# Dimensional analysis of magnetic energy: dimensions of µ0 and H

When calculating the energy difference between the normal and the superconducting state in a superconductor at zero magnetic field, the result is as follows:

Now I'm quite confident of this result, as it is the same as my textbook tells me it is. The problem is that when I do a dimensional analysis, I find

The dimension on the left is energy, the dimension on the right is vacuum permeability times H-field squared. The units of the first are found on Wikipedia, $N/A^2$, as are the units of the second, $A/m$ (Ctrl+F for The H-field is measured in'').

As you can see, the lengths don't cancel out, we have $[L]^2$ on the left and $[L]^{-1}$ on the right. What am I doing wrong? Is the first equation not in SI?

Edit: In case it might be of any help, the derivation of the above equation is (using the Meissner effect in the final step, $M=-H$) as follows:

Remember that your expression gives the energy difference per unit volume. So you need an additional factor of $\left[L^{-3}\right]$ on the left hand side.