Verify that in the solution to part \(a\) of Example 26.6,

(a) the ratio of units \(\mathrm{C}^{2} /(\mathrm{N} \cdot \mathrm{m})\) is equivalent to the unit \(\mathrm{F}\) and \((b)\) the product of units \(\mathrm{F} \cdot \mathrm{V}^{2}\) is equivalent to the unit \(\mathrm{J}\).

Data from Example 26.6

A parallel-plate capacitor consists of two conducting plates with a surface area of \(1.0 \mathrm{~m}^{2}\) and a plate separation distance of \(50 \mu \mathrm{m}\). (a) Determine the capacitance and the energy stored in the capacitor when it is charged by connecting it to a \(9.0-\mathrm{V}\) battery. (b) With the capacitor fully charged and disconnected from the battery, a \(50-\mu \mathrm{m}\)-thick sheet of Mylar is inserted between the plates. Determine the potential difference across the capacitor and the energy stored in it. (c) If the Mylar-filled capacitor is connected to the battery, how much work does the battery do to fully charge the capacitor?