Incoming electrons

Classical Mechanics Level 4

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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