A capacitor is constructed from two square conducting plates of side length ( = 4.0
PHYSICS · Multi-part · A Level · Physics · Fields and their consequences · Fields and their consequences
Problem
A capacitor is constructed from two square conducting plates of side length ℓ=4.0cm. The lower plate is rigidly mounted, while the upper plate is attached to vertical springs (total effective spring constant ks=50N⋅m−1). With only air between the plates, the springs hold the upper plate at an initial equilibrium separation of d0=1.0mm above the lower plate when the capacitor is connected to a 20 V battery.
A mica sheet of thickness 0.30 mm (dielectric constant 6.0) is then carefully inserted so that it completely fills the area between the plates. The sheet compresses the springs until a new static equilibrium separation d is reached (d>0.30mm). Assume the battery remains connected at all times and that the mica sheet does not touch either plate.
(a) Express the new capacitance of the system in terms of the unknown plate separation d, clearly stating all assumptions used.
(b) Using energy considerations that include both the elastic potential energy of the springs and the electrical energy stored in the capacitor, derive an equation that relates d to d0, the dielectric constant, and the given physical parameters. Neglect the gravitational weight of the upper plate.
(c) Solve the equation from part (b) to determine the numerical value of the new separation d. Then calculate the percent change in capacitance relative to its original value before the mica was inserted.