If we consider a simple capacitor with parallel plates of area A, separated by a distance d, we can see that for a capacitor with charge Q, the charge on each plate is +Q and –Q. As the area of the plate is A, the corresponding charge density can be given as ±σ. Where,

\(\sigma =\frac{Q}{A}\)

When the two plates have vacuum between them, potential energy across the capacitor can be given as,

\(V_{0}=E_{0}d=V_{0}\frac{\sigma }{\varepsilon _{0}}d\)

The capacitance of the capacitor can thus be given as,

\(C_{0}=\frac{Q}{V_{0}}=\varepsilon _{0}\frac{A}{d}\)

Let us consider another capacitor with the same specifications as taken before. Let us insert a dielectric between the plates such that it fully occupies the space between the plates. As the dielectric enters the field between the plates, it gets polarized by the field and the charges get arranged such that they act as two charged sheets with a surface charge density of σ_{p} and – σ_{p}, as shown in the figure below.

The net surface charge density then becomes equivalent to ±(σ – σ_{p}).

The potential energy across the capacitor can thus be given as,

\(V=Ed=\frac{\sigma -\sigma _{p}}{\varepsilon _{0}}d\)

In case of linear dielectrics, we can say that σ_{p} is proportional to E_{0 }and hence it is proportional to σ. Thus, we can say that the value (σ – σ_{p}) is also proportional to σ. Mathematically,

\(\sigma -\sigma _{p}=\frac{\sigma }{K}\)

Where K is a constant whose value depends upon the dielectric medium selected. The potential energy across the capacitor can this be written as,

\(V=\frac{\sigma d}{\varepsilon _{0}K}=\frac{Qd}{A\varepsilon _{0}K}\)

And the capacitance between the plates can be given as,

\(C=\frac{Q}{V}=\frac{\varepsilon _{0}KA}{d}\)

Here ε_{0}K is the permittivity of the medium, which can also be given as,

\(\varepsilon =\varepsilon _{0}K\)

Here the value K is the permittivity of the medium such that, for a given medium,

\(K=\frac{\varepsilon }{\varepsilon _{0}}\)

And the ratio of the capacitance of the capacitor with a dielectric medium to the capacitor with vacuum between the plates can be given as,

\(K=\frac{C}{C _{0}}\)

Stay tuned with Byju’s to learn more about the effect of dielectric on a capacitor and other related topics.

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