Can Ampere's Law Still Apply When Whole Solenoid Is Enclosed?

[n77ler]

Homework Statement

I want to know why Ampere's Law can sitll be applied/valid if the Amperian surface is drawn as a rectangle which encloses a whole solenoid. Normally the rectangle would just include one side of the solenoid.

Homework Equations

The Attempt at a Solution

When only one side of the solenoid is enclosed, Ampere's Law will work because B and ds will be parallel on the side of the rectangle which is inside the solenoid so we can intergrate this to an non-zero value. But I'm confused about when the whole solenoid is enclosed by the rectangle. It is an ideal solenoid so there should be no magnetic field outside the solenoid so the integral of the 2 sides which are parallel to B will be 0. The other two sides are perpendicular to B so they will also = 0. The only reason I can think that Ampere's Law still applies is that all of the current is enclosed so that somehow helps with the solution?

I'm really stuck here, because I keep getting the left side of the equation =0 so I cannot see how the law is still valid.

 

[n77ler]

Anyone?

 

[diazona]

Remember that current has a direction. You have to consider not only how much current goes through the rectangle, but in which direction it goes through.

 

[n77ler]

But I thought it only depended on the direction of B and ds for the left side of the laW?

 

[n77ler]

So the current travels in opposite directions on opposite sides of the rectangle and therefore the current will cancel out? So when we evaluate each side of the rectangle with the intergral involving B and ds we are supposed to get 0 because the right side of the equation is also supposed to equal zero?