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\heading{E6040: Electron gas with magnetic field}
\auname{Yaron de Leeuw}
{\bf The problem:}
\Dn
A box with electrons of mass $m$ is subjected to a magnetic field $B$.
The single particle interaction is described by $-\gamma B \sigma_z$.
The chemical potential of the electrons inside the box is $\mu$.
A hole through one of the walls is drilled.
The electrons that are emitted from the hole with
a velocity in the range ${v < v' < v+dv}$ are filtered,
and subsequently their spin is measured.
The measured current is defined as ${I=I_{\uparrow}+I_{\downarrow}}$.
\Dn\Dn
(a) Find the ratio
${\alpha(B;\mu) = (I_{\uparrow}-I_{\downarrow}) / I}$.
\Dn\Dn
(b) Find a linear approximation for $\alpha(B;\mu)$
regarded as a function of the magnetic field.
\Dn\Dn
(c) What is the maximal value of $\alpha(B;\mu)/B$,
and what is the range for which the result is valid.
\Dn\Dn
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{\bf The solution:}
\Dn
The flux for $N$ electrons in volume $V$ with velocity $v$ is :
\[ J = \iint_{|\theta|<\frac{\pi}{2}} \frac{d\Omega}{4\pi}\frac{N}{V}v\cos\alpha = \frac{1}{4}\left(\frac{N}{V}v\right)\]
The number of spin up electrons in the velocity range $v