
Answer:
The capacitance of a parallel plate capacitor is
C = \frac{\epsilon_0\kappa A}{d}
where A=L^2. When a capacitor does not have any dielectrics (free space between plates), it will be
C_0 = \frac{\epsilon_0 A}{d}
We can consider the particular capacitor is like series connection of two different capacitors. Namely,
\frac{1}{C_{\mathrm{Total}}} = \frac{1}{C_1}+\frac{1}{C_2} \\ \rightarrow C_{\mathrm{Total}} = \frac{C_1C_2}{C_1+C_2}
From the condition, we have each capacitance of C_1 and C_2.
C_{1,2} = \frac{\epsilon_0\kappa_{1,2} A}{\frac{1}{2}d} = \frac{2\epsilon_0\kappa_{1,2} A}{d}
Then, we can rewrite it in terms of C_0.
C_{1,2} = 2\kappa_{1,2}C_0
Plug in the above.
C_{\mathrm{Total}} = \frac{2\kappa_1C_0\times 2\kappa_2 C_0}{2\kappa_1C_0+2\kappa_2 C_0} \\ = \frac{4\kappa_1\kappa_2 C^2_0}{2C_0(\kappa_1+\kappa_2)} \\ = \frac{2\kappa_1\kappa_2 C_0}{\kappa_1+\kappa_2}
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