Two capacitors having C₁ and C₂ respectively are connected as shown in figure. Initially, capacitor C₁ is charged to a potential difference V volt by a battery. The battery is then removed and the charged capacitor C₁ is not connected to uncharged capacitor C₂ by closing the switch S. The amount of charge on the capacitor C₂, after equilibrium, is:

 

Correct Option - 1


Detailed Solution:

After switch ‘S’ is closed

$Q_1+Q_2=C_{1}V$ -    (1)

              Using KVL

              $\frac{Q_1}{C_1}-\frac{Q_2}{C_2}=0$  

              $Q_1=Q_2\frac{C_1}{C_2}$                       - (2)

              from (1) & (2)

              $\Rightarrow Q_2\left[\frac{C_1+C_2}{C_2}\right]=C_{1}V\Rightarrow Q_2=\left(\frac{C_1{C}_2}{C_1+C_2}\right)V$

 

 

<p>After switch ‘S’ is closed</p><p><span contenteditable="false"> <math> <mrow> <msub> <mrow> <mi>Q</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>Q</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mi>V</mi> </mrow> </math> </span> -&nbsp; &nbsp; (1)</p><p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Using KVL</p><p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <span contenteditable="false"> <math> <mrow> <mfrac> <mrow> <msub> <mrow> <mi>Q</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> <mo>−</mo> <mfrac> <mrow> <msub> <mrow> <mi>Q</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> </mrow> </math> </span>&nbsp;</p><p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <span contenteditable="false"> <math> <mrow> <msub> <mrow> <mi>Q</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>Q</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mfrac> <mrow> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </math> </span>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; - (2)</p><p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; from (1) &amp; (2)</p><p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <span contenteditable="false"> <math> <mrow> <mo>⇒</mo> <msub> <mrow> <mi>Q</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mrow> <mo> [</mo> <mrow> <mfrac> <mrow> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mi>V</mi> <mo>⇒</mo> <msub> <mrow> <mi>Q</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo> (</mo> <mrow> <mfrac> <mrow> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>V</mi> </mrow> </math> </span></p><div><div><picture><source srcset="" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/1758003585php8YXqjk.jpeg" alt="" width="161" height="152" loading="lazy"></picture></div></div><p>&nbsp;</p><p>&nbsp;</p>

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