Infinitely long cylinder - locate bound currents and calculate field

[blueyellow]

Homework Statement

An infinitely long cylinder, of radius R, carries a frozen-in magnetisation, parallel to the z-axis, M=ks k-hat, where k is a constant and s is the distance from the axis. There is no free current anywhere. Find the magnetic field B inside and outside the cylinder by two different methods:

i) Locate all bound currents, and calculate the field they produce; and
ii) Use Ampere's law, the loop integral of H.dl=I(subscript enc), to find H, then get Bfrom H=(1/mu0)B-M

The Attempt at a Solution

I found the answer on http://www.nhn.ou.edu/~shaferry/41832005_files/finalsol.pdf

but what I don't understand is why far away from the loop the B field must go to zero. I would be grateful if someone could explain please.

 

[rude man]

but what I don't understand is why far away from the loop the B field must go to zero. I would be grateful if someone could explain please.[/QUOTE]

Same reason that B outside an infinitely long solenoid is zero. Looking at a given section of the solenoid, coil currents flow in opposite directions along the two sides of the solenoid, canceling the external field. A permanent-magnet cylinder is no different if magnetization is along the major axis (center). Amperian currents merely assume the role played by a solenoid's current-carrying wire.

So B doesn't just disappear at infinity, it disappears anywhere outside the OD of the rod.

This can be proven more rigorously by application of the Biot-Savart law.

cf. Resnick & Halliday, sect. 34-5.

 

[Liquidxlax]

the current loop need to enclose some sort of current for there to be any sort of magnetic field this is why outside a solenoid the B field is zero

 

[rude man]

That only means that the circulation (closed-path integral) of B is zero. It does not preclude segments of the path being finite, as long as they cancel around the path.

Example: a single loop of curent-carrying wire. Choose a closed path in front of the loop but not intersected by the loop. B is certainly not zero in front of the loop!

 

[blueyellow]

thanks,
for que i), why is the field due to the volume distribution=mu0 k(R-s)z-hat?
The bound current due to the volume is curl M=- k phi-hat, but how does this lead to:

the field due to the volume distribution=mu0 k(R-s)z-hat?

 

[rude man]

I'm sorry no one's replied to that, incl. me. Truth is, I don't know the answer without researching the subject more myself.

 

[blueyellow]

thanks anyway. think i found out in the end, though