Magnetic field of current loop

AriAstronomer

1. Homework Statement
Find magnetic field along the z-axis of a circular loop, radius R with constant current lying in the z=0 plane.

2. Homework Equations
vec(r) = vector of r.
zhat = unit vector z.

3. The Attempt at a Solution
So starting with the definition, B = (u_o I/4pi) (dl x vec(r))/r^3, where:
\vec(r) = z(zhat) - R(s hat), thus:
r = root(z^2 + R^2)

also dl x vec(r) = vec(r)dl(phi x s) = (dl)(-zhat). Since dl = length around current loop = 2piR:
dl x vec(r) = -2piR(zhat) vec(r).

Here's where I get stuck.
If I continue along this line of thought, zhat (from the cross product) dotted with vec(r) will give me z, and my numerator will end up being -2piRz, but the answer says the numerator should be -2piR^2. Basically, I should have an R instead of z, but I don't know where I'm going wrong. I know other approaches will give me the right answer, but can someone identify along my train of thought what the problem with this approach is?

Thanks,
Ari

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Gold Member

AriAstronomer

I'm looking at it, but judging from the diagram it looks like dBz should equal cosine theta, not sine theta which is what they're claiming.... Still kind of confused.

Spinnor

Gold Member
I'm looking at it, but judging from the diagram it looks like dBz should equal cosine theta, not sine theta which is what they're claiming.... Still kind of confused.
Draw your own picture. When theta goes to 90 degrees dB = dB_z as required.

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