# An equation in Laughlin Argument paper (1981)

In his paper Quantized Hall conductivity in two dimensions, Laughlin considers a ribbon of two-dimensional metal bent into a loop of circumference $L$, and pierced everywhere by a magnetic field $H_o$ normal to its surface. He says that the current through the loop is equal to the adiabatic derivative of the electronic energy $U$ of the system wrt the magnetic flux $\phi$ through the loop: $$I=c\frac{\partial U}{\partial \phi}=\frac{c}{L}\frac{\partial U}{\partial A}$$

My question is what is the origin of this equation and why is the current equal to the adiabatic derivative of the electronic energy $U$?

• I think your question might be already answered in here. See the comments right below the question. Laughlin's own Nobel Lecture also has more details on this equation pp 271-272. Aug 28, 2020 at 23:33