Chapter 25: Capacitance

Initially, a single capacitance${\mathit{C}}_{\mathbf{1}}$is wired to a battery. Then capacitance${\mathit{C}}_{\mathbf{2}}$is added in parallel. Are (a) the potential difference across${\mathit{C}}_{\mathbf{1}}$and (b) the charge${\mathit{q}}_{\mathbf{1}}$on${\mathit{C}}_{\mathbf{1}}$now more than, less than, or the same as previously? (c) Is the equivalent capacitance${\mathit{C}}_{\mathbf{12}}$of${\mathit{C}}_{\mathbf{1}}$and${\mathit{C}}_{\mathbf{2}}$more than, less than, or equal to${\mathit{C}}_{\mathbf{1}}$? (d) Is the charge stored on${\mathit{C}}_{\mathbf{1}}$and C2 together more than, less than, or equal to the charge stored previously on${\mathit{C}}_{\mathbf{1}}$?

The chocolate crumb mystery.Explosions ignited by electrostatic discharges (sparks) constitute a serious danger in facilities handling grain or powder. Such an explosion occurred in chocolate crumb powder at a biscuit factory in the 1970s. Workers usually emptied newly delivered sacks of the powder into a loading bin, from which it was blown through electrically grounded plastic pipes to a silo for storage. As part of the investigation of the biscuit factory explosion, the electric potentials of the workers were measured as they emptied sacks of chocolate crumb powder into the loading bin, stirring up a cloud of the powder around themselves. Each worker had an electric potential of about 7.0kVrelative to the ground, which was taken as zero potential.(a)Assuming that each worker was effectively a capacitor with a typical capacitance of 200pF, find the energy stored in that effective capacitor. If a single spark between the worker and any conducting object connected to the ground neutralized the worker, that energy would be transferred to the spark. According to measurements, a spark that could ignite a cloud of chocolate crumb powder, and thus set off an explosion, had to have energy of at least150mJ. (b)Could a spark from a worker have set off an explosion in the cloud of powder in the loading bin?

Figure 25-54 shows capacitor 1 $(C_1=8.00\mathrm{\mu F})$, capacitor 2 $\mathbf{(}{\mathbf{C}}_{\mathbf{2}}\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{\mu F}\mathbf{)}$, and capacitor 3$\mathbf{(}{\mathbf{C}}_3\mathbf{=}\mathbf{8}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{\mu F}\mathbf{)}$ connected to a 12.0 V battery. When switch S is closed so as to connect uncharged capacitor 4 $\mathbf{(}{\mathbf{C}}_4\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{\mu F}\mathbf{)}$, (a) how much charge passes through point Pfrom the battery and (b) how much charge shows up on capacitor 4? (c) Explain the discrepancy in those two results.

Two air-filled, parallel-plate capacitors are to be connected to a 10 V battery, first individually, then in series, and then in parallel. Inthose arrangements, the energy stored in the capacitors turns out tobe, listed least to greatest: $\mathbf{75}\mathbf{}\mathbf{\mu J}\mathbf{}$,$\mathbf{100}\mathbf{}\mathbf{\mu J}$ ,$\mathbf{300}\mathbf{}\mathbf{\mu J}$ , and$\mathbf{400}\mathbf{}\mathbf{\mu J}$ . Of the two capacitors, what is the (a) smaller and (b) greater capacitance?

Two parallel-plate capacitors, $\mathbf{6}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{\mu F}$each, are connected in series to a 10 Vbattery. One of the capacitors is then squeezed so that its plate separation is halved. Because of the squeezing, (a) how much additional charge is transferred to the capacitors by the battery and (b) what is the increase in the total charge stored on the capacitors (the charge on the positive plate of one capacitor plus the charge on the positive plate of the other capacitor)?

In Fig. 25-55,$\mathit{V}\mathbf{=}\mathbf{12}\mathit{V}\mathbf{,}\mathbf{}{\mathit{C}}_{\mathbf{1}}\mathbf{=}{\mathit{C}}_{\mathbf{5}}\mathbf{=}{\mathit{C}}_{\mathbf{6}}\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{\mu F}\mathbf{}\mathbf{and}\mathbf{}{\mathit{C}}_{\mathbf{2}}\mathbf{=}{\mathit{C}}_{\mathbf{3}}\mathbf{=}{\mathit{C}}_{\mathbf{4}}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{\mu F}$. What are (a) the net charge stored on the capacitors and (b) the charge on capacitor 4?

In Fig. 25-56, the parallel-plate capacitor of plate area $\mathbf{2}\mathbf{.}\mathbf{00}\mathit{x}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}\mathbf{}{\mathbf{m}}^{\mathbf{2}}$is filled with two dielectric slabs, each with thickness. One slab has dielectric constant 3.00, and the other, 4.00. How much charge does the 7.00 Vbattery store on the capacitor?

A cylindrical capacitor has radii aand bas in Fig. 25-6. Show that half thestored electric potential energy lieswithin a cylinder whose radius is$\mathit{R}\mathbf{=}\sqrt{\mathbf{a}\mathbf{b}}$.

A capacitor of capacitance ${\mathbf{C}}_{\mathbf{1}}\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{\mu F}$is connected in series with a capacitor of capacitance${\mathbf{C}}_2\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{\mu F}$, and a potential difference of 200 Vis applied across the pair. (a) Calculate the equivalent capacitance. What are (b) charge${\mathit{q}}_{\mathbf{1}}$and (c) potential difference${\mathit{V}}_{\mathbf{1}}$ on capacitor 1 and (d)${\mathit{q}}_{\mathbf{2}}$and (e)${\mathit{V}}_{\mathbf{2}}$on capacitor 2?

Repeat Problem 67 for the same two capacitors but with them now connected in parallel.