  In this work we study the differential aspects of the noncommutative geometry
for the magnetic $C^*$-algebra which is a 2-cocycle deformation of the group
$C^*$-algebra of $\mathbb{R}^2$. This algebra is intimately related to the
study of the Quantum Hall Effect in the continuous, and our results aim to
provide a new geometric interpretation of the related Kubo's formula. Taking
inspiration from the ideas developed by Bellissard during the 80's, we build an
appropriate Fredholm module for the magnetic $C^*$-algebra based on the
magnetic Dirac operator which is the square root (\`a la Dirac) of the quantum
harmonic oscillator. Our main result consist of establishing an important piece
of Bellissard's theory, the so-called second Connes' formula. In order to do
so, we establish the equality of three cyclic 2-cocycles defined on a dense
subalgebra of the magnetic $C^*$-algebra. Two of these 2-cocycles are new in
the literature and are defined by Connes' quantized differential calculus, with
the use of the Dixmier trace and the magnetic Dirac operator.
