Parallel plate capacitor propertie

Capacitance of a P. P. C when a dielectric slab placed between its plate

Consider P. P. C having two plates of charges +Q and -Q, separation between both plates is d and A be the area of cross section of both plates

If Co be the initial capacitance of P. P. C

       Co = EoA/d

Where, Eo = permitivity of medium between P. P. C

                              and 

E0 = charge density of plates/Eo

 Where, E0 = intial electric field  between P. P. C

                Eo = permitivity  of medium between                           P. P. C

W. K. T

           Charge density of plates = Q/A

  E0 = Q/AEo

 If Ep be the E.F inside the slab after introducing it between P. P. C

Which is opposite the direction of E0

So, it means

                  E = E0 - Ep = kappa

               Where, E is the net field inside the dielectric. 

 So,      kappa = E0/E

            E = Eo/kappa                eq. (1)


Now, potential between the plates ( if t be the thickness of slab)

V = E.P in the slab + E.P in the remaining portion

V = E.P in t + E.P in (d-t)
V = E × t + E0 × (d-t)
V = E0/kappa × t + E0 × (d-t)
V = E0[t/kappa + d-t]
V = Q/AEo[t/kappa + d-t]


Now if C be the capacitance after inserting slab between P. P. C.

C = Q/V = Q/Q/AEo[t/kappa + d-t]

C = AEo/[t/kappa + d-t]

                                    Note

If d = t

C = EoA/d/kappa + d-t

C = EoA/d/kappa

C = kappa × EoA/d

C = kappa × Co

Thus, capacitance increases by kappa times. 

            

                                   Or

  Kappa = C/Co

Kappa = capacitance of P. P. C after introducing 

                Slab/

                Capacitance of P. P. C before 

                 Introducing slab