Electrons on potential barrier
There is a beam of protons of $5.0 eV$ on a barrier of potential energy of $6.0 eV$ height and $0.62 nm$ thickness. The equivalent current of the beam is $1175 A$.
(a) If one proton is to be transmitted how long would you have to wait-o?
(b) The same question if the beam consisted of electrons instead of protons?
Transmission coefficient through the barrier is
$T = 16(E/U0) *(1-E/U0)*exp(-2L*\sqrt{m/\hbar^2*(U0-E))}$
(here $\hbar$ stands for $h/(2*\pi$))
$L = 0.62 n m$, $E =5 e V$, $U =6 e V$, $m =1.67*10^{-27} kg$ for protons
$T =1.98*10^{-118}$
This means that one in $1/T =5*10^{117}$ protons pass through barrier
$I = d Q/d t = Ne/t$
it means each second the number of incident protons is
$N = I/e =1175/1.6*10^{-19} =7.34*10^{21} protons/second$
It means the average waiting time for one proton is
$time = 1/NT =(5*10^{117})/7.34*10^{21} =6.8*10^{95} seconds$ !!!!
For electrons $m=9.1*10^{-31}$
$T =2.04*10^{-5}$ (this is a bigger transmission coefficient)
$N = I/e =7.34*10^{21}$ incident electrons/second
$time = 1/NT = 6.64*10^{-18}$ seconds