Q.What are polar and non-polar dielectrics? Find an expression for the capacity of a parallel-plate capacitor with plate separation $d$, in which a dielectric of dielectric constant $K$ is inserted having thickness $t<d$. (2+3=5) OR

Polar dielectric molecules have a permanent dipole moment; non-polar ones acquire an induced dipole moment only under an external field. Splitting the gap between the capacitor plates into the field-reduced dielectric region (thickness $t$, field $E_0/K$) and the remaining air gap (thickness $d-t$, field $E_0$) and adding up the potential drops gives $C=\varepsilon_0A/[(d-t)+t/K]$.

Polar and non-polar dielectrics

• Polar dielectrics: molecules that possess a permanent electric dipole moment even in the absence of any external field, due to an inherently asymmetric distribution of positive and negative charge in the molecule (e.g. water $\text{H}_2\text{O}$, HCl). In the absence of a field these permanent dipoles are randomly oriented (net dipole moment of the bulk sample is zero due to thermal motion), but an external field partially aligns them along its direction.

• Non-polar dielectrics: molecules with a symmetric charge distribution, so the centres of positive and negative charge coincide and there is no permanent dipole moment (e.g. $\text{N}_2$, $\text{O}_2$, $\text{CO}_2$, $\text{CH}_4$, benzene). An external field displaces the positive and negative charge centres slightly relative to each other, inducing a small dipole moment aligned with the field.

In both cases, the resulting alignment/induction of dipole moments produces bound surface charges on the dielectric that create an internal field opposing the applied field, reducing the net field inside the dielectric — the basis for the dielectric constant $K>1$ and the associated increase in capacitance.

Capacitance of a parallel-plate capacitor with a partial dielectric slab

Consider a parallel-plate capacitor of plate area $A$, separation $d$, with a dielectric slab of dielectric constant $K$ and thickness $t$ ($t<d$) inserted between the plates (leaving an air gap of thickness $d-t$). Let the plates carry surface charge density $\sigma=Q/A$.

Field in the two regions

Without any dielectric, the field between the plates would be $E_0=\sigma/\varepsilon_0$. In the air-gap region (thickness $d-t$), the field remains $E_0$ (unaffected by the dielectric elsewhere). Inside the dielectric slab (thickness $t$), the field is reduced by the factor $K$:

$E_{\text{dielectric}}=\frac{E_0}{K}=\frac{\sigma}{K\varepsilon_0}$

Total potential difference

The potential difference across the capacitor is the sum of the potential drops across the air gap and across the dielectric slab:

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

Substituting $E_0=\sigma/\varepsilon_0=Q/(A\varepsilon_0)$:

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

Capacitance

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

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

(As a check: if $t=0$, this reduces to $C=\varepsilon_0A/d$, the capacitance with no dielectric; if $t=d$ — the slab fills the whole gap — it reduces to $C=K\varepsilon_0A/d$, as expected.)

…