Electrostatics 5 Question 42
44. Three identical capacitors $C _1, C _2$ and $C _3$ have a capacitance of $1.0 \mu F$ each and they are uncharged initially. They are connected in a circuit as shown in the figure and $C _1$ is then filled completely with a dielectric material of relative permittivity $\varepsilon _r$. The cell electromotive force (emf) $V _0=8 V$. First the switch $S _1$ is closed while the switch $S _2$ is kept open. When the capacitor $C _3$ is fully charged, $S _1$ is opened and $S _2 \quad$ is closed simultaneously. When all the capacitors reach equilibrium, the charge
on $C _3$ is found to be $5 \mu C$. The value of $\varepsilon _r=\ldots$
(2018 Adv.)
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Solution:
Applying loop rule,
$$ \begin{aligned} \frac{5}{1}-\frac{3}{\varepsilon _r}-\frac{3}{1} & =0 \Rightarrow \frac{3}{\varepsilon _r}=2 \\ \varepsilon _r & =1.50 \end{aligned} $$
