Magnetic field of current loop

[AriAstronomer]

Homework Statement

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

Homework Equations

vec(r) = vector of r.
zhat = unit vector z.

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

 

[Spinnor]

See:

http://230nsc1.phy-astr.gsu.edu/hbase/magnetic/curloo.html

3/4 the way down the page.

Found via Google search,

"magnetic field of current loop along the z axis"

Hope this helps.

 

[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]

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.

 

[rwisz]

elle


Hello Arielle,

Your approach seems to be on the right track, but there are a few errors in your calculations. First, the vector dl should be pointing in the direction of the current, which is in the z-direction. Therefore, dl should be equal to R(phi hat), where phi is the azimuthal angle around the loop. This means that the cross product dl x vec(r) should be equal to R(zhat x phi hat), which gives a result of R(s hat).

Next, when you take the dot product of zhat with vec(r), you should get a result of z, not R. This is because vec(r) is a vector pointing from the origin to a point on the loop, and in this case, the point is located at a distance z above the origin. Therefore, the z component of vec(r) is simply z.

Putting these together, we get a result of -2piR^2 for the numerator, which matches the correct answer. I hope this helps clarify your approach. Keep up the good work in your scientific studies!