A parallel plate capacitor has a dielectric slab of dielectric constant $K$ between its plates that covers $1 / 3$ of the area of its plates, as shown in the figure. The total capacitance of the capacitor is $C$ while that of the portion with dielectric in between is $C _1$. When the capacitor is charged, the plate area covered by the dielectric gets charge $Q_1$ and the rest of the area gets charge $Q_2$. Choose the correct option/options, igonoring edge effects.
$(A)$ $\frac{E_1}{E_2}=1$ $(B)$ $\frac{E_1}{E_2}=\frac{1}{K}$ $(C)$ $\frac{Q_1}{Q_2}=\frac{3}{K}$ $(D)$ $\frac{ C }{ C _1}=\frac{2+ K }{ K }$
IIT 2014, Advanced
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$C =\frac{ K \varepsilon_0 A }{3 d }+\frac{2 \varepsilon_0 A }{3 d } $
$C _1=\frac{ K \varepsilon_0 A }{3 d } $
$\frac{ C }{ C _1}=\frac{2+K}{K} \quad \text { Ans. (D) } $
$E _1= E _2=\frac{ V }{ d } $
$\Rightarrow \quad \frac{ E _1}{ E _2}=1 \quad \text { Ans. (A) } $
$Q _1=C_1 V=\frac{K \varepsilon_0 A }{3 d } V $
$\Rightarrow \quad \frac{Q_2}{ Q _2}= C _2 V =\frac{2 \varepsilon_0 A }{3 d } V$
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