At rst, it was not clear how to model the atom, nor even what it even contained. One of the simpler ideas posed by Thomson soon after he discovered the electron was that an atom only consisted of positively charged stuff and negatively charged electrons. Thomson's model of the atom consisted of a sphere of uniform positive charge with constant charge density throughout the volume. Electrons were embedded in the sphere, much like chocolate chips in a cookie (my favorite), except that the electrons were free to move within the positive stuff (unlike chocolate chips in the cookie). If you learned in high school that the modern view of the atom is that electrons orbit about a dense nucleus of positive and neutral charged particles (protons and neutrons), then Thomson's model sounds ridiculous. Why did physicists take it so seriously? For one intriguing reason. It explained the frequency of light that could be emitted by atoms (kinda). In particular, it explained why visible light was emitted by excited hydrogen atoms. Suppose we only worry about hydrogen, which has 1 electron with mass m and charge -e, and we assume it is much smaller than the atom, so effectively a point charge. It is embedded in a uniform spherical charge +Q and radius L. a) Use Gauss's Law to nd the electric eld as a function of the x from the center of the sphere. That is nd the electric eld vector along the x axis. b) Find the force vector on an electron that is displaced from the center along the x axis by a distance x. c) Write Newton's law to get the acceleration ofthe electron, a=d2xldt1. d) What value of x is equilibrium? e) Is the equilibrium stable? Explain how you know. f) Recall from the homework last week that if you displace the electron from equilibrium by some amount x (which is what you did in part b), and Newtons Law gives an equation of the form de/dt2= -Cx where C is a bunch of constants (well, last week the variable was epsilon, e, and the equation was dze/dt2= -Ce, but it had the same form), then the motion of the object is simple harmonic with a frequency f given by f=[squareroot(C)/(21c)] = CUE/(211). Find the frequency f in terms of m, L, 0. an e. [Aside: Simple harmonic motion means that the position as a function of time is described by a sine or cosine function, for example x(t) = Asin(21tft)] g) Thompson assumed, with good reason, that the oscillation frequency of the electron would be the same frequency of the emitted light. He knew about Maxwell's equations, and those equations did predict that a charged particle in simple harmonic motion would emit electromagnetic radiation with the same frequency as the oscillation frequency. We will touch this idea in the last week of this class. So suppose the radius of the atom was roughly known as 3x10'10m, and the positive charge (1 is the same magnitude as the negative, so Q=+e = +1.6x10'19 C (where C is Coulombs, the Si unit for the electron charge), then what frequency of light is emitted? Is it visible and if so, what color (google it)? Hint: Look up the mass ofthe electron in your book (or google it), and be sure to use SI units for all values