are, and this is important, made up. Sometimes the boundary condition is simply the value of something, like the magnetic field (note that vector fields require more than a single number to specify the value). Other times, the boundary conditions can be more general statements, like "the normal component of the field is continuous across a boundary".

There's two general classes of boundary conditions- the value of something, or the gradient of the something. They can be time-dependent, space-dependent, both, neither. Boundary conditions can be prescribed on a real (material) boundary, or an artificial boundary surface.

But in the end, whatever the specifics of the boundary condition, the role is to generate a specific solution from a class of solutions.
  • #4
If your observation is limited to a finite space, then you can't make any predictions without information what is happening on the boundaries of that space: the future is not determined, since something might come from outside and influence the events in your finite space.

Understanding of initial or boundary conditions required to solve a particular problem is easier if you can imagine a numerical method of solving that problem. For example: you could place a grid in your volume, define one variable for the solution at every point and replace all derivatives in differential equations with finite differences. But you would get more variables than equations, since evaluation of a finite difference also requires known neibour points. Therefore boundary conditions are needed to provide the missing equations.