Incoming electrons

Choose coordinates $x$ and $y$ on a horizontal 2-d graphene sheet. We will inject electrons into the sheet from the origin $O$ such that they enter the upper half-plane ($y>0$) but the direction is otherwise totally random. The speed of electrons in graphene is $v_F$, and the mass of each electron is $m$.

We now put an electron detector along the $x$-axis ($y=0$) and apply an external magnetic field $B_0$ perpendicular to the plane (parallel with the $z$-axis). At what value of $x$ in meters will the detector receive the greatest signal?

Assume that every electron that goes to the lower half plane ($y<0$) will disappear (for example, the lower half plane is made of a special material which absorbs electrons).

Details and assumptions

To make the problem numerically simpler, use the following values for your calculation:

• $v_F=1~\mbox{m/s}$

• $m=1~\mbox{kg}$

• $B_0=1~\mbox{T}$

• The electron charge is $e=1~\mbox{C}$.

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