Capacitance of parallel plate condenser when dielectric medium is partially filled between the plates of condenser”:

#“Capacitance of parallel plate condenser when dielectric medium is partially filled between the plates of condenser”:

In above figure shows the construction of parallel plate condenser by a consisting of two plates M and N of equal area of cross section are placed parallel to each other at a distance d. +Q charge is given to the plate M and outer surface of plate N is earthed. A dielectric medium of thickness (t)  be placed between two plates. Thus the width of space which containing air is (d-t).

Derivation of expression of capacitance:
we know that the electric field intensity between two parallel plates having charge densities +σ and -σ is

1)  In air  E (air)  = σ/εo     N/C
2)  In medium E (medium) = σ/εoK    N/C

we know that the relationship between electric field intensity E and potential difference ∆V

E = ∆V/∆X
∴  ∆V = E×∆X

Thus total pot diff b/w two plates
∆V = ∆V (air) + ∆V (medium)
∆V = E (air) × (d-t) + E (medium) × t
      = (σ/εo)(d-t)  +  (σ/εoK)× t
      = (σ/εo )[d-t + t/K]
∆V = (σ/εo)[d-t (1- 1/K)]

(But the surface charge density  σ = Q/A )
∆V = Q/Aεo [d-t (1-1/k)]

By the theory of condenser
C =  Q/∆V

Putting value of  ∆V

C = Q×A·εo  / Q[d-t (1-1/K)]

C = A·εo / [d-t (1-1/K)]    farad