Chapter 25, Problem 50CP

This problem is a continuation of Problem 45. You are part of a team working in a machine parts mechanic’s shop. An important customer has asked your company to provide springs with a very precise force constant k. You devise the electrical circuit shown in Figure P25.45 to measure the spring constant of each of the springs to be provided to the customer.

The circuit consists of two identical, parallel metal plates connected to identical metal springs, a switch, and a battery with emf ΔV. With the switch open, the plates are uncharged, are separated by a distance d, and have a capacitance C.

To provide a comparison value for the spring constant that you found in Problem 45, you slide a slab of material with dielectric constant κ and thickness t between the plates, so that it is in contact with one of the plates as shown in Figure P25.50. When the switch is closed, the plates become charged and attract each other. The distance between the plates changes by a factor f, after which the plates are in equilibrium between the spring forces and the attractive electric force between the plates. To keep the plates from going into oscillations, you hold each plate with insulating gloves as the switch is closed and apply a force on the plates that allows them to move together at a slow constant speed until they are at the equilibrium separation, at which point you can release the plates. Find an expression for the spring constant in terms of C, d, ΔV, k, t, and f.

Figure P25.50