An iron cylinder inside a solenoid

In summary, this conversation discusses the application of Ampere's Law to a solenoid with an iron core. The value of the current is determined using the number of turns per unit length and the magnetic field is calculated by substituting in for H using the equation H = (B - μ0M)/μ0. However, the given numbers for the problem are small and the non-uniformity of the magnetic field makes it difficult to find a solution. It is also mentioned that the hysteresis curve and saturation point of the iron core are important considerations for this type of problem.
  • #1
lorenz0
148
28
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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  • #2
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.
 
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  • #3
Charles Link said:
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.
Thanks for your very thorough answer
 
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  • #4
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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  • #5
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.
 

Related to An iron cylinder inside a solenoid

1. What is an iron cylinder inside a solenoid?

An iron cylinder inside a solenoid refers to a setup in which a ferromagnetic material, such as iron, is placed inside a coil of wire called a solenoid. When an electric current is passed through the solenoid, it creates a magnetic field that induces a magnetic field in the iron cylinder. This setup is commonly used in experiments and devices that involve electromagnetism.

2. How does an iron cylinder inside a solenoid work?

When an electric current is passed through the solenoid, it creates a magnetic field that is concentrated inside the coil. This magnetic field then induces a magnetic field in the iron cylinder, causing it to become magnetized. The strength of the magnetic field in the iron cylinder can be controlled by adjusting the amount of current flowing through the solenoid.

3. What are the applications of an iron cylinder inside a solenoid?

An iron cylinder inside a solenoid has various applications in science and technology. It is commonly used in devices such as electric motors, generators, and speakers. It is also used in experiments to study the properties of electromagnetism and to demonstrate the principles of electromagnetic induction.

4. What factors affect the strength of the magnetic field in an iron cylinder inside a solenoid?

The strength of the magnetic field in an iron cylinder inside a solenoid depends on several factors. These include the number of turns in the solenoid, the amount of current flowing through the solenoid, and the permeability of the iron cylinder. The strength of the magnetic field can also be affected by the presence of other nearby magnetic fields.

5. How can the magnetic field in an iron cylinder inside a solenoid be increased?

The magnetic field in an iron cylinder inside a solenoid can be increased by increasing the number of turns in the solenoid, increasing the amount of current flowing through the solenoid, or using a material with higher permeability for the iron cylinder. Additionally, placing a ferromagnetic core inside the solenoid can also increase the strength of the magnetic field.

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