# Magnetic Field outside an Iron Core Solenoid

In summary: The equation you have works for any solenoid of radius r and it works for anywhere along the z-axis, both inside and outside the solenoid. (This formula came up previously in a physics forums discussion, and I was able to derive it for the person who asked, does it work? The two terms with the x/sqrt( ) are cosines of angles and this formula containing cosines is often the formula given in E&M textbooks for a solenoid.) For the vector cross product ## K_m=M \times \hat{n}/\mu_o ## the result is ## K_m=M/\mu_o ##. The
I need help in calculating the magnetic field outside of an iron core solenoid at different distances.
I have made an electromagnet by taking a 99.99% pure iron (5cm in length and 1 mm in radius) and wrapping wire over it. I have measured it's magnetic field at the end and cannot find an equation from literature (Biot -Savart) that accurately models the magnetic field as I move the gaussmeter farther from one end (referring to moving the instrument further along the z direction in the picture I uploaded).

I know that the permeability of the iron is 2.5x10e-1 from a wikipedia chart on permeability
https://en.wikipedia.org/wiki/Permeability_(electromagnetism).

I have been using a formula that I found here, but does not seem to be working since it only applies to air core solenoids.

Does anyone have an idea on how I can more closely model/ create an equation for this relationship?
Thanks

Your equation, if I read it correctly is for a solenoid without the iron core. The magnetic field from the iron core, if the magnetization M is uniform, a good assumption, can be calculated from the surface current per unit length ## K_m=M \times \hat{n}/\mu_o ## using Biot-Savart.(## \hat{n} ## is outward pointing unit vector normal to the surface.) The surface currents have a geometry basically identical to those of a (finite length) solenoid, so any equations that work for a finite solenoid would also work for the field generated by the surface currents of the uniformly magnetized core. The magnetization ## M ## can often be found from hysteresis curves (M vs. H ) where H is the field of the solenoid without the iron core.

Thanks!
I have a couple follow up questions.
How does this incorporate moving the sensor farther away on the z plane?
And how do I use a hysteresis curve, since there does not seem to be a standard one available online?

Thanks!
I have a couple follow up questions.
How does this incorporate moving the sensor farther away on the z plane?
And how do I use a hysteresis curve, since there does not seem to be a standard one available online?
I think the equation you have works for any solenoid of radius r and it works for anywhere along the z-axis, both inside and outside the solenoid. (This formula came up previously in a physics forums discussion, and I was able to derive it for the person who asked, does it work? The two terms with the x/sqrt( ) are cosines of angles and this formula containing cosines is often the formula given in E&M textbooks for a solenoid.) For the vector cross product ## K_m=M \times \hat{n}/\mu_o ## the result is ## K_m=M/\mu_o ##. The ## K_m ## replaces ## Ni/L ## in the formula for ## B ##. This gives the iron core contribution. Use whatever radius ## a ## that the iron core has (in place of "r") and whatever length. To be most accurate, the field from the solenoid gets added (superimposed) to that of the iron core. (The field from the iron core is normally many times larger.) M can also be computed from permeability numbers. ## B=\mu H=\mu_o H+M ## where ## H ## is the field from the solenoid without the core. The result of a little algebra is ## M=(\mu-\mu_o)H ## and ## H=B_1/\mu_o ## where ## B_1 ## is the ## B ## field from the solenoid without the iron core. You will find considerable variations in the permeability ## \mu ## for different materials as well as different types of iron.

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## 1. What is a solenoid?

A solenoid is a type of electromagnet that consists of a wire wrapped in a cylindrical shape. When an electric current passes through the wire, it creates a magnetic field around the solenoid.

## 2. How does a solenoid create a magnetic field outside an iron core?

A solenoid creates a magnetic field outside an iron core by aligning the iron atoms in the core in the same direction as the magnetic field created by the electric current in the wire. This amplifies the strength of the magnetic field.

## 3. Is the magnetic field outside an iron core solenoid uniform?

No, the magnetic field outside an iron core solenoid is not uniform. The field is strongest at the poles of the solenoid and decreases in strength as you move away from the center.

## 4. Can the strength of the magnetic field outside an iron core solenoid be changed?

Yes, the strength of the magnetic field outside an iron core solenoid can be changed by adjusting the number of turns in the wire, the amount of current flowing through the wire, or the type of iron used in the core.

## 5. What are some practical applications of a solenoid with an iron core?

Solenoids with iron cores have a wide range of practical applications, including electric motors, generators, loudspeakers, MRI machines, and magnetic separators. They are also used in locks, valves, and other mechanical devices that require a strong and controllable magnetic field.

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