# Magnetic induction by changing permeability of a uniform magnetic field?

As far as I know, magnetic fields are created by either magnet or running current, which both can be changed by changing the permeability of the medium and thus change the magnetic flux through the coil.

So will there be induced current in a coil in a uniform magnetic field with changing permeability? If yes, what permeability should be changed? Is it the space around the coil (eg put an iron bar in the coil), the space between the source of the magnetic field and the coil (which the magnetic field "travels"?), or the permeability of the coil itself? (ie heating it?)

Or does changing permeability by definition conflicts the given "uniform magnetic field"? Thanks a lot :D

If the circle in the sketch is the circular coil you are talking about, there will be an induced current if there is a time-varying magnetic flux through it.

The magnetic flux is defined as: $$\phi_B = \int_{\text{S bounded by loop}} \mathbf{B}\cdot\mathrm{d}\mathbf{S}.$$

Now, when you have a magnetic field in a material (and not just free space), you also need to take into account how the magnetism of the material itself may modify the total net field. The material, in this case, is whatever your coils is wrapped around. It's the material in the region with the 4 crosses inside the loop.

For this reason, you define $$\mathbf{B}$$ to be the net field in the region (external + material's response), and $$\mathbf{H}$$ to be the "magnetising field" i.e. the external field. The two are related by: $$\mathbf{B} = \mu \mathbf{H},$$ where $$\mu$$ is the magnetic permeability (generally a rank-2 tensor, but let's assume a decent material so that it's a scalar here).

$$\mu = \mu_0 \cdot \mu_{\mathrm{r}}$$, where $$\mu_0$$ is the permeability of free space and $$\mu_{\mathrm{r}}$$ is the relative permeability of the material in question.

So all together now, the magnetic flux is:

$$\phi_B = \mu_0 \int_S \mu_{\mathrm{r}} \mathbf{H}\cdot\mathrm{d}\mathbf{S}.$$

For an induced current, current, you need $$\partial_t \phi_B \neq 0$$. In order to do this, you can either vary the size of the cross section $$\mathrm{d}\mathbf{S}_\parallel$$ (by e.g. rotating the loop), vary the external field $$\mathbf{H}$$, or vary the relative permeability $$\mu_{\mathrm{r}}$$. Or all of them at the same time.

So if you (somehow) can control the external field's strength and the material's relative permeability independently, you can indeed keep $$\mathbf{H}$$ fixed and just vary $$\mu_{\mathrm{r}}$$ to get an induced current.

By the way, uniform magnetic field in this case means that it's only in one direction (into the paper). Provided $$\mathbf{B}$$ and $$\mathbf{H}$$ are parallel, i.e. when $$\mu$$ is a scalar and not a tensor, the field is uniform all the time.

• Wow such a thorough response! so if I'm understanding correctly, while placing an iron bar in the coil, the mu_r changes (and the external magnetic source is touched so the external field is fixed) thus there will be induced current right Jul 9, 2020 at 5:32
• Well yes. However the act of moving the iron bar inside the loop causes another flux to be varying with time (the one associated with the iron bar own magnetic field). So the situation is a bit more complex. In the ideal case where the iron bar were non magnetic, you slide it in, and then somehow "activate" its magnetism, thereby changing $\mu = \mu_0 \rightarrow \mu_0\mu_r$, then you'd have an induced current only because of the $\mu_r$. Jul 9, 2020 at 6:20
• got it, thanks! Jul 10, 2020 at 1:06
• I not forgotten; very nice answer. +1. Jul 18, 2020 at 11:37