Several years from now you have graduated with an engineering/physics degree and have been hired by a nanoengineering firm as an intern. You have been assigned to work under a top engineer from the company. Their current project is to design a microscopic oscillator as a time keeping device. The engineering design involves placing a negative charge at the center of a very small positively charged metal ring. Your boss claims that the negative charge will undergo simple harmonic motion if displaced away from the center of the ring. Furthermore, they claim they can change the period (timing) of oscillation by adjusting the amount of charge on the ring. The first task they give you is to check the validity of their design.

a. Consider a charge − located a small distance z above the center of a positively charged ring with total charge +Q and radius R. Write an expression for the net force exerted on the charge − due to the ring of charge. What is the magnitude of the force on the charge − if it is at the location z = 0?

b. Use the binomial expansion as a mathematical tool to understand the motion of the charged particle under the assumption that it is only displaced a small distance z ≪ away from the center of the ring. Using the binomial approximation, simplify your expression from part (a). You should have two terms in your expression for the net force.

c. Suppose that z =1.0 nanometer and R = 1.0 millimeter. Using these values, determine the relative strength of the force that each term in your expression contributes to the total force (calculate the ratio of z/R that appears in each term). Is one term negligible (i. e. is the value of one term very close to zero)?

d. Using results from part (c) show that the relevant force on charge − simplifies to a Hook’s law restoring force: = −x, where k is now a more complicated constant made up of several constants from this problem. What is k?

e. Using Newton’s 2nd Law, write down the differential equation of motion for the particle as it undergoes oscillations due to this Hook’s law force. Determine the period T of oscillations charge − makes about the equilibrium point z = 0 (this should not involve any calculations, see equations 15.32 and 15.37 from your book). Asses the validity of your solution by analyzing the physical units to make sure that your solution has units of time.