A capacitor is formed by two square metal-plates of edge a, separated by a distance d. Dielectrics of dielectric constants K1 and K2 are filled in the gap as shown in figure. Find the capacitance.


These two capacitors are connected in series.



To find out the capacitance, let us consider a small capacitor of


differential width dx at a distance x from


the left end of the capacitor.


The two capacitive elements of dielectric


constants K1 and K2 are with plate


separations as -


and in series,


respectively as seen from fig.


Also, differential plate areas of the capacitors are adx.


We know, capacitance c is given by-



Where,


A= Plate Area


d= separation between the plates,


∈0 = Permittivity of free space = 8.854 × 10-12 m-3 kg-1 s4 A2


k = dielectric strengthof the material


Then, looking into the fig, the capacitances of the capacitive elements of the elemental capacitors are given by –


and



We know that equivalent capacitance of capacitors connected in


series is given by the expression –





Now, integrating both sides to get the actual capacitance,







Looking back into the fig.



Substituting in the expression for capacitance C,



Now,





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