magnetic field rod

[tomfrank]

1. The problem statement, all variables and given/known data
i have a rod of 35 cm and a current of 10 A. the rod is a paramagnetic aluminum rod. how do i find the magnetic field inside the rod?


2. Relevant equations
the fact that the rod is paramagnetic changes anything?


3. The attempt at a solution

[Troels]

the fact that the rod is paramagnetic changes anything?

Yes. You now have a magnetization inside the rod that prohibits use of the biot-savart law there.

Also, I'm not sure what you mean by "35 cm" - are you refering to the length? If so, you should be very careful in applying ampéres law as you do not have a perfect cylindrical symmetry.

Please clarify

[tomfrank]

sorry it is the diameter. and the rod is consider to to infinitely long.

so what should I use?

[Troels]

Okay - that eases thing up a bit :)

Use Ampéres law for the H-field:

\oint_{\partial \mathcal{S}}\vec H \cdot d\vec \ell = \int_{\mathcal{S}}\vec J_\mathrm{free}\cdot d\vec a

For points outside the wire, this of course reduces to the familiar form:

\oint_{\partial \mathcal{S}}\vec H \cdot d\vec \ell = I_\mathrm{free,encl}

but for points inside the wire, you must be careful to only include the part of the current that is enclosed by the loop - assume that the current density is uniform, if not stated otherwise.

THen you should find:

H\cdot2\pi r = J_0 \pi r^2 \quad\Rightarrow\quad H=\frac{J_0 r}{2} for r < 17.5 cm

Where J0 is the totalt current divided by the crossectional area of the wire

(take a moment to verify this)

Now use

\vec B = \mu_0(\vec M + \vec H)

together with

\vec M = \chi \vec H

to find the required result

[tomfrank]

so once i find the H which is the magnetic field i just plugged in the

\vec B = \mu_0(\vec M + \vec H)

along with

\vec M = \chi \vec H


so is
J_0

the current they gave me?

[Troels]

so is
J_0

the current they gave me?

As I stated, it is the current density which is the total current divided by the cross-sectional area of the wire. THat is:


J_0=\frac{10\,\mathrm{A}}{\pi(0.35 \,\mathrm{m})^2}

[tomfrank]

how will I find the bound current density Jb and Kb.


Jb= curl M

Kb= cross product of M and n.

is this right and how do I do the curl?

[Troels]

how will I find the bound current density Jb and Kb.


Jb= curl M

Kb= cross product of M and n.

is this right and how do I do the curl?

It's right - so just plug-in the magnetization you got from \vec M = \chi \vec H