Question

If there is a potential difference v between the metal and the detector, what is the minimum energy emin that an electron must have so that it will reach the detector? express your answer in terms of v and the magnitude of the charge on the electron,
e.

Answer 1

The electrical potential energy of a charge q located at a point at potential V is given by
$U=qV$
Therefore, if the charge must move between two points at potential V1 and V2, the difference in potential energy of the charge will be
$\Delta U = q (V_2 -V_1)=q \Delta V$

In our problem, the electron (charge e) must travel across a potential difference V. So the energy it will lose traveling from the metal to the detector will be equal to 
$\Delta U = e V$
Therefore, if we want the electron to reach the detector, the minimum energy the electron must have is exactly equal to the energy it loses moving from the metal to the detector:
$E_{min} = \Delta U = eV$