GATE 2019 EE – Question 30
A co-axial cylindrical capacitor in Figure (i) has dielectric with relative permittivity $\varepsilon_{r1}=2$. When one-fourth portion of the dielectric is replaced with another dielectric of relative permittivity $\varepsilon_{r2}$ as shown in Figure (ii), the capacitance is doubled. The value of $\varepsilon_{r2}$ is ________.
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Correct answer: 10
Explanation
The two dielectric sectors are in parallel, so the capacitance is proportional to the area-weighted permittivity: $\frac34\varepsilon_{r1}+\frac14\varepsilon_{r2}$. Doubling means $\frac34(2)+\frac{\varepsilon_{r2}}{4}=2\times2=4$, so $\frac{\varepsilon_{r2}}4=2.5$ and $\varepsilon_{r2}=10$.