Question 72: Electrostats & Capacitance

Question

A container has a base of $50 \mathrm{~cm} 	imes 5 \mathrm{~cm}$ and height $50 \mathrm{~cm}$, as shown in the figure. It has two parallel electrically conducting walls each of area $50 \mathrm{~cm} 	imes 50 \mathrm{~cm}$. The remaining walls of the container are thin and non-conducting. The container is being filled with a liquid of dielectric constant 3 at a uniform rate of $250 \mathrm{~cm}^3 \mathrm{~s}^{-1}$. What is the value of the capacitance of the container after 10 seconds?
[Given: Permittivity of free space $\epsilon_0=9 	imes 10^{-12} \mathrm{C}^2 \mathrm{~N}^{-1} \mathrm{~m}^{-2}$, the effects of the non-conducting walls on the capacitance are negligible]

Accepted Answer

Volume of dielectric filled in $10 \mathrm{sec}$
$$
\begin{aligned}
& =250 	imes 10=2500 \mathrm{~cm}^3 \\
h 	imes 50 	imes 5 & =2500 \\
\Rightarrow h & =10 \mathrm{~cm}
\end{aligned}
$$
$C_1=\frac{\varepsilon_0 A_1}{d}, C_2=\frac{\varepsilon_0 A_2}{d} K$
$\begin{aligned} A_1 & =50 	imes(50-h) \\ A_2 & =50 	imes h \\ C_{	ext {eq }} & =C_1+C_2 \\ & =\frac{\varepsilon_0 50(50-h) \mathrm{cm}^2}{5 \mathrm{~cm}}+\frac{3 	imes \varepsilon_0 	imes 50 	imes h}{5 \mathrm{~cm}} \\ & =\varepsilon_0\left[\frac{50 	imes 40}{5}+\frac{3 	imes 50 	imes 10}{5}ight] 	imes 10^{-2} \\ & =7 \varepsilon_0=63 	imes 10^{-12} \\ & =63 \mathrm{pF}\end{aligned}$

Answer

$27 ~\mathrm{pF}$

Answer

$63 ~\mathrm{pF}$

Answer

$81 ~\mathrm{pF}$

Answer

$135 ~\mathrm{pF}$

Electrostats & Capacitance

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