Simple loop and magnetic field

[dingo_d]

Homework Statement

I have a regular 6 sided polygon shaped loop through which flows steady current. I have to find the field at the center, at one of the vertex of the polygon and at the great distance from the loop.

Homework Equations

Biot-Savart law:

$$\vec{B}=\frac{1}{c}I\int\frac{d\vec{\ell}'\times\hat{r}}{r^2}$$

in CGS units.

The Attempt at a Solution

So I found the field at the center, there is a similar problem in Griffiths. And I got:

$$B=\frac{6 I}{c a}$$ where a is the length of a side of a polygon. Because the field for a wire is:

$$B=\frac{I}{cs}(\sin\theta_2-\sin\theta_1)$$, s=a and $$\theta_2=-\theta_1=\frac{\pi}{6}$$. And I multiply by 6 because I have 6 sides.

Now I am not sure if this next part is correct and I need some help with it:

At the vertex the field will be 0? Am I correct? Because the element of the wire is in the same direction as the vector from the source (the wire) to the point in which I'm looking at the field (the vertex).

The last part is the confusing one. Now I read that B-S law is sth like Coulomb law, but for currents and magnetostatics, that is it's inverse square law. That means that it falls as I'm getting away from the source.

When I had, in electrostatics, case with uniformly charged sphere when at great distance I'd get the field that is the same as the field for the single point charge.

But what do I get for this? Since it's closed loop, I certainly can't get the answer for infinite wire. Do I get the field of a magnetic field of a circular loop?

 

[jdc15]

For the first part, I didn't look over it exactly but the method sounds alright. At a vertex, 0 doesn't sound quite right. If you sum up all of the contributions from the different wires you get something into the page. At a great distance, well you have an inverse square relation. What is $$\lim_{x\rightarrow \infty} \frac{C}{x^2}$$ for any constant C?

 

[dingo_d]

Well it's zero, which I though at first, but wasn't really sure.

And for the vertex one I looked only at one vertex and from only one point, so that might be wrong...

 

[jdc15]

Yes, the field is 0 very far away.

If you look at the vertex, try considering the effect of each wire segment to the magnetic field to that point. The two segments attached to the vertex won't contribute (if I'm not mistaken), but all of the other segments will.

 

[dingo_d]

Thanks about this far away part :) I'll try to work that other thing out ^^