Derive Equation 26.33 for the induced surface charge density on the dielectric in a capacitor.

Polystyrene has dielectric constant $2.6$ and dielectric strength $2.0\times {10}^7$ V/m. A piece of polystyrene is used as a dielectric in a parallel-plate capacitor, filling the volume between the plates.

(a) When the electric field between the plates is $80$% of the dielectric strength, what is the energy density of the stored?

(b) When the capacitor is connected to a battery with voltage $500.0$ V, the electric field between the plates is $80\%$ of the dielectric strength. What is the area of each plate if the capacitor stores $0.200$ mJ of energy under these conditions?

A constant potential difference of $12$ V is maintained between the terminals of a $0.25$-$\mu$F, parallel-plate, air capacitor.

(a) A sheet of Mylar is inserted between the plates of the capacitor, completely filling the space between the plates. When this is done, how much additional charge flows onto the positive plate of the capacitor (see Table $24.1$)?

(b) What is the total induced charge on either face of the Mylar sheet?

(c) What effect does the Mylar sheet have on the electric field between the plates? Explain how you can reconcile this with the increase in charge on the plates, which acts to increase the electric field.

A parallel-plate capacitor has capacitance $C_0=8.00$ pF when there is air between the plates. The separation between the plates is $1.50$ mm.

(a) What is the maximum magnitude of charge $Q$ that can be placed on each plate if the electric field in the region between the plates is not to exceed $3.00\times {10}^4$ V/m?

(b) A dielectric with $K=2.70$ is inserted between the plates of the capacitor, completely filling the volume between the plates. Now what is the maximum magnitude of charge on each plate if the electric field between the plates is not to exceed $3.00\times {10}^4$ V/m?

A vacuum-insulated parallel-plate capacitor with plate separation d has capacitance C₀. What is the capacitance if an insulator with dielectric constant κ and thickness d/2 is slipped between the electrodes without changing the plate separation?

50 pJ of energy is stored in a 2.0 cm×2.0 cm×2.0 cm region of uniform electric field. What is the electric field strength?

A parallel-plate capacitor with plate area A = 2.0 m² and plate separation d = 3.0 mm is connected to a 45-V battery (Fig. 24–39a). With the capacitor still connected to the battery, a slab of plastic with dielectric constant K = 3.2 is placed between the plates of the capacitor, so that the gap is completely filled with the dielectric. What are the new values of charge, electric field, capacitance, and the energy U stored in the capacitor?