Question

A parallel plate capacitor with plate separation 5mm is charged up by a battery. It is found that on introducing a dielectric sheet of thickness 2 mm, while keeping the battery connections intact, the capacitor draws 25% more charge from the battery than before. The dielectric constant of the sheet is ____.

Answer 1

Without the dielectric, the charge stored is given by:

$Q = \frac{A \epsilon_0 V}{d}$

With the dielectric inserted:

$Q' = \frac{A \epsilon_0 V}{d - t + \frac{t}{K}}$

Given:

$Q' = 1.25 Q \implies \frac{A \epsilon_0 V}{d - t + \frac{t}{K}} = 1.25 \cdot \frac{A \epsilon_0 V}{d}$

Canceling common terms:

$\frac{d}{d - t + \frac{t}{K}} = 1.25$

Substituting $d = 5 \, \text{mm}$ and $t = 2 \, \text{mm}$:

$1.25 \left( 5 - 2 + \frac{2}{K} \right) = 5$

Simplifying:

$1.25 \left( 3 + \frac{2}{K} \right) = 5$

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

$\frac{2}{K} = 1 \implies K = 2$