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

1. Mar 18, 2013

### 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!

2. Mar 18, 2013

### Staff: Mentor

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?

3. Mar 19, 2013

### Jormungandr

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?