Calculus Based Physics 3 Chapter 41 Question 5

Leaming Goal: To understand and be able to perform simple calculations relating to the Zeeman effect. The Zeeman effect is a change in the energy levels of electrons in an atom in the presence of a magnetic field. It can be observed as the spliting of spectral lines when the atom is in a strong magnetic field. To understand this effect, you need to recall a few facts about magnetic moments. When a loop of wire with a current running through it is placed in a magnetic field, it acquires a potential energy I/ = pi - B, where B is the magnetic field and g is the magnetic moment of the loop. The magnetic moment is a vector perpendicular to the plane of the loop of wire, with its direction determined by applying the right-hand rule to the direction of current flow. The magnitude of g is given by || = I'A, where I is the current in the loop and A is the area enclosed by the loop. In the Bohr model of the atom, electrons follow circular orbits around the nucleus. Such an orbit would constitute a loop of current, since the electron has charge and moves repeatedly through the same circular path. While the Bohr model is fundamentally incorrect, its conclusions about electrons interacting with magnetic fields can be used as a basis for correctly understanding the Zeeman effect. Consider an electron moving in a circular orbit of radius r at a speed v. The orbit will serve as the loop of current as you calculate the magnetic moment. Part A What is the area A enclosed by the electron's orbit? Express your answer in terms of r. A = 172 Submit Previous Answers Correct PartB v What is the current I due to the orbit of the electron? Recall that current is defined as the amount of charge passing a point in one unit of time. Use e for the charge on an electron. Express your answer in terms of r, v, and e. P View Available Hint(s) Previous Answers +" Correct Part C What is the magnitude / = pi of the magnetic moment for the orbiting electron? Express your answer in terms of e, v, and r. CUT 2 Submit Previous Answers V Correct* PartD v In the Schradinger picture of the hydrogen atom, the electrons are waves with associated probability distributions. Thus, v and r are not well-defined quantities. However, the angular momentum is a well-defined quantity. Use the classical definition of angular momentum L = mur, where m is the mass of the electron, to find an expression for the magnitude u of the magnetic moment of the electron. Express your answer in terms of L, e, and m. Previous Answers v Cormrect * PartE Consider a magnetic field of magnitude B oriented along the z axis. The energy associated with the interaction of this field with an orbiting electron with magnetic moment g is I/ = pi - B. Since you know that the magnetic field is oriented along the z axis, this inner product will be equal to pu. B, where . is the z component of the magnetic moment. Since the only part of the expression for the electron's magnetic momentum that has direction is the angular momentum L, we conclude that p: = eL;/(2m). You know that in the Schrodinger picture of the hydrogen atom, the angular momentum component in a specific direction (here we are concemed with the z direction) is equal to Ay, where ymy is the magnetic quantum number of the electron's state. \\What is the magnitude of the magnetic potential energy U7 for an electron with magnetic quantum number 1 in a magnetic field of strength B? Express your answer in terms of my, e, B, h, and m. \"# 0 m Previous Answers Request