# I Bring 1 kg of Iron to a Flux Density of 1 T. How Much Energy Does That Take?

I'm an EE, not a physicist, so please forgive if this question is dumb.

I learned a bit of magnetics when I took motors 20 years ago, but I don't remember much.

I'm reaching out to the physics community because finding EE's who know the answer to this question can be tough. (Take me, for example.)

I bring 1 kg of iron to a flux density of 1 T. How much energy does that take?

Usually we can answer questions like this using $$\frac{U}{V} = \frac{B^2}{2 \mu}$$ (magnetic energy stored per unit volume), but since iron is nonlinear and ferromagnetic, we need to use its magnetisation curve. In this case $$\frac{U}{V} = \int_{0}^{B} H \ dB$$, so the energy required per unit volume of iron is equal to the area between this curve, the vertical axis, and the line $$B = 1$$. It's quite linear in this region so you could approximate it as a triangle (which is equivalent to just using the original formula above). I estimate it as
$$\text{energy per unit volume} = \frac{1}{2} \times {1} \times {2000} = 1000 J = 1 \ kJ.$$