# 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