What is a parallel plate capacitor? Derive an expression for the capacitance of a parallel plate capacitor?

Explanation

Capacitor : An arrangement of two metallic conductors separated by some non conducting medium which can store large amount of charge in small space is called a capacitor.
Capacitor is also known as a condenser.
`C_(0) = (epsilon_(0)A)/d`,
Di-electrics : Those non-conducting materials in which equal and opposite induced charges are produced on their opposite faces, when electric field is applied, are called dielectrics.
Capacitance of a parallel plate capacitor having vacuum (or air) between the plates of the capacitor is given by-
`C_(0) = (epsilon_(0)A)/d`,
where A - Area of each plate
d = Distance or separation between the plates of a capacitor
Now, let us introduce a dielectric slab of thickness .t. between the plates of the capacitor.
Due to polarization, equal and opposite induced charges appear on the two faces of the dielectric slab. These induced charges give rise to induced electric field (`vec(E_p)`) which is in a direction opposite to the applied field `(vecE_(0))`. The reduced
value of the electric field in the dielectric, `E=(E_(0)-E_(p))`.
Now, `E=(E_(0)-E_(p))` exists in a region of thickness .t. and `E_(0)` field exists in a region of thickness (d - t).
Potential difference between the two plates of the capacitor
is given by:
`V = E_(0) (d-t) +ET`
Since, `E_(0)/E = K` or `E=E_(0)/K`
`therefore V =E_(0)(d-t) +E_(0)/Kt =E_(0)[(d-t) +t/K]`......(2)
We know, `E_(0) =sigma/epsilon_(0) =q/(A epsilon_(0))` [`therefore sigma = q//A]`
`therefore` Eqn (2) becomes `V =q/(A epsilon_(0))[(d-t)+t/K]`......(3)
Now, Capacitance of the capacitor
`C=q/V =(qA epsilon_(0))/(q[(d-t)+t/K])` or `C = (epsilon_(0)A)/((d-t) + t/K)`
`C=C_(0)/(1-t/d(1-1/K))`
Clearly, `C gt C_(0)`
So when a dielectric slab is introduced.between the plates of a parallel plate capacitor, then its capacity increases.
If the dielectric fills the whole space between the plates of a capacitor i.e., t = d
Then `C=C_(0)/(1//K) = KC_(0)`
The capacity of a capacitor increases by dielectric constant (K) times.