A parallel plate capacitor (area $6\ \text{cm}^2$, separation $3\ \text{mm}$) has its gap filled with three dielectric slabs of equal thickness placed side-by-side (not stacked), with $K_1=10$, $K_2=12$, $K_3=14$. What single dielectric constant $K$ gives the same capacitance when it fully fills the capacitor?
The three slabs are placed side-by-side (parallel combination), each occupying one-third of the area $A/3$ and the full separation $d$.
Total capacitance:
$C = \frac{\varepsilon_0 (A/3)}{d}(K_1 + K_2 + K_3) = \frac{\varepsilon_0 A}{3d}(10+12+14) = \frac{36\varepsilon_0 A}{3d} = \frac{12\varepsilon_0 A}{d}$
For a single dielectric $K$:
$C = \frac{K\varepsilon_0 A}{d}$
$\Rightarrow K = 12$
The equivalent dielectric constant is simply the arithmetic mean: $(10+12+14)/3 = 12$.
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