In the figure below, $20.0 \mathrm{V}$ battery is connected across capacitors of capacitance $C_1=C_6=6.00 \mathrm{\mu F}$ and $C_3=C_5=4.00 \mathrm{\mu F}, C_2=C_4=2.00 \mathrm{\mu F}$. What are (a) the equivalent capacitance $C_{eq}$ of the capacitors and (b) the charge stored by $C_{\mathrm{eq}}?$ What are (c)$V_1$ and (d)  $q_1$ of capacitor $1,$ (e)$V_2$  and (f) $q_2$ of capacitor $2$  and (g) $V_3$ and (h) $q_3$ of capacitor $3$?

Figure represents two air-filled cylindrical capacitors connected in series across a battery with potential $V=10 \mathrm{V}.$ Capacitor $1$ has an inner plate radius of $3.00 \mathrm{mm},$ an outer plate radius of $1.5 \mathrm{cm},$ and a length of $5.0 \mathrm{cm}$. Capacitor $2$ has an inner plate radius of $2.5 \mathrm{mm},$ an outer plate radius of $1.0 \mathrm{cm},$  and a length of $9.0 \mathrm{cm}.$ The outer plate of capacitor $2$ is a conducting organic membrane that can be stretched, and the capacitor can be inflated to increase the plate separation. If the outer plate radius is increased to $2.5 \mathrm{cm}$ by inflation, (a) how many electrons move through point $P$ and (b) do they move toward or away from the battery?

For the arrangement of figure, given below, suppose that the battery remains connected while the dielectric slab is being introduced. Calculate (a) the capacitance, (b) the charge on the capacitor plates, (c) the electric field in the gap, and (d) the electric field in the slab, after the slab is in place.

Assume $A=115 {\mathrm{cm}}^2$, $d=1.24 \mathrm{cm}$, $V_0=85.5 \mathrm{V}$, $b=0.780 \mathrm{cm}$, and $k=2.61$.

In the following figure, the capacitance are $C_1=1.0 \mu \mathrm{F}$ and $C_2=3.0 \mu \mathrm{F}$ and both capacitors are charged to a potential difference of $V=200 \mathrm{V}$ but with opposite polarity as shown. Switches $S_1$ and $S_2$ are now closed. (a) What is now the potential difference between points $a$ and $b$?

Capacitor $3$ in Fig. (a) is a variable capacitor (its capacitance $C_3$ can be varied). Figure (b) gives the electric potential $V_1$ across capacitor $1$ versus $C_3.$ The horizontal scale is set by $C_{3s}=12.0 \mu \mathrm{F}.$ Electric potential $V_1$ approaches an asymptote of $8.0\mathrm{V}$ as $C_3\to \infty .$ What are (a) the electric potential $V$ across the battery, (b) $C_1,$and (c) $C_2?$

(a)                                                                             (b)

The capacitors in Fig. are initially uncharged. The capacitance are $C_1=4.0 \mathrm{\mu F}, C_2=8.0 \mathrm{\mu F}$ and $C_3=12 \mathrm{\mu F}$, and the battery's potential difference is $V=6.0 \mathrm{V}.$ When switch $\mathrm{S}$ is closed, how many electrons travel through (a) point $a$ (b) point $b$,(c) point $c$ and (d) point $d ?$ In the figure, do the electrons travel up or down through (e) point $b$ and (f) point $c$?

The figure shows a circuit section of four air-filled capacitors that is connected to a larger circuit. The graph below the section shows the electric potential $V\left(x\right)$ as a function of position $x$ along the lower part of the section, through capacitor $4$. Similarly, the graph above the section shows the electric potential $V\left(x\right)$ as a function of position $x$ along the upper part of the section, through capacitors$1,2,$ and $3$. Capacitor $3$ has a capacitance of $1.60 \mathrm{\mu F}$. What are the capacitance of (a) capacitor $1$ and (b) capacitor $2 ?$

A charged isolated metal sphere of diameter$15 \mathrm{cm}$ has a potential of $6500 \mathrm{V}$ relative to $V=0$ at infinity.

(a) Calculate the energy density in the electric field near the surface of the sphere.

(b) If the diameter is decreased, does the energy density near the surface increase, decrease or remain the same?