A parallel plate capacitor with plate separation 5 mm is charged by a battery. On introducing a mica sheet of 2 mm and maintaining the connections of the plates with the terminals of the battery, it is found that it draws 25% more charge from the battery. The dielectric constant of mica is _______.

Let initial capacitance be:

$$C_0=\frac{\varepsilon_0A}{5}$$

(since separation = 5 mm)

When mica of thickness 2 mm is inserted, remaining air gap = 3 mm

So capacitor becomes series combination:

$$\frac{1}{C}=\frac{3}{\varepsilon_0A}+\frac{2}{k\varepsilon_0A}$$

Given: charge increases by 25%

$$Q'=1.25Q$$

Since battery connected ⇒ voltage constant ⇒

$$C'=1.25C_0$$

step 1: equate

$$\frac{\varepsilon_0A}{3+\frac{2}{k}}=1.25\cdot\frac{\varepsilon_0A}{5}$$

Cancel $$\varepsilon_0A$$:

$$\frac{1}{3+\frac{2}{k}}=\frac{1.25}{5}=\frac{1}{4}$$

step 2: solve

$$3+\frac{2}{k}=4$$