Two metal plates form a parallel plate capacitor. The distance of separation between the plates is $\mathrm{d}$. A metal sheet of thickness $\frac{d}{2}$ and of the same area is inserted completely between the plates. What is the ratio of the capacitances in the cases$?$

The section$\mathrm{AB}$ of a circuit is shown in Figure The e.mf of the cell is $\mathrm{E}=15 \mathrm{V}$ and the capacitances of $C_1=1\mathrm{\mu F}$ and $C_2=2\mu \mathrm{F}$. The potential difference of $\mathrm{A}$ and $\mathrm{B}$ is $6 \mathrm{V}$. Find the potential difference across each capacitor. 

Each of the capacitors shown in Figure has capacitance of $3\mu \mathrm{F}$ Find the equivalent capacitance of the system between the points $\mathrm{A}$ and $\mathrm{B}$. Suppose a battery of emf $60$ volts is connected between the $\mathrm{A}$ and $\mathrm{B}$ Find the potential difference across each capacitor.

Find the equivalent capacitance of the system of capacitors between the two points $A$ and $B$, Fig. $4.87.$

A parallel-plate capacitor with a plate separation of  $d_1=1\mathrm{cm}$ has $29\mathrm{kV}$ applied across the plates. Show why the air breaks down when a thin piece of glass $(K=6.5)$ with a dielectric strength of $290\mathrm{kV}/\mathrm{cm}$ and thickness $d_2=0.20 \mathrm{cm}$ is inserted as shown in figure. An air has a dielectric strength of $30\mathrm{kV}/\mathrm{cm}.$

 One plate of a capacitor is fixed and the other plate is connected to a spring, other end of which is attached to a right wall Fig. The plate area of the capacitor is $A$ and the separation between the plates is $d$. When the switch $S$ is closed, the distance between the plates is $d$. When the switch $S$ is closed, the distance between the plates reduces to a steady value $0.8 d$. Find the force constant $k$ of the spring.