Maxwell's equations: a quick check

[DivGradCurl]

Folks,

I believe the direction of the circulation in the curl equations must always be considered counterclockwise (positive), because of Stokes' theorem and the right-hand rule. Right? (I'm asking because my books do not have a direction specified in the circulation integral) Thanks.

 

[leright]

When you take the curl of a vector field you get a vector. The direction of the vector describes the direction of the circulation by the right hand rule.

Now, if you are taking about the circulation integral, which is NOT The curl, you are thinking about a vector field integrated along a closed path. The result of a circulation integral is a SCALAR. A negative circulation integral around a closed path means that the direction of the differential path length is taken to be opposite the direction of the vector field (the vector field is opposite of the dl path length vector, so the dot product is negative). The direction of the path depends on the direction you assume the current to be in (in the case of ampere's law) by the right hand rule. When you take the circulation integral of the B-field along this path and you get a negative number, then the current is actually in the opposite direction.

 

[DivGradCurl]

I'm asking about the circulation integral (not the curl operator). My question is the following: Is there an initial assumption on its direction? I know the results (CW -> - and CCW -> +). Sometimes authors leave the positive direction as an implicit assumption (e.g. Stokes' theorem in some texts). It seems that the assumption is CCW direction. Maybe it doesn't matter since the sign "pops out" of the curl operation. Just wondering... Thanks.

 

[leright]

I'm asking about the circulation integral (not the curl operator). My question is the following: Is there an initial assumption on its direction? I know the results (CW -> - and CCW -> +). Sometimes authors leave the positive direction as an implicit assumption (e.g. Stokes' theorem in some texts). It seems that the assumption is CCW direction. Maybe it doesn't matter since the sign "pops out" of the curl operation. Just wondering... Thanks.

yes, there is an initial assumption. I answered this question also. What do you mean CW -> - and CCW -> +?

The direction of the path you are calculating the circulation integral for depends on the assumed direction of the current. Use the right hand rule to find the direction of the path. If your resulting circulation integral is positive then you assumed your current direction correctly. IF your resulting circulation integral is negative then you assumed your current direction incorrectly, and the actual current is in the opposite direction. It's as simple as that.

Why do you bring up 'curl'? I thought you weren't asking about the curl operation. At any rate, when you calculate the curl of a vector field and then evaluate the curl at a certain point you get a vector whose direction describes the direction of the circulation about that point. You use the right hand rule to determine the direction of circulation based on the direction of the curl vector.

Stoke's theorem simply says that if you determine the net flux of the curl vector field through an open surface defined by a closed path then you get the circulation integral of the original vector field around that closed path. To me, this is intuitively pleasing.

 

[Hurkyl]

If you have an oriented surface, then its boundary is oriented counter-clockwise around the surface.

"Clockwise" and "counterclockwise" make absolutely no sense for the loop itself; those terms only have meaning in relation to an oriented surface.


To drive the point home, here is a drawing of the oriented boundary of an annulus, which is given the default orientation for being part of your computer screen. (i.e. it's orientation points from the computer screen towards you)

   /---<---\
  /#########\
 /###########\
/#############\
|####/->-\####|
|####|   |####|
V####^   V####^
|####|   |####|
|####\-<-/####|
\#############/
 \###########/
  \#########/
   \--->---/

In particular, note the orientation of the inner part of the boundary. That is correct.