Two parallel plate air capacitors of same capacity ' C ' are connected in series to a battery of emf ' $E$ '…

Two parallel plate air capacitors of same capacity ' C ' are connected in series to a battery of emf ' $E$ '. Then one of the capacitors is completely filled with dielectric material of constant ' $K$ '. The change in the effective capacity of the series combination is
  1. $\frac{\mathrm{C}}{2}\left[\frac{\mathrm{~K}+1}{\mathrm{~K}-1}\right]$
  2. $\frac{2}{\mathrm{C}}\left[\frac{\mathrm{K}-1}{\mathrm{~K}+1}\right]$
  3. $\frac{\mathrm{C}}{2}\left[\frac{\mathrm{~K}-1}{\mathrm{~K}+1}\right]$
  4. $\frac{\mathrm{C}}{2}\left[\frac{\mathrm{~K}-1}{\mathrm{~K}+1}\right]^2$

Solution

$\begin{aligned} \quad \frac{1}{\mathrm{C}_1} & =\frac{1}{\mathrm{C}}+\frac{1}{\mathrm{C}}=\frac{2}{\mathrm{C}} \\ \therefore \quad \mathrm{C}_1 & =\frac{\mathrm{C}}{2} \\ \frac{1}{\mathrm{C}_2} & =\frac{1}{\mathrm{C}}+\frac{1}{\mathrm{KC}} \\ \frac{1}{\mathrm{C}_2} & =\frac{1}{\mathrm{C}}\left[1+\frac{1}{\mathrm{~K}}\right] \\ \therefore \quad \mathrm{C}_2 & =\frac{\mathrm{CK}}{\mathrm{K}+1} \\ \therefore \mathrm{C} & =\frac{\mathrm{CK}}{\mathrm{K}+1}-\frac{\mathrm{C}}{2}=\mathrm{C}\left[\frac{\mathrm{K}}{\mathrm{K}+1}-\frac{1}{2}\right] \\ \therefore \quad \Delta \mathrm{C} & =\frac{\mathrm{C}}{2}\left[\frac{\mathrm{~K}-1}{\mathrm{~K}+1}\right]\end{aligned}$

Asked in: MHT CET 2024 (11 May Shift 1)

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