Thomson’s Model of the Atom, Continued. Using Thomson’s (outdated) model of the atom...

Thomson’s Model of the Atom, Continued. Using Thomson’s (outdated) model of the atom described in Problem 22.50, consider an atom consisting of two electrons, each of charge -e, embedded in a sphere of charge +2e and radius R. In equilibrium, each electron is a distance d from the center of the atom (Fig. P22.51). Find the distance d in terms of the other properties of the atom.

22.50 .. CP Thomson’s Model of the Atom. Early in the 20th century, a leading model of the structure of the atom was that of English physicist J. J. Thomson (the discoverer of the electron). In Thomson’s model, an atom consisted of a sphere of positively charged material in which were embedded negatively charged electrons, like chocolate chips in a ball of cookie dough. Consider such an atom consisting of one electron with mass m and charge -e, which may be regarded as a point charge, and a uniformly charged sphere of charge + e and radius R. (a) Explain why the electron’s equilibrium position is at the center of the nucleus. (b) In Thomson’s model, it was assumed that the positive material provided little or no resistance to the electron’s motion. If the electron is displaced from equilibrium by a distance less than R, show that the resulting motion of the electron will be simple harmonic, and calculate the frequency of oscillation. (Hint: Review the definition of SHM in Section 14.2. If it can be shown that the net force on the electron is of this form, then it follows that the motion is simple harmonic. Conversely, if the net force on the electron does not follow this form, the motion is not simple harmonic.) (c) By Thomson’s time, it was known that excited atoms emit light waves of only certain frequencies. In his model, the frequency of emitted light is the same as the oscillation frequency of the electron(s) in the atom. What radius would a Thomson-model atom need for it to produce red light of frequency 4.57 X 10¹⁴ Hz? Compare your answer to the radii of real atoms, which are of the order of 10^-10 m (see Appendix F). (d) If the electron were displaced from equilibrium by a distance greater than R, would the electron oscillate? Would its motion be simple harmonic? Explain your reasoning. (Historical note: In 1910, the atomic nucleus was discovered, proving the Thomson model to be incorrect. An atom’s positive charge is not spread over its volume, as Thomson supposed, but is concentrated in the tiny nucleus of radius 10^-14 to 10^-15 m.)