SATHEE: Electrostatics 5 Question 15

Electrostatics 5 Question 15

17. A parallel plate capacitor is made of two square plates of side ’ $a$ ’ separated by a distance $d (d « a)$ . The lower triangular portions is filled with a dielectric of dielectric constant $k$ , as shown in the figure. Capacitance of this capacitor is

(Main 2019, 9 Jan I)

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(a) $\frac{K ε_{0} a^{2}}{d} \ln K$

(b) $\frac{K ε_{0} a^{2}}{d (K - 1)} \ln K$

(d) $\frac{1}{2} \cdot \frac{K ε_{0} a^{2}}{d}$

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Answer:

Correct Answer: 17. (b)

Solution:

Let’s consider a strip of thickness ’ $d x$ ’ at a distance of ’ $x$ ’ from the left end as shown in the figure. From the figure, $△ A B C$ and $△ A D E$ are similar triangles,

$\Rightarrow \frac{y}{x} = \frac{d}{a} \Rightarrow y = \frac{d}{a} x$

We know that, the capacitance of parallel plate capacitor,

$\begin{aligned} C & = \frac{ε_{0} A}{d} \\ C_{1} = \frac{ε_{0} (a d x)}{(d - y)} and C_{2} & = \frac{K ε_{0} (a d x)}{y} \end{aligned}$

Here, two capacitor are placed in series with variable thickness, therefore

$C_{eq} = \frac{C_{1} C_{2}}{C_{1} + C_{2}} \Rightarrow C_{eq} = \frac{K ε_{0} a d x}{K d + (1 - K) y}$

Now, integrate it from 0 to $a$

$C = \int_{0}^{a} \frac{K ε_{0} a d x}{K d + (1 - K) y}$

Using Eq. (i), $y = \frac{d}{a} x$ , we get

$\begin{aligned} C & = ε_{0} a \int_{0}^{a} \frac{d x}{d + \frac{1}{K} - 1 \frac{d}{a} x} \\ \Rightarrow C & = \frac{ε_{0} a}{\frac{1 - K}{K} \frac{d}{a}} \ln \frac{1}{K} \\ \Rightarrow C & = \frac{ε_{0} a^{2} K \ln K}{(K - 1) d} \end{aligned}$

Electrostatics 5 Question 15

17. A parallel plate capacitor is made of two square plates of side ’ $a$ ’ separated by a distance $d (d « a)$ . The lower triangular portions is filled with a dielectric of dielectric constant $k$ , as shown in the figure. Capacitance of this capacitor is

Answer:

Solution:

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Electrostatics 5 Question 15

17. A parallel plate capacitor is made of two square plates of side ’ a ’ separated by a distance «d(d«a). The lower triangular portions is filled with a dielectric of dielectric constant k, as shown in the figure. Capacitance of this capacitor is

Answer:

Solution:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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17. A parallel plate capacitor is made of two square plates of side ’ $a$ ’ separated by a distance $d (d « a)$ . The lower triangular portions is filled with a dielectric of dielectric constant $k$ , as shown in the figure. Capacitance of this capacitor is