A parallel-plate capacitor is filled with two dielectrics as shown in the figure. We have to show that the capacitance is given by

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Let q be the free charge on the capacitor. Let the area of the plates be A and the separation between the plates be d. Space between the plates has been filled with dielectrics with dielectric constants and as shown in the figure.

The electric field inside the dielectric with dielectric constant will be

and the electric field inside the dielectric with dielectric constant will be

Therefore, the potential difference between the plates will be

As

we have