# How can I calculate the relative permeability of iron?

I'm a high school physics student, and part of our project requires us to make an electromagnet. We have an iron core, and it will go inside a solenoid. The problem is, we don't know what the relative permeability constant of iron is so we can calculate the magnetic field with the iron. How can we get this constant?

As you don't know the purity of the iron, I don't think you can calculate the permeability. Therefore, if you are not going to buy another core with known properties, you should measure the properties of the existing one, for example, by using the core in the electromagnet:-) If the properties of the test electromagnet are not satisfactory, you can redesign the coil based on the test results.

This Wikipedia page has the permeability value for iron, and other materials. Obtain the relative permeability by dividing by the fundamental constant $\mu_0$ (defined elsewhere on that page).

• I've already been on that page, and I am looking for another way because I don't know the purity of the iron. Jun 4 '14 at 13:24

We could make use of some of the inductance properties for resonating potentials.

$$f = \frac{1}{2pi\sqrt{LC}}$$

The inductance of rectangular coil of sizes $L$ x $W$ x $H$ with N turns of wire (spire) is : $$L = \mu_0 \mu_r \frac{N^2 A}{l}$$

Here is our test circuit:

We could make use of alternative current from a transformer (in Europe the frequency $f$ would be $60$Hz but you can use the frequency from your own region).

$$f = \frac{1}{2pi \sqrt{\mu_0 \mu_r \frac{N^2 A}{l} C}}$$

How ? Well, the circuit will resonate with the AC source only when the LC circuit has the same frequency with it. If the frequency of the LC is higher or lower than of the AC then the ampermeter will show irregular current.

We will add/subtract more turns(spires; will increase/decrease $N$) to the core and/or a smaller/bigger capacitor until our circuit resonates (we could really use an oscilloscope or a low frequency AC such as 2Hz, but I leave this for you to solve).
Since we know all the other parameters we can extract our coil magnetic permeability $\mu_r$ : $$\mu_r = \frac{1}{4pi^2 f^2 \mu_0 \frac{N^2 A}{l} C}$$

$L$ - coil inductance
$C$ - capacitance in Farads
$N$ - number of turns (spire) of wire
$A$ - core's surface (aria = height * width)
$l$ - core's length
$f$ - resonance frequency [Hertz]
$\mu_0$ - vacuum magnetic permeability (= $4 \pi 10^{-7})$
$\mu_r$ - core's magnetic permeability