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.

The electrical potential energy of a charge q located at a point at potential V is given by

Therefore, if the charge must move between two points at potential V1 and V2, the difference in potential energy of the charge will be


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 

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: