A parallel-plate capacitor of area $A,$ plate separation $d$ and capacitance $C$ is filled with four dielectric materials having dielectric constants $K_1,K_2,K_3$ and $K_4$ as shown in the figure. If a single dielectric material is to be used to have the same capacitance $C$ in this capacitor, then its dielectric constant $K$ is given by
NEET 2016, Medium
Download our app for free and get started
Here, $C_{1}=\frac{2 \varepsilon_{0} k_{1} A}{3 d}, C_{2}=\frac{2 \varepsilon_{0} k_{2} A}{3 d}$
$C_{3}=\frac{2 \varepsilon_{0} k_{3} A}{3 d}, C_{4}=\frac{2 \varepsilon_{0} k_{4} A}{d}$
Given system of $C_{1}, C_{2}, C_{3}$ and $C_{4}$ can be simplified as $\therefore \frac{1}{C_{A B}}=\frac{1}{C_{1}+C_{2}+C_{3}}+\frac{1}{C_{4}}$
Suppose, $C_{A B}=\frac{k \varepsilon_{0} A}{d}$
$\quad \frac{1}{k\left(\frac{\varepsilon_{0} A}{d}\right)}=\frac{1}{\frac{2}{3} \frac{\varepsilon_{0} A}{d}\left(k_{1}+k_{2}+k_{3}\right)}+\frac{1}{\frac{2 \varepsilon_{0} A}{d} k_{4}}$
$\Rightarrow \frac{1}{k}=\frac{3}{2\left(k_{1}+k_{2}+k_{3}\right)}+\frac{1}{2 k_{4}} \therefore \frac{2}{k}=\frac{3}{k_{1}+k_{2}+k_{3}}+\frac{1}{k_{4}}$
Download our appand get started for free
Experience the future of education. Simply download our apps or reach out to us for more information. Let's shape the future of learning together!No signup needed.*
Similar Questions
- 1An uncharged parallel plate capacitor having a dielectric of constant $K$ is connected to a similar air cored parallel capacitor charged to a potential $V$. The two capacitors share charges and the common potential is $V$. The dielectric constant $K$ isView Solution
- 2In moving from $A$ to $B$ along an electric field line, the electric field does $6.4 \times {10^{ - 19}}\,J$ of work on an electron. If ${\phi _1},\;{\phi _2}$ are equipotential surfaces, then the potential difference $({V_C} - {V_A})$ is.....$V$View Solution
- 3A charge $3$ coulomb experiences a force $3000$ $N$ when placed in a uniform electric field. The potential difference between two points separated by a distance of $1$ $cm$ along the field lines is.....$V$View Solution
- 4A capacitor of $10\,\mu F$ charged up to $250\, volts$ is connected in parallel with another capacitor of $5\,\mu F$ charged up to $100\, volts$. The common potential is.....$V$View Solution
- 5A solid sphere of radius $R$ carries a charge $(Q+q)$ distributed uniformly over its volume. A very small point like piece of it of mass $m$ gets detached from the bottom of the sphere and falls down vertically under gravity. This piece carries charge $q.$ If it acquires a speed $v$ when it has fallen through a vertical height $y$ (see figure), then :View Solution
(assume the remaining portion to be spherical).
- 6View SolutionIn the circuit shown in figure
- 7Two batteries and two condensers are connected as shown in the figure. The charge on $2\,\mu F$ capacitor is.....$\mu F$View Solution
- 8If the plates of a parallel plate capacitor connected to a battery are moved close to each other, thenView Solution
$A$. the charge stored in it, increases.
$B$. the energy stored in it, decreases.
$C$. its capacitance increases.
$D$. the ratio of charge to its potential remains the same.
$E$. the product of charge and voltage increases.
Choose the most appropriate answer from the options given below:
- 9A finite ladder is constructed by connecting several sections of $2\,\mu F,\;4\,\mu F$ capacitor combinations as shown in the figure. It is terminated by a capacitor of capacitance $C$. What value should be chosen for $C$ such that the equivalent capacitance of the ladder between the points $A$ and $B$ becomes independent of the number of sections in between......$\mu F$View Solution
- 10Figure shows a solid metal sphere of radius $‘a’$ surrounded by a concentric thin metal shell of radius $2a$ . Initially both are having charges $Q$ each. When the two are connected by a conducting wire as shown in the figure, then amount of heat produced in this process will beView Solution
