Physics-
General
Easy

Question

A parallel plate capacitor of area A, plate separation d and capacitance C is filled with three different dielectric materials having dielectric constants K subscript 1 end subscript comma K subscript 2 end subscript blank a n d blank K subscript 3 end subscript as shown. If a single dieletric material is to be used to have the same capacitance C is this capacitors, then its dielectric constant K is given by

  1. fraction numerator 1 over denominator K end fraction equals fraction numerator 1 over denominator K subscript 1 end subscript end fraction plus fraction numerator 1 over denominator K subscript 2 end subscript end fraction plus fraction numerator 1 over denominator 2 K subscript 3 end subscript end fraction    
  2. A parallel plate capacitor of area A, plate separation d and capacitance C is filled with three different dielectric materials having dielectric constants K subscript 1 end subscript comma K subscript 2 end subscript blank a n d blank K subscript 3 end subscript as shown. If a single dieletric material is to be used to have the same capacitance C is this capacitors, then its dielectric constant K is given by    
  3. K equals fraction numerator K subscript 1 end subscript K subscript 2 end subscript over denominator K subscript 1 end subscript plus K subscript 2 end subscript end fraction plus 2 K subscript 3 end subscript    
  4. K equals K subscript 1 end subscript plus K subscript 2 end subscript plus 2 K subscript 3 end subscript    

Hint:

A parallel plate capacitor of area A, plate separation d and capacitance C is filled with three different dielectric materials having dielectric constants K subscript 1 end subscript comma K subscript 2 end subscript blank a n d blank K subscript 3 end subscript as shown. If a single dieletric material is to be used to have the same capacitance C is this capacitors, then its dielectric constant K is given by

The correct answer is: A parallel plate capacitor of area A, plate separation d and capacitance C is filled with three different dielectric materials having dielectric constants K subscript 1 end subscript comma K subscript 2 end subscript blank a n d blank K subscript 3 end subscript as shown. If a single dieletric material is to be used to have the same capacitance C is this capacitors, then its dielectric constant K is given by


    Capacitance of two capacitors each of area fraction numerator A over denominator 2 end fraction blank comma plate separation d but dielectric constants K subscript 1 end subscript and K subscript 2 end subscriptrespectively joined in parallel
    C subscript 1 end subscript equals fraction numerator K subscript 1 end subscript epsilon subscript 0 end subscript open parentheses fraction numerator A over denominator 2 end fraction close parentheses over denominator d divided by 2 end fraction plus fraction numerator K subscript 2 end subscript epsilon subscript 0 end subscript open parentheses fraction numerator A over denominator 2 end fraction close parentheses over denominator d divided by 2 end fraction equals fraction numerator open parentheses K subscript 1 end subscript plus K subscript 2 end subscript close parentheses epsilon subscript 0 end subscript A over denominator d end fraction
    It is in series with a capacitor of plate area A comma plate separation d divided by 2 blankand dielectric constantK subscript 3 end subscript blank i e comma blank C subscript 2 end subscript equals fraction numerator K subscript 3 end subscript epsilon subscript 0 end subscript A over denominator d divided by 2 end fraction.
    If resultant capacitance be taken as C equals fraction numerator K subscript blank epsilon subscript 0 end subscript A over denominator d end fraction,
    Then fraction numerator 1 over denominator C end fraction equals fraction numerator 1 over denominator C subscript 1 end subscript end fraction plus fraction numerator 1 over denominator C subscript 2 end subscript end fraction
    therefore fraction numerator d over denominator K epsilon subscript 0 end subscript A end fraction equals fraction numerator d over denominator open parentheses K subscript 1 end subscript plus K subscript 2 end subscript close parentheses epsilon subscript 0 end subscript A end fraction plus fraction numerator d divided by 2 over denominator table row cell K subscript 3 end subscript epsilon subscript 0 end subscript A end cell row cell end cell end table end fraction
    rightwards double arrow fraction numerator 1 over denominator K end fraction equals fraction numerator 1 over denominator K subscript 1 end subscript plus blank K subscript 2 end subscript end fraction plus fraction numerator 1 over denominator 2 K subscript 3 end subscript end fraction

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