Effect of Dielectric on Capacitance

Effect of a Dielectric on Capacitance

When a dielectric material is placed between the plates of a capacitor, the capacitance increases. This happens because the dielectric reduces the effective electric field between the plates for the same amount of free charge.

Vacuum Between Plates (Baseline)

Consider a parallel plate capacitor with plates of area $A$, separation $d$, and free charge $\pm Q$ on the plates. The surface charge density is $\sigma = Q/A$.

• With vacuum, the electric field between the plates is:

• The potential difference is:

• The capacitance in vacuum is:

Dielectric Fully Inserted

Now, a dielectric completely fills the space between the plates. The dielectric gets polarised by the external field $E_0$. This polarisation produces bound surface charge densities $\pm \sigma_p$ on the faces of the dielectric normal to the field.

• The net surface charge density on the plates effectively becomes $\pm (\sigma - \sigma_p)$.
• The electric field inside the dielectric is therefore:

• The potential difference across the plates becomes:

For a linear dielectric, the induced charge density $\sigma_p$ is proportional to the applied field $E_0$, and hence to $\sigma$. This allows us to write:

where $K$ is a constant characteristic of the dielectric, called the dielectric constant. Since $\sigma_p > 0$, we have $K > 1$.

• Substituting this into the expression for $V$:

• The capacitance with the dielectric is:

Key Results and Definitions

• Permittivity of the medium: The product $\varepsilon_0 K$ is defined as the permittivity $\varepsilon$ of the dielectric: $\varepsilon = \varepsilon_0 K$ …