A dielectric slab of thickness $\frac{d}{4}$ and relative permittivity ${\epsilon }_{\mathrm{r}}=2$ is inserted between two parallel conducting plates each of area $A$ kept at a separation $d$. Find the

(a) The charge density in the inner side of conductors

(b) The charge density on the dielectric slab

(c) The electric field in the dielectric

(d) The potential difference between the conductor

(e) The capacitance

The separation between the plates of a parallel-plate capacitor is $0.05 \mathrm{m}$. A field of $3\times {10}^4 \mathrm{V} {\mathrm{m}}^{-1}$ is established between the plates. It is disconnected from the battery and an uncharged metal plate of thickness $0.01 \mathrm{m}$ is inserted between the plates of the capacitor.

(a) What would be the potential difference across the capacitor, before the introduction of the metal plate and after its introduction.

(b) What would be the potential difference if a plate of dielectric constant $K=2$ is introduced in place of metal plate?

A dielectric slab is filled in the space between the parallel plate capacitor whose relative permittivity varies as, ${\epsilon }_r=a+bx$

Find: (a) capacitance of the system

(b) potential at a point situated at a distance $x$ from the origin $O$ inside of capacitor.

If you have several $2.0 \mathrm{\mu F}$ capacitors, each capable of withstanding $200 \mathrm{V}$ without breakdown, how would you assemble a combination that has an equivalent capacitance of:

(a) $0.4 \mathrm{\mu F}$

(b) $1.2 \mu \mathrm{F},$ each withstanding $1000 \mathrm{V}?$

Find the equivalent capacitance between points $A$ and $B$ as shown in figure.

Find the capacitance between $A$ and $B$ if three dielectric slabs of dielectric constants $K_1$ (area $A_1$and thickness $\mathrm{d}$), $K_2$ (area $A_2$ and thickness $d_2$), and $K_3$ (area $A_2$ and thickness $d_2$) are inserted between the plates of a parallel plate capacitor of plate area $A$(figure). (Given that the distance between the two plates is $d=d_1+d_2$)

Three dielectrics of relative permittivity’s ${\epsilon }_{{\mathrm{r}}_1}=1, {\epsilon }_{{\mathrm{r}}_2}=2,$ and ${\epsilon }_{{\mathrm{r}}_3}=3$ are introduced in a parallel plate capacitor of plate area $A$ and separation $d$.

Find the effective capacitance between $A$ and $B$.