SATHEE: electrostatic-potential-and-capacitance Question 43

electrostatic-potential-and-capacitance Question 43

Question: Q. 17. Two capacitors of unknown capacitances $C_{1}$ and $C_{2}$ are connected first in series and then in parallel across a battery of $100 V$ . If the energy stored in the two combinations is $0.045 J$ and $0.25 J$ respectively, determine the value of $C_{1}$ and $C_{2}$ . Also calculate the charge on each capacitor in paraller combination.

A. Delhi I, II, III 2015]

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Solution:

Ans. Energy stored in a capacitor

In series combination

$\begin{matrix} (i) & 0.045 = \frac{1}{2} \frac{C_{1} C_{2}}{C_{1} + C_{2}} (100)^{2} \end{matrix}$

$\Rightarrow \frac{C_{1} C_{2}}{C_{1} + C_{2}} = 0.09 \times 10^{- 4}$

In parallel combination

$\begin{matrix} (ii) & 0.25 = \frac{1}{2} (C_{1} + C_{2}) (100)^{2} \end{matrix}$

$\Rightarrow C_{1} + C_{2} = 0.5 \times 10^{- 4}$

On simplifying (i) and (ii)

$\begin{aligned} (iii) & C_{1} C_{2} & = 0.045 \times 10^{- 8} {(C_{1} - C_{2})}^{2} & = {(C_{1} + C_{2})}^{2} - 4 C_{1} C_{2} & = {(0.5 \times 10^{- 4})}^{2} - 4 \times 0.045 \times 10^{- 8} & = 0.25 \times 10^{- 8} - 0.180 \times 10^{- 8} {(C_{1} - C_{2})}^{2} & = 0.07 \times 10^{- 8} (C_{1} - C_{2}) & = 2.6 \times 10^{- 5} = 0.25 \times 10^{- 4} \dots (iii) \end{aligned}$

From (ii) and (iii) we have

$1 / 2$ Charges on capacitors $C_{1}$ and $C_{2}$ in parallel combination

$Q_{1}=\mathrm{C}{1} V=\left(0.38 \times 10^{-4} \times 100\right)=0.38 \times 10^{-2} \mathrm{C} \quad 1 / 2Q{2}=C_{2} V=\left(0.12 \times 10^{-4} \times 100\right)=0.12 \times 10^{-2} \mathrm{C} \quad 1 / 2$ Alternatively,

and

$\begin{aligned} U & = \frac{1}{2} C V^{2} 0.045 & = \frac{1}{2} (\frac{C_{1} C_{2}}{C_{1} + C_{2}}) (100)^{2} 0.25 & = \frac{1}{2} (C_{1} + C_{2}) (100)^{2} \end{aligned}$

If the student is unable to calculate $C_{1}$ and $C_{2}$ , award him/her full 2 marks.

Also if the student just writes

$Q_{1} = C_{1} V = C_{1} (100)$ and $Q_{2} = C_{2} V = C_{2} (100)$

award him/her one mark for this part of the question.

[CBSE Marking Scheme, 2015]

Electrostatic Potential And Capacitance

electrostatic-potential-and-capacitance Question 43

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Electrostatic Potential And Capacitance

electrostatic-potential-and-capacitance Question 43

Table of Contents

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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