Ampere's Law -- What is the meaning behind each part?

[np115]

Homework Statement

So I know that SB · dl = u0I (sorry this is the only way that was working). But I was wondering what each part meant? Cause in Gauss's law, the SE · dA is the object that is being used to calculate electric field and in Ampere's, it is meant to be similar. So if we had a wire of radius a. This wire creates a uniform field. If we had to calculate the field at point b outside the wire, what would the equation look like? From what I have learned, we would use a circle for this. So B(2pi a) or B(2pi b)?

Relevant Equations

S B · dl = u0I

I believe it would be B(2pi b) but I'm not sure how exactly to explain why.

 

[rude man]

Homework Statement:: So I know that SB · dl = u0I (sorry this is the only way that was working). But I was wondering what each part meant? Cause in Gauss's law, the SE · dA is the object that is being used to calculate electric field and in Ampere's, it is meant to be similar. So if we had a wire of radius a. This wire creates a uniform field. If we had to calculate the field at point b outside the wire, what would the equation look like? From what I have learned, we would use a circle for this. So B(2pi a) or B(2pi b)?
Relevant Equations:: S B · dl = u0I

I believe it would be B(2pi b) but I'm not sure how exactly to explain why.

You need to read Ampere's law carefully.

 

[kuruman]

This wire creates a uniform field. If we had to calculate the field at point b outside the wire, what would the equation look like?

No wire creates a uniform magnetic field.
If you wanted to calculate the electric field due to a charged sphere of radius $a$ at point $b$ outside the sphere, would you use $E (4 \pi a^2)$ or $E (4 \pi b^2)$ on the left hand side of the equation for Gauss's law? Why?

I agree with @rude man: study Ampere's law some more and pay attention to how it is used in your textbook's examples.

 

[vanhees71]

Of course a single current-conducting wire doesn't create a uniform magnetic field, but you can use symmetry for the simple case of an infinitely long wire. You know by symmetry that the magnetic field is always of the form $\vec{B}(\vec{r})=B(R) \vec{e}_{\varphi}$, where I've put the wire along the $z$-axis of a cylinder-coordinate system $(R,\varphi,z)$.

To get $B(R)$ just use Ampere's circuital law with a circle of radius $R$ around the $z$-axis in a plane perpendicular to the $z$-axis.