# An iron cylinder inside a solenoid

lorenz0
Homework Statement:
An iron cylinder with section ##S = 10 cm^2## and length ##d = 20 cm## is uniformly magnetized being placed into a solenoid with ##200## turns around the surface of the cylinder and traversed by a current ##i##. The magnetic field that is measured inside the cylinder is ##H = 10^3 A / m## and ##\oint_{\Gamma} \vec{B}\cdot d\vec{l}= 8\cdot 10^{-4} T m##.
Calculate the current ##i## flowing in the circuit and the magnetization vector ##\vec{M}##.
Relevant Equations:
##\oint_{\Gamma}\vec{H}\cdot d\vec{l}=\sum I##, ##\vec{H}=\frac{\vec{B}-\mu_0\vec{M}}{\mu_0}##
From ##\oint_{\Gamma}\vec{H}\cdot d\vec{l}=\sum I## by Ampere's Law which gives ##H \Delta l=\Delta N\cdot i\Leftrightarrow H=n i## where ##n=## number of turns per unit length so ##i=\frac{H}{n}=\frac{10^3 A / m}{\frac{200}{0.2m}}=1 A##.

Since ##\vec{H}=\frac{\vec{B}-\mu_0\vec{M}}{\mu_0}## we also get ##\oint_{\Gamma}\vec{H}\cdot d\vec{l}=\frac{1}{\mu_0}\oint_{\Gamma}(\vec{B}-\mu_0\vec{M})\cdot d\vec{l}\Leftrightarrow \oint_{\Gamma}\vec{M}\cdot d\vec{l}=\frac{1}{\mu_0}\oint_{\Gamma}\vec{B}\cdot d\vec{l}-ni\Leftrightarrow M=\frac{1}{\mu_0}\oint_{\Gamma} \vec{B}\cdot d\vec{l}-\frac{n}{l}i=\frac{1}{4\pi\cdot 10^{-7} H/m}\cdot (8\cdot 10^{-4}Tm)-\frac{200}{0.2m}1 A## ... this last part doesn't really convince me, even dimensionally, even if it looks like the initial idea to use Ampere's Law and then make the substitution ##\vec{H}=\frac{\vec{B}-\mu_0\vec{M}}{\mu_0}## does make sense?

Is there a way to amend my work? I would like to understand how to work with magnetic fields in matter like in this case and I would be grateful for an explanation about how to do that. Thanks

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Homework Helper
Gold Member
This one doesn't seem to be the best problem of this type. The numbers for ## B ## are too small, and both ## H ## and ## B ## will not be constant along the loop ## \Gamma ## shown for the integral. It looks like you computed the current ## i ## correctly, but I don't know that much else can be done with the numbers they give you.

In addition, the magnetic field that gets measured is ## B ##. The "field" ## H ## does not get measured unless you measure ## B ## without the iron in the solenoid. It is also incorrect to say that ## \oint B \cdot dl ## was measured, unless a complete mapping was done.

I don't know of a very good way of treating the finite length iron cylinder inside a solenoid, unless you are given the value of the magnetization ## M ## and or the value of ## B ## in the iron. Writing ##\oint H \cdot dl=NI ## does not lead to a straightforward solution, because ## H ## is very non-uniform. Complex numerical methods using the pole method could be useful, but that is a somewhat advanced treatment of the problem, and would take a lot of work.

Last edited:
• lorenz0
lorenz0
This one doesn't seem to be the best problem of this type. The numbers for ## B ## are too small, and both ## H ## and ## B ## will not be constant along the loop ## \Gamma ## shown for the integral. It looks like you computed the current ## i ## correctly, but I don't know that much else can be done with the numbers they give you.

In addition, the magnetic field that gets measured is ## B ##. The "field" ## H ## does not get measured unless you measure ## B ## without the iron in the solenoid. It is also incorrect to say that ## \oint B \cdot dl ## was measured, unless a complete mapping was done.

I don't know of a very good way of treating the finite length iron cylinder inside a solenoid, unless you are given the value of the magnetization ## M ## and or the value of ## B ## in the iron. Writing ##\oint H \cdot dl=NI ## does not lead to a straightforward solution, because ## H ## is very non-uniform. Complex numerical methods using the pole method could be useful, but that is a somewhat advanced treatment of the problem, and would take a lot of work.

• Homework Helper
Gold Member
I gave it a little more thought: To a somewhat good approximation, you could assume that ## B\approx 0 ## in the exterior part of the loop for the integral. This is not ideal, but it could work for some estimates. In any case, ## B ## in the iron should be in the neighborhood of ## 1 ## T, (considering the value of ## H ## and typical magnetic susceptibility values for iron), so that the integral ## \oint B \cdot dl ## should be about .2 Tm. This is where the problem is really lacking, and without a better number for the integral, it limits us in what we can do with it.

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Homework Helper
Gold Member
It is perhaps worthwhile to mention a couple additional things about this problem. It seems to not be treated in detail in a lot of the E&M textbooks. In the case of iron, such as that which is used in transformers, the hysteresis curve is such that when the applied ## H ## from the solenoid is just slightly negative (approximately zero), the magnetization ## M ## will also be zero. Meanwhile for material that would make a permanent magnet, the applied ## H ## needs to be very large and negative to reverse the direction of magnetization.

One other thing worth mentioning is that the magnetization will saturate in the iron at somewhere around ## \mu_o M=2 ## T. With the numbers that were given here for ## H ##, I believe the ## M ## would be rather close to saturation, rather than giving a number that they gave for ## \oint B \cdot dl ## of ## 8E-4 ## with an ## L=.2 ##.

The available literature on the subject of permanent magnets as well as the case of iron in a solenoid that makes an electromagnet seems to have improved in the last couple of years, but previous to that, it seemed to be rather deficient.