Question

Two plates of parallel plate capacitve transducer are $30mm$ apart and the space is filled with two different dielectric materials, one material is $1cm$ thick with a dielectric constant of $5$ and the other material is $20mm$ thick with a dielectirc constant $10$. If the capacitive transducer were to be made up of a single dielectric material then the dielectric constant of the signle material is

• A. 7.5

Answer 1

The correct option is A 7.5

$C=\frac{\epsilon WX}{d}$

The given figure (a) is nothing but the series combination of the two capacitances $C_1$ & $C_2$


So,
$C=\frac{C_1{C}_2}{C_1+C_2}$

$\frac{1}{C}=\frac{1}{C_1}+\frac{1}{C_2}$

$\frac{d}{\epsilon WX}=\frac{d_1}{{\epsilon }_{1}WX}+\frac{d_2}{{\epsilon }_{2}WX}$

$\frac{d}{\epsilon }=\frac{d_1}{{\epsilon }_1}+\frac{d_2}{{\epsilon }_2}$

$\frac{3}{\epsilon }=\frac{1}{5}+\frac{2}{10}=\frac{2}{5}$

$\epsilon =\frac{3\times 5}{2}=7.5$

$\epsilon =7.5$