A parallel plate capacitor with plate area A and plate separation d is filled with a dielectric material of dielectric constant K = 4. The thickness of the dielectric material is $x$ , where $x < d$ .

Let $C_{1}$ and $C_{2}$ be the capacitance of the system for $x = \frac{d}{4}$ and $x = \frac{d}{2}$ , respectively. If $C_{1} = 10 μ F$ the value of $C_{2}$ is ______ $μ F$ .

Question

A parallel plate capacitor with plate area A and plate separation d is filled with a dielectric material of dielectric constant K = 4. The thickness of the dielectric material is $x$ , where $x < d$ .

Let $C_{1}$ and $C_{2}$ be the capacitance of the system for $x = \frac{d}{4}$ and $x = \frac{d}{2}$ , respectively. If $C_{1} = 10 μ F$ the value of $C_{2}$ is ______ $μ F$ .

Smart Achievers · Accepted Answer

(13)

$C_{1} = \frac{ε_{0} A}{\frac{d}{4 K} + \frac{3 d}{4}} = \frac{ε_{0} A}{\frac{13 d}{16}}$ , $C_{2} = \frac{ε_{0} A}{\frac{d}{2 K} + \frac{d}{2}} = \frac{ε_{0} A}{\frac{5 d}{8}}$ $\Rightarrow$ $\frac{C_{2}}{C_{1}} = \frac{13}{10}$

Solution

Ans.
(13)

$C_{1} = \frac{ε_{0} A}{\frac{d}{4 K} + \frac{3 d}{4}} = \frac{ε_{0} A}{\frac{13 d}{16}}$ , $C_{2} = \frac{ε_{0} A}{\frac{d}{2 K} + \frac{d}{2}} = \frac{ε_{0} A}{\frac{5 d}{8}}$ $\Rightarrow$ $\frac{C_{2}}{C_{1}} = \frac{13}{10}$

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Solution

Ans. (13) C1=ε0Ad4K+3d4=ε0A13d16, C2=ε0Ad2K+d2=ε0A5d8 ⇒ C2C1=1310

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Ans.
(13)

$C_{1} = \frac{ε_{0} A}{\frac{d}{4 K} + \frac{3 d}{4}} = \frac{ε_{0} A}{\frac{13 d}{16}}$ , $C_{2} = \frac{ε_{0} A}{\frac{d}{2 K} + \frac{d}{2}} = \frac{ε_{0} A}{\frac{5 d}{8}}$ $\Rightarrow$ $\frac{C_{2}}{C_{1}} = \frac{13}{10}$