The space between the plates of a parallel plate capacitor is com… - Vidyadip

Question

The space between the plates of a parallel plate capacitor is completely filled in two ways. In the first case, it is filled with a slab of dielectric constant K . In the second case, it is filled with two slabs of equal thickness and dielectric constants $\mathrm{K}_1$ and $\mathrm{K}_2$ respectively as shown in the figure. The capacitance of the capacitor is same in the two cases. Obtain the relationship between $\mathrm{K}, \mathrm{K}_1$ and $\mathrm{K}_2$.

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Answer

$\text{C}_1=\frac{\text{K}\varepsilon_0\text{A}}{\text{d}}C_2=$ parallel combination of two capacitors,
$=\frac{\text{K}_1\varepsilon_0\big(\frac{\text{A}}{2}\big)}{\text{d}}+\frac{\text{K}_2\varepsilon_0\big(\frac{\text{A}}{2}\big)}{\text{d}}$
$\frac{\varepsilon_0\text{A}}{2\text{d}}\big(\text{K}_1+\text{K}_2\big)$
$\because\text{C}_1=\text{C}_2$
$\Rightarrow\text{K}=\frac{\text{K}_1+\text{K}_2}{2}$

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