Question

A parallel plate capacitor of area A, plate separation d and capacitance C is filled with three different dielectric materials having dielectric constants as shown. If a single dielectric material is to be used to have the same capacitance C in this capacitor then its dielectric constant K is given by

769866
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A

=

B

=

C

=

D

K =

Medium

Solution

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Correct option is D)

 
The common mistake students do in this problem is that they assume  and in parallel and their combination in series with . Which is completely wrong method. [Refer Fig. 1]

For two capacitors to be taken in parallel, their both ends should be at same potential.
Here, the upper ends of   and are at same potential as they are connected to A through a metal plate.
But their lower ends being connected to a dielectric are not at same potential. As Electric field will be different in both dielectrics and , since  .      

 
Hence, To solve this problem, we divide dielectric in two equal parts as shown in figure 2. So,
                   

                 

                 

                 

Now, From figure, Clearly, we can assume and in series and  and , further their combinations will be in parallel.

Combining series combinations in figure 3, we get:

                 

Putting values of and from Step  

              

,    
                   

         

               

Hence equivalent dielectric constant will be as given in Option

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