Derivation : Capacitance of a Parallel Plate Capacitor with a Dielectric Slab :

For a parallel plate capacitor with vacuum (or air) between plates:

$C_0 = \frac{\varepsilon_0 A}{d} \quad ...(i)$

where (A) = area of each plate , (d) = separation between plates , ($\varepsilon_0$) = permittivity of free space .

Let a dielectric slab of thickness (t) and dielectric constant (K) be inserted between the plates.

- Electric field in free space region (thickness (d - t)): ($E_0$)

- Electric field in dielectric region (thickness (t)): ($E = \frac{E_0}{K}$)

Potential difference between plates:

$V = E_0(d-t) + \frac{E_0 t}{K}$

$V= E_0 \left[(d-t) + \frac{t}{K}\right] ...(ii)$

Since

$E_0 = \frac{q}{A \varepsilon_0}$

we get

$V = \frac{q}{A \varepsilon_0} \left[(d - t) + \frac{t}{K}\right] \quad ...(iii)$

By definition,

$C = \frac{q}{V}$

$C = \frac{q}{ \frac{q}{A \varepsilon_0} \left[(d - t) + \frac{t}{K}\right]}$

$C= \frac{\varepsilon_0 A}{(d - t) + \frac{t}{K}}$

$C = \frac{\varepsilon_0 A}{d[\left(1 - \frac{t}{d}\right) + \frac{t}{dK}]}\quad ...(iv)$

From equation (i)

$C_0 = \frac{\varepsilon_0 A}{d}$

$C = \frac{C_0}{\left(1 - \frac{t}{d}\right) + \frac{t}{dK}} \quad ...(v)$

$\frac{C}{C_0} = \frac{1}{\left(1 - \frac{t}{d}\right) + \frac{t}{dK}}$

$\Rightarrow C = \frac{C_0}{1 - \frac{t}{d}\left(1 - \frac{1}{K}\right)} \quad ...(vi)$

Since ($K = \varepsilon_r$) (relative permittivity):

$C = \frac{C_0}{1 - \frac{t}{d}\left(1 - \frac{1}{K}\right)}$

If the dielectric slab completely fills the space ((t = d)):

$C = \frac{C_0}{1 - \frac{t}{t}\left(1 - \frac{1}{K}\right)}$

$C = \frac{C_0}{1 - 1 +\frac{1}{K}}$

$C = \frac{C_0}{\frac{1}{K}}$

$C = KC_0 \quad ...(vii)$

Thus, capacitance increases by a factor equal to the dielectric constant.

From equation (vii):

$K = \frac{C}{C_0} \quad ...(viii)$

So, dielectric constant is defined as the ratio of capacitance with dielectric to capacitance without dielectric.

✅ Final Result:

$C = \frac{\varepsilon_0 A}{(d - t) + \frac{t}{K}} = \frac{C_0}{1 - \tfrac{t}{d}\left(1 - \tfrac{1}{K}\right)}$

Derivation : Capacitance of a Parallel Plate Capacitor with a Dielectric Slab