Find all the bound currents. What is the net bound current flowing down the wire?

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Homework Statement

A current I flows down a long straight wire of radius a. The wire is made of linear material with susceptibility chi(subscript m), and the current is distributed uniformly.

i) what is the magnetic field a distance s from the axis?
ii) Find all the bound currents. What is the net bound current flowing down the wire?

The Attempt at a Solution

i) I found a similar homework question on this website:

http://maxwell.uncc.edu/gjgbur/coursepages/4231assets/homework11answers.pdf

which says that B=[mu Is/(2pi a^2)]phi-hat
ii) J(subscript M)= curl M
= [(chi(subscript M) I)/(pi a^2)] z-hat

integral of [(chi(subscript M) I)/(pi a^2)] z-hat dS
= loop integral of [(chi(subscript M) I)/(pi a^2)] z-hat. dl
=0

why does J(subscript M) have to be integrated over the surface, though?

j(subscript m)=M cross n-hat
=[- chi(subscript M) Is/(2 pi a^2)]z-hat ?

why does http://maxwell.uncc.edu/gjgbur/coursepages/4231assets/homework11answers.pdf
say that j(subscripr m) has to be integrated over the perimeter? How do I do this? And why is the result of it supposed to be zero? Please help

 

[mark726]

Thank you for your question. Let me address each of your concerns separately:

i) Your solution for the magnetic field a distance s from the axis is correct. This formula can be derived using the Biot-Savart law, which relates the magnetic field at a point to the current element passing through that point.

ii) To find the bound currents, we need to use the definition of magnetization, which is the magnetic moment per unit volume of a material. In this case, since the wire is made of linear material, we can write the magnetization as M = chi(subscript m)H, where H is the applied magnetic field. Using this definition, we can then find the bound currents by taking the curl of the magnetization, as you have done. This gives us an expression for the bound current density J(subscript M) at any point in the wire.

Now, to find the net bound current flowing down the wire, we need to integrate this current density over the entire surface of the wire. This is because the bound current is distributed over the entire volume of the wire and not just on the surface. The result of this integral should give us the total bound current flowing down the wire.

The reason why the result of the integral is zero is due to the fact that the magnetization is uniform throughout the wire and there is no net current flowing in the wire. This means that for every bound current flowing in one direction, there is an equal and opposite bound current flowing in the opposite direction, resulting in a net current of zero.

I hope this helps clarify your doubts. Let me know if you have any further questions. Good luck with your studies!