Question

The plates of a parallel plate capacitor are charged up to $200V$ . A dielectric slab of thickness $4mm$ is inserted between the plates. Then, to maintain the same potential difference between the plates of the capacitor, the distance between the plates is increased by $3.2mm$ . The dielectric constant of a dielectric slab is :
$(A)1$
$(B)4$
$(C)5$
$(D)6$

Answer 1

When we introduced a thick plate between the plates of the capacitor, therefore, the electric field value was reduced, and the capacitance of the parallel plate capacitor increased by a factor $k$ (dielectric constant).

Complete step by step solution:
Parallel Plate Capacitors are the type of capacitors which have an arrangement of electrodes and insulating material (dielectric). The two conducting plates act as electrodes. There is a dielectric between them. This acts as a separator for the plates.
The simplest design for a capacitor is a parallel-plate, which consists of two metal plates with a gap between them: electrons are placed onto one plate (the negative plate), while an equal amount of electrons are removed from the other plate (the positive plate).
Initial voltage $= 200V$
$Q = CV$
We know that $C = \dfrac{{A{\varepsilon _o}}}{d}$
So, $Q = \dfrac{{A{\varepsilon _o}}}{d}V.......(1)$
When a dielectric is inserted, $Q$ remains constant and $V$ decreases by $k$
$Q = \dfrac{{A{\varepsilon _o}}}{{{d'} + t - \dfrac{t}{k}}}V........(2)$
From equation (1) and (2), we get,
$d = {d'} + t - \dfrac{t}{k}$
$d - {d'} = t - \dfrac{t}{k}$
On putting the required values, we get,
$1.6mm = 2 - \dfrac{2}{k}$
On simplifying the above equation,
$k = 5$
So, the final answer is $(C)\;5$ .

Note:
When a dielectric is put inside the plates then the value of the electric field increases and the value of voltage decreases and hence the capacitance increases. The plates store the equal charge one before and after introducing the dielectric but with a lower voltage.
We must know that polarization is very good for a dielectric apart from being insulators. It is very good if a dielectric gets polarized. If a certain voltage is applied to a dielectric, then the polarization takes place which really affects the electric field inside it.