Answer: High-speed electrons are used to probe the

Chapter 39, Problem 39.69

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High-speed electrons are used to probe the interior structure of the atomic nucleus. For such electrons the expression l = h>p still holds, but we must use the relativistic expression for momentum, p = mv>21 - v2>c2 . (a) Show that the speed of an electron that has de Broglie wavelength l is v = c 21 + 1mcl>h22 (b) The quantity h>mc equals 2.426 * 10-12 m. (As we saw in Section 38.3, this same quantity appears in Eq. (38.7), the expression for Compton scattering of photons by electrons.) If l 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 = 11 - 2c and express the speed of the electron in terms of rather than v. Find an expression for valid when l V h>mc. [Hint: Use the binomial expansion 11 + z2n = 1 + nz + 3n1n - 12z 2>24 + g, valid for the case 0z0 6 1.4 (c) How fast must an electron move for its de Broglie wavelength to be 1.00 * 10-15 m, comparable to the size of a proton? Express your answer in the form v = 11 - 2c, and state the value of .