Explain the effect of dielectric capacitance of parallel plate capacitor and obtain the formula of dielectric constant.

Explanation

Let us have two large plates each of area A separated by a distance d. The charge on the plates is `pm` Q corresponding to the charge density `pm sigma` .
When there ls vacuum between the plates
`E_(0)= (sigma)/(in_(0))`
and the potential difference `V_(0)`.
`:. V_(0)=E_(0)d`
If `C_(0)` is the capacitance ,
`C_(0)=(Q)/(V_(0))`
`= (sigmaA)/(E_(0)d) " " [because Q = sigmaA]`
`C_(0)= (in_(0)A)/(d) " [ because E_(0)= (sigma)/(in_(0))implies in_(0)= (sigma)/(E_(0)]`
Consider a dielectric inserted between the plate fully occupying the intervening region. So, th dielectric is polarised by the field and surfac charge density `pm sigma_(p)` arise on the plates. Hence, electric field between two plates,
`E= E_(0)-E_(p)`
`:. E =(sigma)/(in_(0))-(sigma_(p))/(in_(0)) " " [because E_(p)=(sigma_(p))/(in_(0))]`
`:. E = (sigma-sigma_(p))/(in_(0))`
and electric potential difference
V= Ed
`= (sigma-sigma_(p))/(in_(0)).d`
For linear dielectric ap is proportional to `sigma` Hence `sigma- sigma_(p)` is also proportional to `sigma` so we can write `sigma- sigma_(p)= (sigma)/(K)`
Where K is a oonstant characteristic of dielectric.
`:. V= (sigmad)/(in_(0)K)=(Qd)/(A in_(0)K)[because sigma = (Q)/(A)` and from (2)]
The capacitance with dielectric between the plate is then,
`C = (Q)/(V)= (A in_(0)K)/(d)= K(Ain_(0))/(d)`
`:. C = KC_(0)` [ From equation ]
Hence, capacitance of capacitor with dielectric medium of dielectric constant K becomes K times the capacitance of capacitor with vacuum as medium. The product to `epsilon_(0)K` is called the permittivity of the medium and is denoted by `epsilon` .
`:. epsilon = epsilon_(0)K`
`:. K = (epsilon_(0))/(epsilon_(0))= (C)/(C_(0))= (E_(0))/(E)`
Deflnldon of dielectric constant : "The ratio of 1. permittivity of the medium to the permittivity of vacuum is known as dielectric constant".
`:. K = (epsilon)/(epsilon_(0))`
OR
"The ratio of intensity of electric field in vacuum to the intensity of electric field in dielectric medium is known as dielectric constant".
`:. K = (E_(0))/(E)`