# Energy conservation and electromagnets

It is written that:

"Electromagnets are usually in the form of iron core solenoids. The ferromagnetic property of the iron core causes the internal magnetic domains of the iron to line up with the smaller driving magnetic field produced by the current in the solenoid. The effect is the multiplication of the magnetic field by factors of tens to even thousands."

So, if I have understood it correctly, the amplification of the magnetic field (with respect to a "normal" solenoid) is due to the very high magnetic permeability of the iron core. But I have a question about this: it seems to me that it is a free way to increase the magnetic field, i.e. I provide the same current, but I get the same magnetic field simply because of the presence of the iron core.

It seems strange to me, because if I can get a higher magnetic field with the same input current, I am saying that I can get higher magnetic energy $$(1/2) BH$$ with simply putting iron inside the solenoid. Where does this energy amount (absent in case of an ordinary solenoid) come from? How do we see that the energy conservation principle is respected?

• How much energy does it take to create the field? Current is not energy. Commented Jun 16, 2020 at 22:59

The energy stored in the magnetic field is indeed higher when you use an iron core, provided you pass the same current through the solenoid. However, this does not violate the conservation of energy in any way. The energy stored in the solenoid is $$\frac{1}{2}Li^2$$, where $$L$$ is the self inductance and $$i$$ is the current. The iron core increases the self inductance. When you have an iron core, you just need to put more energy into the solenoid to establish the magnetic field (or equivalently, to establish the current).
As a concrete example, suppose you connect the solenoid without the iron core to a battery of voltage $$V$$ in series with a resistor $$R$$ (which might model the series resistance of the battery or the solenoid). At steady state, the current through the solenoid is $$V/R$$. Now if you quickly insert an iron core into the solenoid, the current suddenly drops as a consequence of Lenz's law. Then the current gradually rises again to the steady state value $$V/R$$, as more energy is delivered to the solenoid.
• The iron core certainly increases the time constant and so the time it takes to reach steady state. You can view the greater energy transfer as being caused by the increase in the time constant. I don't really like this description because it doesn't take into account how the rate at which energy is delivered to the inductor (i.e. the power) changes when $L$ is different. But turns out the peak power after "turning on" the battery doesn't depend on $L$ so your description is still valid.