$A$ parallel plate capacitor is made up of a stair-like structure with a plate area $A$ for each stair,and the plates are separated by distances $b$,$2b$,and $3b$ as shown in the figure. The capacitance of the arrangement is $\frac{ x }{15} \frac{\varepsilon_{0} A }{ b }$. The value of $x$ is ............

The distance between the circular plates of a parallel plate capacitor $40\,mm$ in diameter,in order to have the same capacity as a sphere of radius $1\,m$,is....$mm$.

Show that the force on each plate of a parallel plate capacitor has a magnitude equal to $\frac{1}{2} Q E$,where $Q$ is the charge on the capacitor,and $E$ is the magnitude of the electric field between the plates. Explain the origin of the factor $\frac{1}{2}$.

In a parallel plate capacitor,the capacity can be increased by decreasing

The area of each plate of a parallel plate capacitor is $20 \, cm^2$ and the separation between the plates is $2 \, mm$. If the dielectric strength of air is $3 \times 10^6 \, V/m$,the maximum possible value of the emf of the battery that can be connected across the plates of this capacitor and the corresponding charge on the plates are:

Four metallic plates are arranged as shown in the figure. If the distance between each plate is $d$ and the area of each plate is $A$, then the capacitance of the given system between points $A$ and $B$ is (Given $d << \sqrt{A}$):