  We present a simple method that allows to calculate the electromagnetic
response of non-interacting electrons in strong magnetic field to arbitrary
order in the gradients of external electric and magnetic fields. We illustrate
the method on both non-relativistic and massless Dirac electrons filling $N$
Landau levels. First, we derive an exact relation between the electromagnetic
response of the non-relativistic and Dirac electrons in the lowest Landau
level. Next, we obtain a closed form expression for the polarization operator
in the large $N$ (or weak magnetic field) limit. We explicitly show that in the
large $N$ limit the random phase approximation (RPA) computation of the
polarization tensor agrees - in leading and sub-leading order in $N$ - with a
Fermi liquid computation to {\it all} orders in the gradient expansion and for
arbitrary value of the $\mathrm{g}$-factor. Finally, we show that in the large
$N$ limit the non-relativistic polarization tensor agrees with Dirac's in the
leading and sub-leading orders in $N$, provided that Berry phase of the Dirac
cone is taken into account via replacement $N\longrightarrow N+\frac{1}{2}$.
