Question

Two parallel plate air capacitors of the same capacity $C$ are connected in series to a battery of emf $E$ . Then one of the capacitors is completely filled with the dielectric material of constant $K$ . The change in the effective capacitance of the series combination is

• A. $\frac{C}{2}\left[\frac{K - 1}{K + 1}\right]$
• B. $\frac{C}{4}\left[\frac{K - 1}{K + 1}\right]$
• C. $\frac{C}{2}\left[\frac{K + 1}{K - 1}\right]$
• D. $\frac{C}{2}\left[\frac{K - 1}{K + 1}\right]^{2}$

Answer 1

Answer: (A)

$\frac{1}{C_{1}}=\frac{1}{C}+\frac{1}{C}=\frac{2}{C}\Rightarrow C_{1}=\frac{C}{2}$
$\frac{1}{C_{2}}=\frac{1}{C}+\frac{1}{K C}$
$\frac{1}{C_{2}}=\frac{1}{C}\left[1 + \frac{1}{K}\right]$
$\therefore \, \, C_{2}=\frac{C}{\left(1 + \frac{1}{K}\right)}=\frac{C K}{\left(\right. K + 1 \left.\right)}$
$\Delta C= \, \frac{C K}{\left(\right. K + 1 \left.\right)}-\frac{C}{2}=C\left[\frac{K}{K + 1} - \frac{1}{2}\right]$
$=C\left[\frac{2 K - K - 1}{2 \left(\right. K + 1 \left.\right)}\right]=\frac{C}{2}\left[\frac{K - 1}{K + 1}\right] \, $