Energy dispersive spectroscopy (EDS) is a technique for identifying and quantifying the elemental composition of a sample. Volumes as small as a few cubic micrometers can be probed using EDS. The characteristic X-rays of the elements in a sample are produced when the sample is bombarded with electrons in an electron beam instrument such as a scanning electron microscope. For elements heavier than hydrogen, we can approximate the energy of an electron in the n = 1 state (called a K-shell electron) using 

with n = 1. Here Z is the atomic number of the element, and the (Z = 1) term corrects for the partial cancellation of the charge the electron sees as a result of the other K-shell electrons. Suppose we want to detect the presence of aluminum, copper, and tungsten in a sample. 

(a) Estimate the minimum potential difference through which the bombarding electron beam must be accelerated to produce the characteristic X-rays of each of these elements. 

(b) From the data provided in Table 29.1, estimate the energy of an electron in the n = 2 state for each of these elements.