A model of the magnetic properties of materials is based upon small magnetic moments generated by each atom in the material. One source of this magnetic moment is the magnetic field generated by the electron in its orbit around the nucleus. For simplicity, we will assume that each atom consists of a single electron of charge -e and mass m_{e}, a single proton of charge +e and mass m_{p} \gg m_{e}, and that the electron orbits in a circular orbit of radius R about the proton.
a. Magnetic Moments.
Assume that the electron orbits in the x - y plane.
(3 pts) i. Calculate the net electrostatic force on the electron from the proton. Express your answer in terms of any or all of the following parameters: e, m_{e}, m_{p}, R, and the permittivity of free space, \varepsilon_{0}, where
(k is the Coulomb’s Law constant).
(5 pts) ii. Determine the angular velocity \omega_{0} of the electron around the proton in terms of any or all of the following parameters: e, m_{e}, R, and \varepsilon_{0}.
(8 pts) iii. Derive an expression for the magnitude of the magnetic field B_{e} due to the orbital motion of the electron at a distance z \gg R from the x-y plane along the axis of orbital rotation of the electron. Express your answer in terms of any or all of the following parameters: e, m_{e}, R, \omega_{0}, z, and the permeability of free space \mu_{0}.
(4 pts) iv. A small bar magnet has a magnetic field far from the magnet given by
where z is the distance from the magnet on the axis connecting the north and south poles, m is the magnetic dipole moment, and \mu_{0} is the permeability of free space. Assuming that an electron orbiting a proton acts like a small bar magnet, find the dipole moment m for an electron orbiting an atom in terms of any or all of the following parameters: e, m_{e}, R, and \omega_{0}.
b. Diamagnetism.
We model a diamagnetic substance to have all atoms oriented so that the electron orbits are in the x-y plane, exactly half of which are clockwise and half counterclockwise when viewed from the positive z axis looking toward the origin. Some substances are predominantly diamagnetic.
(3 pts) i. Calculate the total magnetic moment of a diamagnetic substance with N atoms. Write your answer in terms of any or all of the following parameters: e, m_{e}, R, N, and \mu_{0}.
(6 pts) ii. An external magnetic field \vec{B}_{0}=B_{0} \hat{z} is applied to the substance. Assume that the introduction of the external field doesn’t change the fact that the electron moves in a circular orbit of radius R. Determine \Delta \omega, the change in angular velocity of the electron, for both the clockwise and counterclockwise orbits. Throughout this entire problem you can assume that \Delta \omega \ll \omega_{0}. Write your answer in terms of e, m_{e}, and B_{0} only.
(6 pts) iii. Assume that the external field is turned on at a constant rate in a time interval \Delta t. That is to say, when t=0 the external field is zero and when t=\Delta t the external field is \vec{B}_{0}. Determine the induced emf \mathcal{E} experienced by the electron. Write your answer in terms of any or all of the following parameters: e, m_{e}, R, N, B_{0}, \omega_{0}, and \mu_{0}.
(6 pts) iv. Verify that the change in the kinetic energy of the electron satisfies \Delta K=e \mathcal{E}. This justifies our assumption in (ii) that R does not change.
(6 pts) v. Determine the change in the total magnetic moment \Delta m for the N atoms when the external field is applied, writing your answer in terms of e, m_{e}, R, N, \mu_{0} and B_{0}.
(3 pts) vi. Suppose that the uniform magnetic field used in the previous parts of this problem is replaced with a bar magnet. Would the diamagnetic substance be attracted or repelled by the bar magnet? How does your answer show this?