Quick question about Ampere's Law and how to use it?

[Jormungandr]

I just have a quick question about how to use Ampere's Law. It says that ∫B ds = u0(i_enc), which I suppose is easy enough to understand. But I'm having trouble reconciling it with the notion of, say, the magnetic field at the center of a loop of wire.

The magnetic field at the center of a loop of wire is B = (u0 * i) / (2R). But what if I were to draw an Amperian loop inside the loop of wire? Not enclosing the loop itself, but just an arbitrary circle within the loop's boundaries. From Ampere's Law, there's no i_enc here, which makes the right side of Ampere's Law equal to 0, which implies there is no magnetic field. And yet the earlier formula says that there is a magnetic field here, and we know there is. So clearly, I'm either using Ampere's Law wrong, or it's not applicable here. I'm not sure. Help is appreciated!

 

[jtbell]

there's no i_enc here, which makes the right side of Ampere's Law equal to 0, which implies there is no magnetic field.

No. it implies that $\int {\vec B \cdot d\vec s} = 0$. What is the direction of $\vec B$ and what is the direction of $d\vec s$ along the loop that you're integrating over?

 

[Jormungandr]

No. it implies that $\int {\vec B \cdot d\vec s} = 0$. What is the direction of $\vec B$ and what is the direction of $d\vec s$ along the loop that you're integrating over?

Hmm. I'm guessing since they're orthogonal, $\int {\vec B \cdot d\vec s} = 0$, right? Okay. That makes perfect sense, thank you!

Actually, on the subject, I was wondering about something that I saw on a website somewhere. Apparently, the graph of B vs radius of a solid, cylindrical conducting material was a straight line from the origin until the surface radius and then it decreased nonlinearly according to (u0 * I) / (2*pi*r).

However, for a hollow, cylindrical conductor, the graph was 0 until the inner radius, after which time B increased nonlinearly as well until the outer radius, after which it too decreased nonlinearly. I understand why it increases nonlinearly in this case, because the i_enc term has some radius squared terms in it since the Amperian loop doesn't always enclose all of the current.

I don't, however, understand why B increases linearly for a solid material. Doesn't this also have the case where the loops don't encircle the entire current? Shouldn't this also be nonlinearly increasing, but just from the origin?