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\sect{Effusion of electrons from a box in magnetic field}

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

(a) Find the ratio ${\alpha(B;\mu) =  (I_{\uparrow}-I_{\downarrow}) / I}$.

\Dn

(b) Find a linear approximation for $\alpha(B;\mu)$
regarded as a function of the magnetic field.

\Dn

(c) What is the maximal value of $\alpha(B;\mu)/B$,
and what is the range for which the result is valid.

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