Hi, I am currently working on Ex. 3 of the Drude model, and I am curious as to why we see a dependence on B for longitudinal conductivity, but not for longitudinal resistivity. Is there some underlying physical explanation for this? Thanks in advance for your reply!
I agree this is quite puzzling. The way I see this is that, when we apply a current J_x, then the electric field along x is given by E_x=\rho_{xx}J_x. Note that in this case, there is no contribution of \rho_{xy} because J_y is zero. In contrast, when we apply an electric field E_x, then the current along x is given by J_x=\sigma_{xx} E_x+\sigma_{xy}E_y. I.e., there is a contribution from \sigma_{xy} because E_y is non-zero (because of the magnetic field).
So we see that when we apply E_x and measure J_x, the quantity we can extract directly is \rho_{xx}. In contrast, we cannot extract \sigma_{xx} without additional knowledge because J_x=\sigma_{xx} E_x+\sigma_{xy}E_y. So we see that \sigma_{xx}=\rho_{xx}^{-1} cannot be true as it would not allow us to satisfy this equation (because J_x must be independent of B). As such, the \sigma_{xx} term needs to incorporate a B dependence.
Thank you!