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)

$\frac{1}{C_1}=\frac{1}{C}+\frac{1}{C}=\frac{2}{C}\Rightarrow C_1=\frac{C}{2}$
$\frac{1}{C_2}=\frac{1}{C}+\frac{1}{KC}$
$\frac{1}{C_2}=\frac{1}{C}\left[1+\frac{1}{K}\right]$
$\therefore C_2=\frac{C}{\left(1+\frac{1}{K}\right)}=\frac{CK}{(K+1)}$
$\Delta C=\frac{CK}{(K+1)}-\frac{C}{2}=C\left[\frac{K}{K+1}-\frac{1}{2}\right]$
$=C\left[\frac{2K-K-1}{2(K+1)}\right]=\frac{C}{2}\left[\frac{K-1}{K+1}\right]$