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Explain the effect of dielectriccapacitance of parallel plate capacitor and obtainthe formula of dielectric constant. |
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Answer» Solution :Let us have two large plates each of area Aseparated 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 surfaccharge 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 proportionalto `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)` |
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