Question

prove kinetic energy = eV where v is stopping potential​

Answer 1

We know that electric potential difference between two points is V

also V is the work done per unit charge to move a charged object from one point to other .

so, for an electron of charge e , the work done is eV .

The KE of an object is the work that it can do in coming to rest . so if a pd V just stop an electron of KE equal to E

then, E = qV

Answer 2

Answer:

The proper way to treat this is to say: the kinetic energy of the emitted electron is hν−eϕemitterhν−eϕemitter, as the electron flies towards the collector it sees an uphill potential and so decreases its energy by e[Vstop+(ϕcollector−ϕemitter)]e[Vstop+(ϕcollector−ϕemitter)]. Now, by definition of the stopping potential, hν−eϕemitter−e[Vstop+(ϕcollector−ϕemitter)]=0hν−eϕemitter−e[Vstop+(ϕcollector−ϕemitter)]=0 (i.e. electrons arrive with zero K.E.). So we see the emitter's work function cancel out and rearranging we get Einstein's equation with the collector's work function instead!

The energy diagram you are drawing plots the total energy of the electron (K.E. + P.E.). So if you think about it, energy conservation is the same as saying that the level of hνhν matches that of e(Vstop+ϕcollector)e(Vstop+ϕcollector). The emitter's work function simply does not play a role in determining the stopping potential!

There are some subtleties: we assumed that all the electrons emitted were near the Fermi level, but this is of course not a bad approximation at room temperatures, so it's fine. Now here's the thing, even in theory there are unequal numbers of electrons sitting at a given energy inside the collector (the density of states for a free particle scales as ϵ√ϵ in 3D), so experimentally you will find that the I-V curve does not have a sharp stopping potential, but has a so called 'Fermi tail' tending to zero current. This complaint of Einstein's theory is well documented, with articles dating back to the 1920s (Millikan).

So while Einstein was technically wrong in this aspect of the photoelectric effect, he did get the photon bit right which won him the Nobel!

Hope that clears things up!

P.S. For future generations: if you are from a certain London college, your professor is indeed correct!