Suppose a beam of 5.0 eV protons strikes a potential energy barrier of height 5.

Question

Suppose a beam of 5.0 eV protons strikes a potential energy barrier of height 5.5 eV and thickness 0.59 nm, at a rate equivalent to a current of 1005 A.
(a) How long would you have to wait-on average-for one proton to be transmitted?
1 . s

(b) How long would you have to wait if the beam consisted of electrons rather than protons?
2 s

Explanation / Answer

(a) Transmission Coefficient , (T=|t|^2= rac{1}{1+rac{V_0^2sinh^2(k_1 a)}{4E(V_0-E)}}) where, (k_1=sqrt{2m (V_0-E)/hbar^{2}}) by using the values, V0 = 5.5 eV; E = 5 eV; m = 1.67 x 10-27 kg; e = 1.6 x 10-19 C ==> T = 1/668633534.088 = Probability of Proton passing through ==> We require about N_req = 668633534 protons to get one proton to pass through the barrier. From the current, I = 1005 A we get the number of protons coming n = 628.125 x 10+19 per sec So the time req., t = N_req / n = 0.1064491 ps (b) For electron, T =1/1.1779871957923720756658442586451 N_req = 1.1779871957923720756658442586451 (pprox) 2 t = N_req / n = 0.31840796 x 10-20 sec

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