High-speed electrons are used to probe the interior structure of the atomic nucleus. For such electrons the expression λ = h/p still holds, but we must use the relativistic expression for momentum, p = mv/√1 – v2/c2. (a) Show that the speed of an electron that has de Broglie wavelength λ is (b) The quantity h/mc equals 2.426 x 10-12 m. (As we saw in Section 38.7, this same quantity appears in Eq. (38.23), the expression for Compton scattering of photons by electrons.) If A is small compared to h/mc, the denominator in the expression found in part
(a) is close to unity and the speed v is very close to c. In this case it is convenient to write v = (1 - ∆)c and express the speed of the electron in terms of ∆ rather than v. Find an expression for Il valid when A « h/me.
(c) How fast must an electron move for its de Broglie wavelength to be 1.00 x 10-15 m, comparable to the size of a proton? Express your answer in the form v = (I - ∆)c, and state the value of ∆.