Solving Ampere's Law Homework with Curve C

[saubhik]

Homework Statement

In Ampere's Law, we consider an amperian loop (suppose a curve C), and any surface with boundary C can be chosen.
1.If the circulation of B is zero along the curve. Does this directly imply B=0 at points on the curve C?

2.Also in some cases there may be surfaces with boundary C through which a current can penetrate while through others not. Which surface will we take?

Homework Equations

$$\oint \vec{B} \cdot \vec{dr} = \mu_0 \ I_{enclosed}$$

The Attempt at a Solution

1.I don't think so since the B-field can be perpendicular to the points of C and thus even if B is present the circulation comes 0. However, if in a problem we have to show that B at a point is 0, how can we do it using Ampere's Law?

2. I think that if a current does not pass through the boundary C, we should take enclosed current=0. Only a current passing through C and penetrating S should be taken into account.(I think)

Please help.Thanks

 

[SammyS]

...
2.Also in some cases there may be surfaces with boundary C through which a current can penetrate while through others not. Which surface will we take?
...

The Attempt at a Solution

2. I think that if a current does not pass through the boundary C, we should take enclosed current=0. Only a current passing through C and penetrating S should be taken into account.(I think)

Please help.Thanks

This does not answer the second question.

If there's a choice? Which surface should you take?

 

[saubhik]

This does not answer the second question.

If there's a choice? Which surface should you take?

If a current does pass through a curve C (we are dealing with magnetostatics so no question of any change in electric fields) then it must also penetrate through surface S.
But my problem is: what if a current does not pass through C but penetrates S? Clearly there's a B-field at all points on the loop, but $$I_{enclosed}=0$$, so that circulation comes 0.

Here, S is any surface with boundary C.