- #1
Mr_Allod
- 42
- 16
- Homework Statement
- A conducting wire is placed along the z axis, carries a current I in the +z direction and
is embedded in a nonconducting magnetic material with permeability ##\mu##. Find the
magnitude and direction of ##\vec H##, ##\vec B##, ##\vec M## and the equivalent current ##\vec J_b## at the point (x,0,0).
- Relevant Equations
- Ampere's Law: ##\oint \vec B \cdot d\vec l##
##\vec H## and ##\vec B## Relation: ##\vec H = \frac 1 \mu_0 \vec B - \vec M##
Permeability: ##\mu = \mu_0 (1+ \chi_m)##
Magnetisation: ##\vec M = \chi_m \vec H##
Hello there, I've worked through this problem and I would just like to check whether I've understood it correctly. I found ##\vec H##, ##\vec B## and ##\vec M## using Ampere's Law and the above relations as I would for any thin current carrying wire and these were my answers:
$$\vec H = \frac I {2\pi s} \hat \phi$$ $$\vec B = \frac {\mu I} {2\pi s} \hat \phi$$ $$\vec M = \chi_m \frac I{2\pi s} \hat \phi$$
For the point (x,0,0) I would simply swap ##x## for ##s##.
Then using ##\vec J_b = \nabla \times \vec M## I tried to calculate ##\vec J_b##. All but one of the terms of the cross product evaluate to 0 leaving:
$$\vec J_b = \frac 1 s \frac \partial {\partial s}(s M_\phi) \hat z = \frac 1 s \frac \partial {\partial s}(s \chi_m \frac {\mu I}{2\pi s}) \hat z$$
Which as it turns out also evaluates to 0. This leads me to believe I must have misunderstood something about the question since I don't expect I would be asked to find ##\vec J_b## if it was simply 0. If somebody could help me figure out what I've done wrong I'd appreciate it.
$$\vec H = \frac I {2\pi s} \hat \phi$$ $$\vec B = \frac {\mu I} {2\pi s} \hat \phi$$ $$\vec M = \chi_m \frac I{2\pi s} \hat \phi$$
For the point (x,0,0) I would simply swap ##x## for ##s##.
Then using ##\vec J_b = \nabla \times \vec M## I tried to calculate ##\vec J_b##. All but one of the terms of the cross product evaluate to 0 leaving:
$$\vec J_b = \frac 1 s \frac \partial {\partial s}(s M_\phi) \hat z = \frac 1 s \frac \partial {\partial s}(s \chi_m \frac {\mu I}{2\pi s}) \hat z$$
Which as it turns out also evaluates to 0. This leads me to believe I must have misunderstood something about the question since I don't expect I would be asked to find ##\vec J_b## if it was simply 0. If somebody could help me figure out what I've done wrong I'd appreciate it.
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