Two identical capacitors are connected in parallel and each acquires a charge Q₀ when connected to a source of voltage V₀. The voltage source is disconnected and then a dielectric (K = 3.6) is inserted to fill the space between the plates of one of the capacitors. Determine the voltage now across each capacitor.

A general rule for estimating the capacitance C of an isolated conducting sphere with radius r is C (in pF) ≈ r (in cm). That is, the numerical value of C in pF is about the same as the numerical value of the sphere’s radius in cm. Justify this rule.

Capacitors can be used as “electric charge counters.” Consider an initially uncharged capacitor of capacitance C with its bottom plate grounded and its top plate connected to a source of electrons. If N electrons flow onto the capacitor’s top plate, show that the resulting potential difference V across the capacitor is directly proportional to N.

Capacitors can be used as “electric charge counters.” Consider an initially uncharged capacitor of capacitance C with its bottom plate grounded and its top plate connected to a source of electrons. Assume a voltage-measuring device can accurately resolve voltage changes of about 1 mV. What value of C would be necessary to resolve the arrival of an individual electron?

Suppose it takes 75 kW of power for your car to travel at a constant speed on the highway. If this capacitor were to be made from activated carbon (Section 24–2), the voltage would be limited to no more than 10 V. In this case, how many grams of activated carbon would be required?

A large metal sheet of thickness ℓ is placed between, and parallel to, the plates of the parallel-plate capacitor of Fig. 24–4. It does not touch the plates, and extends beyond their edges. What is now the net capacitance in terms of A, d, and ℓ?

A large metal sheet of thickness ℓ is placed between, and parallel to, the plates of the parallel-plate capacitor of Fig. 24–4. It does not touch the plates, and extends beyond their edges. If ℓ = 0.40 d, by what factor does the capacitance change when the sheet is inserted?

Determine the capacitance of the Earth, assuming it to be a spherical conductor.