Q116 / 180 · AIIMS 2017 · Physics

A parallel plate capacitor with air as a dielectric has capacitance $C$. A slab of dielectric constant $K$, having same thickness as the separation between the plates is introduced so as to fill one-fourth of the capacitor as shown in the figure. The new capacitance will be Figure

  • Correct answer
    A
    $(K+3) \frac{C}{4}$
  • B
    $(K+2) \frac{C}{4}$
  • C
    $(K+1) \frac{C}{4}$
  • D
    $\frac{K C}{4}$

Solution

The two condensers with $K$ and with air are in parallel. With air, $C_1=\frac{\varepsilon_0}{d}\left(\frac{3 A}{4}\right)=\frac{3 \varepsilon_0 A}{4 d}$ With medium, $C_2=\frac{\varepsilon_0 K}{d}\left(\frac{A}{4}\right)=\frac{\varepsilon_0 A K}{4 d}$ $\therefore C^{\prime}=C_1+C_2$ or $C^{\prime}=\frac{3 \varepsilon_0 A}{4 d}+\frac{\varepsilon_0 A K}{4 d}=\frac{\varepsilon_0 A}{d}\left[\frac{3}{4}+\frac{K}{4}\right]$ or $C^{\prime}=\frac{C}{4}(K+3) \quad\left[\because C=\frac{A \varepsilon_0}{d}\right]$

Hence, the correct answer is option: 'A'.

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