Derive an expression for effective capacitance of three parallel plate capacitors connected in series.

The space between the plates of a parallel plate capacitor is completely filled in two ways. In the first case, it is filled with a slab of dielectric constant $K$. In the second case, it is filled with two slabs of equal thickness and dielectric constants $K_1$ and $K_2$ respectively as shown in the figure. The capacitance of the capacitor is same in the two cases. Obtain the relationship between $K$, $K_1$ and $K_2$.

A sphere of radius $0.25 \mathrm{m}$ has positive charge $+8.8 \mathrm{\mu C}$ uniformly distributed on its surface. A small pellet of mass $0.075 \mathrm{kg}$ and charge $+2.4 \mathrm{\mu C}$ is directed radially at the sphere. When the pellet is at a distance of 0.75 m from the centre of the sphere its speed is $3.2 \mathrm{m} {\mathrm{s}}^{-1}$.

$\mathrm{a}.$ Determine the distance from the centre of the sphere at which the pellet will stop.

A parallel-plate capacitor consists of two identical metal plates. Two dielectric slabs having dielectric constants $K_1$ and $K_2$ are introduced in the space between two plates as shown below. Show that capacity of capacitor in the arrangement is given by $C=\frac{2{\epsilon }_0{K}_1{K}_{2}A}{\left(K_1+K_2\right)d}$