# Inductance of a coil with a core

Does inductance of a coil increase if we add a material with good magnetic permeability only because of its magnetization, or also because of changing the path of magnetic field lines?

We can say in a closed loop,

$$\Phi=LI$$

defines the inductance of the loop. Where $I$ is the current in the loop, $\Phi$ is the magnetic flux through the surface of the loop (the loop was closed, therefore acting as the boundary of an open surface), and $L$ is the inductance.

So for the inductance to maximize, we need to have the most magnetic field lines penetrating the surface of the loop, with the least current flowing along it.

What a core does is actually strengthening the field lines. So that we have a more $\Phi$ with the same current.

Which yields higher inductance.

I am not much good with your sentence "because of its magnetization, or also because of changing the path of magnetic field lines?". Adding a core to a coil changes the magnetic field both via field line pattern and magnitude.

The better saying would be just "adding a core increases the flux.", in my opinion.

• This is why I asked a quation the way I did. I hoped for a Yes or No answer. I already know what you explained. Jun 6, 2017 at 11:13
• @MaDrung then the answer is yes. This is why a core twice the length of the coil has an even greater $L$ than a core with just the length of the coil. In case you have access, try to look into Sensors, A Comprehensive Survey, Volume 5, Magnetic Sensors Jun 6, 2017 at 12:05
• @MaDrung I implicitly answered yes. It's just like bringing some magnets inside the coil. The field pattern will obviously change.
– AHB
Jun 6, 2017 at 14:38
• I know it will change. But is the field pattern responsible for increase of inductance or does only magnetization because of material increase it? More material, more magnetization. That much is clear. But is inductance also affected by the change of path of magnetic field lines? Jun 7, 2017 at 7:34
• @MaDrung I think I got what you mean. Hmmm. Then, yes of course! but, to say whether it will increase or decrease, that is a difficult question to answer, because it totally depends on how the pattern has changed. And Because it is a VECTOR FIELD! if, instead, it was a number or a vector, it would be much easier to identify the increase and decrease regimes.... But, by the way, everything we want (to say whether the inductance will increase or decrease) is summarized in the concept of flux! I myself am convinced with "Higher the flux (not due to external sources), Higher the inductance."
– AHB
Jun 7, 2017 at 10:38

The magnetic H Field is calculated via: $\oint \vec{H}~d\vec{s} = I$

This means the H field is directly caused by the current I.

The inductance is dependent on the electric filed that the magnetic field induces.

According to Maxwell's equations:

$\text{rot}~\vec{E} = -\frac{d\vec{B}}{dt}$

This equation uses the B field which is bound to the H field through this equation:

$\vec{B} = \mu \vec{H}$.

In the simplest case $\mu$ is only a constant value.

Therefore inserting a material into the space, where the magnetic filed is in, scales the magnetic B field and therefore the induced E field and inductance.

Another approach is seeing the magnetic resistance of a system. The resistance of a piece of material (metal) is:

$R_m = \frac{l}{\mu A}$, where l is the lenght and A is the area perpendicular to the magnetic flux.

The inductance of a coil wound around a medium with the resistance $R_m$ is:

$L = \frac{n^2}{R_m} = \frac{n^2\mu A}{l}$, assuming that all of the magnetic flux is inside the medium.

You can see the proportionality.

Materials with high permeability guide the magnetic flux, and therefore improve the coupling / inductance between coils. This image shows the magnetic flux inside a transformer. You can see that most of the magnetic filed couples both coils together and therefore increases the coupling inductance.