SATHEE: electrostatic-potential-and-capacitance Question 24

इलेक्ट्रोस्टैटिक क्षमता और धारिता

Question: Q. 2. A parallel plate capacitor is made of two dielectric blocks in series. One of the blocks has thickness $d_{1}$ and dielectric constant $κ_{1}$ and the other has thickness $d_{2}$ and dielectric constant $κ_{2}$ as shown in Figure. This arrangement can be thought as a dielectric slab of thickness $d (= d_{1} + d_{2})$ and effective dielectric constant $κ$ . The $κ$ is :

(a) $\frac{k_{1} d_{1} + k_{2} d_{2}}{d_{1} + d_{2}}$ (b) $\frac{k_{1} d_{1} + k_{2} d_{2}}{k_{1} + k_{2}}$ (c) $\frac{k_{1} k_{2} (d_{1} + d_{2})}{(k_{1} d_{1} + k_{2} d_{2})}$ (d) $\frac{2 k_{1} k_{2}}{k_{1} + k_{2}}$

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Solution:

Ans. Correct option : (c)

Explanation: Capacitance of a parallel plate capacitor filled with dielectric of constant $k_{1}$ and thickness $d_{1}$ is,

$C_{1} = \frac{k_{1} ε_{0} A}{d_{1}}$

Similarly, for other capacitance of a parallel plate capacitor filled with dielectric of constant $k_{2}$ and thickness $d_{2}$ is,

$C_{2} = \frac{k_{2} ε_{0} A}{d_{2}}$

Both capacitors are in series so equivalent capacitance $C$ is related as :

(2) $C_{1}+\frac{1}{C_{2}}=\frac{d_{1}}{k_{1} \varepsilon_{0} A}+\frac{d_{2}}{\mathrm{k}{2} \varepsilon{0} A}$

$\begin{matrix} (i) & = \frac{1}{ε_{0} A} [\frac{k_{2} d_{1} + k_{1} d_{2}}{k_{1} k_{2}}] \end{matrix}$

So, $C = \frac{k_{1} k_{2} ε_{0} A}{(k_{1} d_{2} + k_{2} d_{1})}$

$\begin{matrix} (ii) & C^{'} = \frac{k ε_{0} A}{d} \end{matrix}$

where, $d = (d_{1} + d_{2})$

So, multiply the numerator and denominator of eqn. (i) with $(d_{1} + d_{2})$ ,

$\begin{matrix} (iii) & C = \frac{k_{1} k_{2} ε_{0} A}{(k_{1} d_{2} + k_{2} d_{1})} \cdot \frac{(d_{1} + d_{2})}{(d_{1} + d_{2})} = \frac{k_{1} k_{2}}{(k_{1} d_{2} + k_{2} d_{1})} \cdot \frac{ε_{0} A}{(d_{1} + d_{2})} . \end{matrix}$

Comparing eqns. (ii) and (iii), the dielectric constant of new capacitor is :

$k = \frac{k_{1} k_{2} (d_{1} + d_{2})}{(k_{1} d_{2} + k_{2} d_{1})}$

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इलेक्ट्रोस्टैटिक क्षमता और धारिता

विषयसूची

इंजीनियरिंग की तैयारी

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एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

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